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Integral Vector Calculus and parametrization of...

by: ecterrab 14455 Maple
December 31 2022
9  6

This is about functionality introduced in Maple 2022, which however is still not well known: Integral Vector Calculus and parametrization using symbolic (algebraic) vector notation. Four new commands were added to the Physics:-Vectors package, implementing the parametrization of curves, surfaces and volumes, as well as the computation of path, surface and volume vector integrals. Those are integrals where the integrand is a scalar or vector function. The computation is done from any description (algebraic, parametric, vectorial) of the region of integration - a path, surface or volume.
 
There are three kinds of line or path integrals:

NOTE Jan 1: Updated the worksheet linked below; it runs in Maple 2022.
Download Integral_Vector_Calculus_and_Parametrization.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

New Maple application center search doesn't work

by: Christopher2222 5980
appcenter search
October 16 2022
1  0

The search query in the new Maple Application center is broken.  There is no advanced search options and a search for mapleflow or maple flow brings up 0 results.  There should at least be one found, for example the search should have at least brought up The Liquid Volume in a Partially-Filled Horizontal Tank".

Maple, please fix.

 

New coordinate-system labels in Physics

by: ecterrab 14455 Maple
tensor relativity physics
August 25 2022
9  0

Notation is one of the most important things to communicate with others in science. It is remarkable how many people use or do not use a computer algebra package just because of its notation. For those reasons, in the context of the Physics package, strong emphasis is put on using textbook notation as much as possible regarding input and output, including, for that purpose, as people here know, significant developments in Maple typesetting.

Still, for historical reasons, when using the Physics package, the labels used to refer to a coordinate system had been a single Capital Letter, as in X, Y, ...It was not possible to use, e.g. X', or x.

That has changed. Starting with the Maplesoft Physics Updates v.1308, any symbol can be used as a coordinate system label. The lines below demo this change.

 

Download new_coordinates_labels.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Deriving 4D relativistic Lorentz transformations

by: ecterrab 14455 Maple
transformation relativity 4-dimensional
August 04 2022
7  0

Following the previous post on The Electromagnetic Field of Moving Charges, this is another non-trivial exercise, the derivation of 4-dimensional relativistic Lorentz transformations,  a problem of a 3rd-year undergraduate course on Special Relativity whose solution requires "tensor & matrix" manipulation techniques. At the end, there is a link to the Maple document, so that the computation below can be reproduced, and a link to a corresponding PDF file with all the sections open.

Deriving 4D relativistic Lorentz transformations

Freddy Baudine(1), Edgardo S. Cheb-Terrab(2)

(1) Retired, passionate about Mathematics and Physics

(2) Physics, Differential Equations and Mathematical Functions, Maplesoft

 

Lorentz transformations are a six-parameter family of linear transformations Lambda that relate the values of the coordinates x, y, z, t of an event in one inertial reference system to the coordinates diff(x, x), diff(y(x), x), diff(z(x), x), diff(t(x), x) of the same event in another inertial system that moves at a constant velocity relative to the former. An explicit form of Lambda can be derived from physics principles, or in a purely algebraic mathematical manner. A derivation from physics principles is done in an upcoming post about relativistic dynamics, while in this post we derive the form of Lambda mathematically, as rotations in a (pseudo) Euclidean 4 dimensional space. Most of the presentation below follows the one found in Jackson's book on Classical Electrodynamics [1].

 

The computations below in Maple 2022 make use of the Maplesoft Physics Updates v.1283 or newer.

Formulation of the problem and ansatz Lambda = exp(`𝕃`)

 

 

The problem is to find a group of linear transformations,

  "x^(' mu)=(Lambda^( mu))[nu] x^(nu)" 

that represent rotations in a 4D (pseudo) Euclidean spacetime, and so they leave invariant the norm of the 4D position vector x^mu; that is,

"x^(' mu) (x')[mu]=x^( mu) (x^())[mu]"

For the purpose of deriving the form of `#msubsup(mi("Λ",fontstyle = "normal"),mi("ν",fontstyle = "normal"),mrow(mo("⁢"),mi("μ",fontstyle = "normal")))`, a relevant property for it can be inferred by rewriting the invariance of the norm in terms of `#msubsup(mi("Λ",fontstyle = "normal"),mi("ν",fontstyle = "normal"),mrow(mo("⁢"),mi("μ",fontstyle = "normal")))`. In steps, from the above,

"g[alpha,beta] x^(' alpha) (x^(' beta))[]=g[mu,nu] x^( mu) (x^( nu))[]"
 

g[alpha, beta]*`#msubsup(mi("Λ",fontstyle = "normal"),mi("μ",fontstyle = "normal"),mrow(mo("⁢"),mi("α",fontstyle = "normal")))`*x^mu*`#msubsup(mi("Λ",fontstyle = "normal"),mi("ν",fontstyle = "normal"),mrow(mo("⁢"),mi("β",fontstyle = "normal")))`*x^nu = g[mu, nu]*x^mu*`#msup(mi("x"),mrow(mo("⁢"),mi("ν",fontstyle = "normal")))`
 

g[alpha, beta]*`#msubsup(mi("Λ",fontstyle = "normal"),mi("μ",fontstyle = "normal"),mrow(mo("⁢"),mi("α",fontstyle = "normal")))`*x^mu*`#msubsup(mi("Λ",fontstyle = "normal"),mi("ν",fontstyle = "normal"),mrow(mo("⁢"),mi("β",fontstyle = "normal")))`*x^nu = g[mu, nu]*x^mu*`#msup(mi("x"),mrow(mo("⁢"),mi("ν",fontstyle = "normal")))`

from where,

g[alpha, beta]*`#msubsup(mi("Λ",fontstyle = "normal"),mi("μ",fontstyle = "normal"),mrow(mo("⁢"),mi("α",fontstyle = "normal")))`*`#msubsup(mi("Λ",fontstyle = "normal"),mi("ν",fontstyle = "normal"),mrow(mo("⁢"),mi("β",fontstyle = "normal")))` = g[mu, nu]``

or in matrix (4 x 4) form, `#mrow(msubsup(mi("Λ",fontstyle = "normal"),mi("μ",fontstyle = "normal"),mrow(mo("⁢"),mi("α",fontstyle = "normal"))),mo("⁢"),mo("≡"),mo("⁢"),mo("⁢"),mi("Λ",fontstyle = "normal"))`, `≡`(g[alpha, beta], g)

Lambda^T*g*Lambda = g

where Lambda^T is the transpose of Lambda. Taking the determinant of both sides of this equation, and recalling that det(Lambda^T) = det(Lambda), we get

 

det(Lambda) = `&+-`(1)

 

The determination of Lambda is analogous to the determination of the matrix R (3D tensor R[i, j]) representing rotations in the 3D space, where the same line of reasoning leads to det(R) = `&+-`(1). To exclude reflection transformations, that have det(Lambda) = -1 and cannot be obtained through any sequence of rotations, because they do not preserve the relative orientation of the axes, the sign that represents our problem is +. To explicitly construct the transformation matrix Lambda, Jackson proposes the ansatz

  Lambda = exp(`𝕃`)   

Summarizing: the determination of `#msubsup(mi("Λ",fontstyle = "normal"),mi("ν",fontstyle = "normal"),mrow(mo("⁢"),mi("μ",fontstyle = "normal")))` consists of determining `𝕃`[nu]^mu entering Lambda = exp(`𝕃`) such that det(Lambda) = 1followed by computing the exponential of the matrix `𝕃`.

Determination of `𝕃`[nu]^mu

 

In order to compare results with Jackson's book, we use the same signature he uses, "(+---)", and lowercase Latin letters to represent space tensor indices, while spacetime indices are represented using Greek letters, which is already Physics' default.

 

restart; with(Physics)

Setup(signature = "+---", spaceindices = lowercaselatin)

[signature = `+ - - -`, spaceindices = lowercaselatin]

(1)

Start by defining the tensor `𝕃`[nu]^mu whose components are to be determined. For practical purposes, define a macro LM = `𝕃` to represent the tensor and use L to represent its components

macro(LM = `𝕃`, %LM = `%𝕃`); Define(Lambda, LM, quiet)

LM[`~mu`, nu] = Matrix(4, symbol = L)

`𝕃`[`~mu`, nu] = Matrix(%id = 36893488153289603060)

(2)

"Define(?)"

{Lambda, `𝕃`[`~mu`, nu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-LeviCivita[alpha, beta, mu, nu]}

(3)

Next, from Lambda^T*g*Lambda = g (see above in Formulation of the problem) one can derive the form of `𝕃`. To work algebraically with `𝕃`, Lambda, g representing matrices, set these symbols as noncommutative

Setup(noncommutativeprefix = {LM, Lambda, g})

[noncommutativeprefix = {`𝕃`, Lambda, g}]

(4)

From

Lambda^T*g*Lambda = g

Physics:-`*`(Physics:-`^`(Lambda, T), g, Lambda) = g

(5)

it follows that

(1/g*(Physics[`*`](Physics[`^`](Lambda, T), g, Lambda) = g))/Lambda

Physics:-`*`(Physics:-`^`(g, -1), Physics:-`^`(Lambda, T), g) = Physics:-`^`(Lambda, -1)

(6)

eval(Physics[`*`](Physics[`^`](g, -1), Physics[`^`](Lambda, T), g) = Physics[`^`](Lambda, -1), Lambda = exp(LM))

Physics:-`*`(Physics:-`^`(g, -1), Physics:-`^`(exp(`𝕃`), T), g) = Physics:-`^`(exp(`𝕃`), -1)

(7)

Expanding the exponential using exp(`𝕃`) = Sum(`𝕃`^k/factorial(k), k = 0 .. infinity), and taking into account that the matrix product `𝕃`^k/g*g can be rewritten as(`𝕃`/g*g)^k, the left-hand side of (7) can be written as exp(`𝕃`^T/g*g)

exp(LM^T/g*g) = rhs(Physics[`*`](Physics[`^`](g, -1), Physics[`^`](exp(`𝕃`), T), g) = Physics[`^`](exp(`𝕃`), -1))

exp(Physics:-`*`(Physics:-`^`(g, -1), Physics:-`^`(`𝕃`, T), g)) = Physics:-`^`(exp(`𝕃`), -1)

(8)

Multiplying by exp(`𝕃`)

(exp(Physics[`*`](Physics[`^`](g, -1), Physics[`^`](`𝕃`, T), g)) = Physics[`^`](exp(`𝕃`), -1))*exp(LM)

Physics:-`*`(exp(Physics:-`*`(Physics:-`^`(g, -1), Physics:-`^`(`𝕃`, T), g)), exp(`𝕃`)) = 1

(9)

Recalling that  "g^(-1)=g[]^(mu,alpha)", g = g[beta, nu] and that for any matrix `𝕃`, "(`𝕃`^T)[alpha]^( beta)= `𝕃`(( )^(beta))[alpha]",  

"g^(-1) `𝕃`^T g= 'g_[~mu,~alpha]*LM[~beta, alpha] g_[beta, nu] '"

Physics:-`*`(Physics:-`^`(g, -1), Physics:-`^`(`𝕃`, T), g) = Physics:-`*`(Physics:-g_[`~mu`, `~alpha`], `𝕃`[`~beta`, alpha], Physics:-g_[beta, nu])

(10)

subs([Physics[`*`](Physics[`^`](g, -1), Physics[`^`](`𝕃`, T), g) = Physics[`*`](Physics[g_][`~mu`, `~alpha`], `𝕃`[`~beta`, alpha], Physics[g_][beta, nu]), LM = LM[`~mu`, nu]], Physics[`*`](exp(Physics[`*`](Physics[`^`](g, -1), Physics[`^`](`𝕃`, T), g)), exp(`𝕃`)) = 1)

Physics:-`*`(exp(Physics:-g_[`~alpha`, `~mu`]*Physics:-g_[beta, nu]*`𝕃`[`~beta`, alpha]), exp(`𝕃`[`~mu`, nu])) = 1

(11)

To allow for the combination of the exponentials, now that everything is in tensor notation, remove the noncommutative character of `𝕃```

Setup(clear, noncommutativeprefix)

[noncommutativeprefix = none]

(12)

combine(Physics[`*`](exp(Physics[g_][`~alpha`, `~mu`]*Physics[g_][beta, nu]*`𝕃`[`~beta`, alpha]), exp(`𝕃`[`~mu`, nu])) = 1)

exp(`𝕃`[`~beta`, alpha]*Physics:-g_[beta, nu]*Physics:-g_[`~alpha`, `~mu`]+`𝕃`[`~mu`, nu]) = 1

(13)

Since every tensor component of this expression is real, taking the logarithm at both sides and simplifying tensor indices

`assuming`([map(ln, exp(`𝕃`[`~beta`, alpha]*Physics[g_][beta, nu]*Physics[g_][`~alpha`, `~mu`]+`𝕃`[`~mu`, nu]) = 1)], [real])

`𝕃`[`~beta`, alpha]*Physics:-g_[beta, nu]*Physics:-g_[`~alpha`, `~mu`]+`𝕃`[`~mu`, nu] = 0

(14)

Simplify(`𝕃`[`~beta`, alpha]*Physics[g_][beta, nu]*Physics[g_][`~alpha`, `~mu`]+`𝕃`[`~mu`, nu] = 0)

`𝕃`[nu, `~mu`]+`𝕃`[`~mu`, nu] = 0

(15)

So the components of `𝕃`[`~mu`, nu]

LM[`~μ`, nu, matrix]

`𝕃`[`~μ`, nu] = Matrix(%id = 36893488151939882148)

(16)

satisfy (15). Using TensorArray  the components of that tensorial equation are

TensorArray(`𝕃`[nu, `~mu`]+`𝕃`[`~mu`, nu] = 0, output = setofequations)

{2*L[1, 1] = 0, 2*L[2, 2] = 0, 2*L[3, 3] = 0, 2*L[4, 4] = 0, -L[1, 2]+L[2, 1] = 0, L[1, 2]-L[2, 1] = 0, -L[1, 3]+L[3, 1] = 0, L[1, 3]-L[3, 1] = 0, -L[1, 4]+L[4, 1] = 0, L[1, 4]-L[4, 1] = 0, L[3, 2]+L[2, 3] = 0, L[4, 2]+L[2, 4] = 0, L[4, 3]+L[3, 4] = 0}

 (17)

Simplifying taking these equations into account results in the form of `𝕃`[`~mu`, nu] we were looking for

"simplify(?,{2*L[1,1] = 0, 2*L[2,2] = 0, 2*L[3,3] = 0, 2*L[4,4] = 0, -L[1,2]+L[2,1] = 0, L[1,2]-L[2,1] = 0, -L[1,3]+L[3,1] = 0, L[1,3]-L[3,1] = 0, -L[1,4]+L[4,1] = 0, L[1,4]-L[4,1] = 0, L[3,2]+L[2,3] = 0, L[4,2]+L[2,4] = 0, L[4,3]+L[3,4] = 0})"

`𝕃`[`~μ`, nu] = Matrix(%id = 36893488153606736460)

(18)

This is equation (11.90) in Jackson's book [1]. By eye we see there are only six independent parameters in `𝕃`[`~mu`, nu], or via

"indets(rhs(?), name)"

{L[1, 2], L[1, 3], L[1, 4], L[2, 3], L[2, 4], L[3, 4]}

(19)

nops({L[1, 2], L[1, 3], L[1, 4], L[2, 3], L[2, 4], L[3, 4]})

6

(20)

This number is expected: a rotation in 3D space can always be represented as the composition of three rotations, and so, characterized by 3 parameters: the rotation angles measured on each of the space planes x, y, y, z, z, x. Likewise, a rotation in 4D space is characterized by 6 parameters: rotations on each of the three space planes, parameters L[2, 3], L[2, 4] and L[3, 4],  and rotations on the spacetime planest, x, t, y, t, z, parameters L[1, j]. Define now `𝕃`[`~mu`, nu] using (18) for further computing with it in the next section

"Define(?)"

{Lambda, `𝕃`[`~mu`, nu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], Physics:-LeviCivita[alpha, beta, mu, nu]}

(21)

Determination of Lambda[`~mu`, nu]

 

From the components of `𝕃`[`~mu`, nu] in (18), the components of Lambda[`~mu`, nu] = exp(`𝕃`[`~mu`, nu]) can be computed directly using the LinearAlgebra:-MatrixExponential command. Then, following Jackson's book, in what follows we also derive a general formula for `𝕃`[`~mu`, nu]in terms of beta = v/c and gamma = 1/sqrt(-beta^2+1) shown in [1] as equation (11.98), finally showing the form of Lambda[`~mu`, nu] as a function of the relative velocity of the two inertial systems of references.

 

An explicit form of Lambda[`~mu`, nu] in the case of a rotation on thet, x plane can be computed by taking equal to zero all the parameters in (19) but for L[1, 2] and substituting in "?≡`𝕃`[nu]^(mu)"  

`~`[`=`](`minus`({L[1, 2], L[1, 3], L[1, 4], L[2, 3], L[2, 4], L[3, 4]}, {L[1, 2]}), 0)

{L[1, 3] = 0, L[1, 4] = 0, L[2, 3] = 0, L[2, 4] = 0, L[3, 4] = 0}

(22)

"subs({L[1,3] = 0, L[1,4] = 0, L[2,3] = 0, L[2,4] = 0, L[3,4] = 0},?)"

`𝕃`[`~μ`, nu] = Matrix(%id = 36893488153606695500)

(23)

Computing the matrix exponential,

"Lambda[~mu,nu]=LinearAlgebra:-MatrixExponential(rhs(?))"

Lambda[`~μ`, nu] = Matrix(%id = 36893488151918824492)

(24)

"convert(?,trigh)"

Lambda[`~μ`, nu] = Matrix(%id = 36893488151918852684)

(25)

This is formula (4.2) in Landau & Lifshitz book [2]. An explicit form of Lambda[`~mu`, nu] in the case of a rotation on thex, y plane can be computed by taking equal to zero all the parameters in (19) but for L[2, 3]

`~`[`=`](`minus`({L[1, 2], L[1, 3], L[1, 4], L[2, 3], L[2, 4], L[3, 4]}, {L[2, 3]}), 0)

{L[1, 2] = 0, L[1, 3] = 0, L[1, 4] = 0, L[2, 4] = 0, L[3, 4] = 0}

(26)

"subs({L[1,2] = 0, L[1,3] = 0, L[1,4] = 0, L[2,4] = 0, L[3,4] = 0},?)"

`𝕃`[`~μ`, nu] = Matrix(%id = 36893488151918868828)

(27)

"Lambda[~mu, nu]=LinearAlgebra:-MatrixExponential(rhs(?))"

Lambda[`~μ`, nu] = Matrix(%id = 36893488153289306948)

(28)

NULL

Rewriting `%𝕃`[`~mu`, nu] = K[`~i`]*Zeta[i]+S[`~i`]*omega[i]

 

Following Jackson's notation, for readability, redefine the 6 parameters entering `𝕃`[`~mu`, nu] as

'{LM[1, 2] = `ζ__1`, LM[1, 3] = `ζ__2`, LM[1, 4] = `ζ__3`, LM[2, 3] = `ω__3`, LM[2, 4] = -`ω__2`, LM[3, 4] = `ω__1`}'

{`𝕃`[1, 2] = zeta__1, `𝕃`[1, 3] = zeta__2, `𝕃`[1, 4] = zeta__3, `𝕃`[2, 3] = omega__3, `𝕃`[2, 4] = -omega__2, `𝕃`[3, 4] = omega__1}

(29)

(Note in the above the surrounding backquotes '...' to prevent a premature evaluation of the left-hand sides; that is necessary when using the Library:-RedefineTensorComponent command.) With this redefinition, `𝕃`[`~mu`, nu] becomes

Library:-RedefineTensorComponent({`𝕃`[1, 2] = zeta__1, `𝕃`[1, 3] = zeta__2, `𝕃`[1, 4] = zeta__3, `𝕃`[2, 3] = omega__3, `𝕃`[2, 4] = -omega__2, `𝕃`[3, 4] = omega__1})

LM[`~μ`, nu, matrix]

`𝕃`[`~μ`, nu] = Matrix(%id = 36893488151939901668)

(30)

where each parameter is related to a rotation angle on one plane. Any Lorentz transformation (rotation in 4D pseudo-Euclidean space) can be represented as the composition of these six rotations, and to each rotation, corresponds the matrix that results from taking equal to zero all of the six parameters but one.

 

The set of six parameters can be split into two sets of three parameters each, one representing rotations on the t, x__j planes, parameters `ζ__j`, and the other representing rotations on the x__i, x__j planes, parameters `ω__j`. With that, following [1], (30) can be rewritten in terms of four 3D tensors, two of them with the parameters as components, the other two with matrix as components, as follows:

Zeta[i] = [`ζ__1`, `ζ__2`, `ζ__3`], omega[i] = [`ω__1`, `ω__2`, `ω__3`], K[i] = [K__1, K__2, K__3], S[i] = [S__1, S__2, S__3]

Zeta[i] = [zeta__1, zeta__2, zeta__3], omega[i] = [omega__1, omega__2, omega__3], K[i] = [K__1, K__2, K__3], S[i] = [S__1, S__2, S__3]

(31)

Define(Zeta[i] = [zeta__1, zeta__2, zeta__3], omega[i] = [omega__1, omega__2, omega__3], K[i] = [K__1, K__2, K__3], S[i] = [S__1, S__2, S__3])

{Lambda, `𝕃`[mu, nu], Physics:-Dgamma[mu], K[i], Physics:-Psigma[mu], S[i], Zeta[i], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], omega[i], Physics:-LeviCivita[alpha, beta, mu, nu]}

(32)

The 3D tensors K[i] and S[i] satisfy the commutation relations

Setup(noncommutativeprefix = {K, S})

[noncommutativeprefix = {K, S}]

(33)

Commutator(S[i], S[j]) = LeviCivita[i, j, k]*S[k]

Physics:-Commutator(S[i], S[j]) = Physics:-LeviCivita[i, j, k]*S[`~k`]

(34)

Commutator(S[i], K[j]) = LeviCivita[i, j, k]*K[k]

Physics:-Commutator(S[i], K[j]) = Physics:-LeviCivita[i, j, k]*K[`~k`]

(35)

Commutator(K[i], K[j]) = -LeviCivita[i, j, k]*S[k]

Physics:-Commutator(K[i], K[j]) = -Physics:-LeviCivita[i, j, k]*S[`~k`]

(36)

The matrix components of the 3D tensor K__i, related to rotations on the t, x__j planes, are

K__1 := matrix([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]])

array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (0), ( 1, 2 ) = (1), ( 3, 2 ) = (0), ( 1, 3 ) = (0), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (1), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (0), ( 3, 4 ) = (0) ] )

(37)

K__2 := matrix([[0, 0, 1, 0], [0, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, 0]])

array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (1), ( 4, 2 ) = (0), ( 1, 2 ) = (0), ( 3, 2 ) = (0), ( 1, 3 ) = (1), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (0), ( 3, 4 ) = (0) ] )

(38)

K__3 := matrix([[0, 0, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 0]])

array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (0), ( 1, 2 ) = (0), ( 3, 2 ) = (0), ( 1, 3 ) = (0), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (1), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (1), ( 3, 4 ) = (0) ] )

(39)

The matrix components of the 3D tensor S__i, related to rotations on the x__i, x__j 3D space planes, are

S__1 := matrix([[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1], [0, 0, 1, 0]])

array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (0), ( 1, 2 ) = (0), ( 3, 2 ) = (0), ( 1, 3 ) = (0), ( 4, 3 ) = (1), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (0), ( 3, 4 ) = (-1) ] )

(40)

S__2 := matrix([[0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, -1, 0, 0]])

array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (-1), ( 1, 2 ) = (0), ( 3, 2 ) = (0), ( 1, 3 ) = (0), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (1), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (0), ( 3, 4 ) = (0) ] )

(41)

S__3 := matrix([[0, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [0, 0, 0, 0]])

array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (0), ( 1, 2 ) = (0), ( 3, 2 ) = (1), ( 1, 3 ) = (0), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (-1), ( 4, 1 ) = (0), ( 3, 4 ) = (0) ] )

(42)

NULL

Verifying the commutation relations between S[i] and K[j]

   

The `𝕃`[`~mu`, nu] tensor is now expressed in terms of these objects as

%LM[`~μ`, nu] = omega[i].S[i]+Zeta[i].K[i]

`%𝕃`[`~μ`, nu] = K[`~i`]*Zeta[i]+S[`~i`]*omega[i]

(50)

where the right-hand side, without free indices, represents the matrix form of `%𝕃`[`~mu`, nu]. This notation makes explicit the fact that any Lorentz transformation can always be written as the composition of six rotations

SumOverRepeatedIndices(`%𝕃`[`~μ`, nu] = K[`~i`]*Zeta[i]+S[`~i`]*omega[i])

`%𝕃`[`~μ`, nu] = zeta__1*K[`~1`]+zeta__2*K[`~2`]+zeta__3*K[`~3`]+omega__1*S[`~1`]+omega__2*S[`~2`]+omega__3*S[`~3`]

(51)

Library:-RewriteInMatrixForm(`%𝕃`[`~μ`, nu] = zeta__1*K[`~1`]+zeta__2*K[`~2`]+zeta__3*K[`~3`]+omega__1*S[`~1`]+omega__2*S[`~2`]+omega__3*S[`~3`])

`%𝕃`[`~μ`, nu] = (array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (0), ( 1, 2 ) = (zeta__1), ( 3, 2 ) = (0), ( 1, 3 ) = (0), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (zeta__1), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (0), ( 3, 4 ) = (0) ] ))+(array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (zeta__2), ( 4, 2 ) = (0), ( 1, 2 ) = (0), ( 3, 2 ) = (0), ( 1, 3 ) = (zeta__2), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (0), ( 3, 4 ) = (0) ] ))+(array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (0), ( 1, 2 ) = (0), ( 3, 2 ) = (0), ( 1, 3 ) = (0), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (zeta__3), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (zeta__3), ( 3, 4 ) = (0) ] ))+(array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (0), ( 1, 2 ) = (0), ( 3, 2 ) = (0), ( 1, 3 ) = (0), ( 4, 3 ) = (omega__1), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (0), ( 3, 4 ) = (-omega__1) ] ))+(array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (-omega__2), ( 1, 2 ) = (0), ( 3, 2 ) = (0), ( 1, 3 ) = (0), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (omega__2), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (0), ( 3, 4 ) = (0) ] ))+(array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (0), ( 4, 2 ) = (0), ( 1, 2 ) = (0), ( 3, 2 ) = (omega__3), ( 1, 3 ) = (0), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (0), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (0), ( 2, 2 ) = (0), ( 2, 3 ) = (-omega__3), ( 4, 1 ) = (0), ( 3, 4 ) = (0) ] ))

 (52)

Library:-PerformMatrixOperations(`%𝕃`[`~μ`, nu] = zeta__1*K[`~1`]+zeta__2*K[`~2`]+zeta__3*K[`~3`]+omega__1*S[`~1`]+omega__2*S[`~2`]+omega__3*S[`~3`])

`%𝕃`[`~μ`, nu] = (array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (zeta__2), ( 4, 2 ) = (-omega__2), ( 1, 2 ) = (zeta__1), ( 3, 2 ) = (omega__3), ( 1, 3 ) = (zeta__2), ( 4, 3 ) = (omega__1), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (zeta__1), ( 3, 3 ) = (0), ( 2, 4 ) = (omega__2), ( 1, 4 ) = (zeta__3), ( 2, 2 ) = (0), ( 2, 3 ) = (-omega__3), ( 4, 1 ) = (zeta__3), ( 3, 4 ) = (-omega__1) ] ))

(53)

NULL

which is the same as the starting point (30)NULL

The transformation Lambda[`~mu`, nu] = exp(`%𝕃`[`~mu`, nu]), where  `%𝕃`[`~mu`, nu] = K[`~i`]*Zeta[i], as a function of the relative velocity of two inertial systems

 

 

As seen in the previous subsection, in `𝕃`[`~mu`, nu] = K[`~i`]*Zeta[i]+S[`~i`]*omega[i], the second term, S[`~i`]*omega[i], corresponds to 3D rotations embedded in the general form of 4D Lorentz transformations, and K[`~i`]*Zeta[i] is the term that relates the coordinates of two inertial systems of reference that move with respect to each other at constant velocity v.  In this section, K[`~i`]*Zeta[i] is rewritten in terms of that velocity, arriving at equation (11.98)  of Jackson's book [1]. The key observation is that the 3D vector Zeta[i], can be rewritten in terms of arctanh(beta), where beta = v/c and c is the velocity of light (for the rationale of that relation, see [2], sec 4, discussion before formula (4.3)).

 

Use a macro - say ub - to represent the atomic variable `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))` (this variable can be entered as `#mover(mi("β"),mo("ˆ")`. In general, to create atomic variables, see the section on Atomic Variables of the page 2DMathDetails ).

 

macro(ub = `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`)

ub[j] = [ub[1], ub[2], ub[3]], Zeta[j] = ub[j]*arctanh(beta)

`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j] = [`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1], `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2], `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]], Zeta[j] = `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j]*arctanh(beta)

(54)

Define(`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j] = [`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1], `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2], `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]], Zeta[j] = `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j]*arctanh(beta))

{Lambda, `𝕃`[mu, nu], Physics:-Dgamma[mu], K[i], Physics:-Psigma[mu], S[i], Zeta[i], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], omega[i], Physics:-LeviCivita[alpha, beta, mu, nu], `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j]}

(55)

With these two definitions, and excluding the rotation term S[`~i`]*omega[i] we have

%LM[`~μ`, nu] = Zeta[j]*K[j]

`%𝕃`[`~μ`, nu] = Zeta[j]*K[`~j`]

(56)

SumOverRepeatedIndices(`%𝕃`[`~μ`, nu] = Zeta[j]*K[`~j`])

`%𝕃`[`~μ`, nu] = arctanh(beta)*(K[`~1`]*`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]+K[`~2`]*`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]+K[`~3`]*`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3])

(57)

Library:-PerformMatrixOperations(`%𝕃`[`~μ`, nu] = arctanh(beta)*(K[`~1`]*`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]+K[`~2`]*`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]+K[`~3`]*`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]))

`%𝕃`[`~μ`, nu] = (array( 1 .. 4, 1 .. 4, [( 3, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]*arctanh(beta)), ( 4, 2 ) = (0), ( 1, 2 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]*arctanh(beta)), ( 3, 2 ) = (0), ( 1, 3 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]*arctanh(beta)), ( 4, 3 ) = (0), ( 4, 4 ) = (0), ( 1, 1 ) = (0), ( 2, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]*arctanh(beta)), ( 3, 3 ) = (0), ( 2, 4 ) = (0), ( 1, 4 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]*arctanh(beta)), ( 2, 2 ) = (0), ( 2, 3 ) = (0), ( 4, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]*arctanh(beta)), ( 3, 4 ) = (0) ] ))

(58)

 

From this expression, the form of "Lambda[nu]^(mu)" can be obtained as in (24) using LinearAlgebra:-MatrixExponential and simplifying the result taking into account that `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j] is a unit vector

SumOverRepeatedIndices(ub[j]^2) = 1

`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]^2 = 1

(59)

exp(lhs(`%𝕃`[`~μ`, nu] = (array( 1 .. 4, 1 .. 4, [( 3, 3 ) = (0), ( 2, 3 ) = (0), ( 4, 2 ) = (0), ( 1, 1 ) = (0), ( 1, 2 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]*arctanh(beta)), ( 4, 4 ) = (0), ( 4, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]*arctanh(beta)), ( 3, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]*arctanh(beta)), ( 3, 4 ) = (0), ( 4, 3 ) = (0), ( 1, 4 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]*arctanh(beta)), ( 3, 2 ) = (0), ( 1, 3 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]*arctanh(beta)), ( 2, 4 ) = (0), ( 2, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]*arctanh(beta)), ( 2, 2 ) = (0) ] )))) = simplify(LinearAlgebra:-MatrixExponential(rhs(`%𝕃`[`~μ`, nu] = (array( 1 .. 4, 1 .. 4, [( 3, 3 ) = (0), ( 2, 3 ) = (0), ( 4, 2 ) = (0), ( 1, 1 ) = (0), ( 1, 2 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]*arctanh(beta)), ( 4, 4 ) = (0), ( 4, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]*arctanh(beta)), ( 3, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]*arctanh(beta)), ( 3, 4 ) = (0), ( 4, 3 ) = (0), ( 1, 4 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]*arctanh(beta)), ( 3, 2 ) = (0), ( 1, 3 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]*arctanh(beta)), ( 2, 4 ) = (0), ( 2, 1 ) = (`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]*arctanh(beta)), ( 2, 2 ) = (0) ] )))), {`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]^2 = 1})

exp(`%𝕃`[`~μ`, nu]) = Matrix(%id = 36893488153234621252)

 (60)

It is useful at this point to analyze the dependency on the components of `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j] of this matrix

"map(u -> indets(u,specindex(ub)), rhs(?))"

Matrix(%id = 36893488151918822812)

(61)

We see that the diagonal element [4, 4] depends on two instead of only one component of  `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j]. That is due to the simplification with respect to side relations , performed in (60), that constructs an elimination Groebner Basis that cannot reduce at once, using the single equation (59), the dependency of all of the elements [2, 2], [3, 3] and [4, 4] to a single component of  `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j]. So, to reduce further the dependency of the [4, 4] element, this component of (60) requires one more simplification step, using a different elimination strategy, explicitly requesting the elimination of "{(beta)[1],(beta)[2]}"

"rhs(?)[4,4]"

((`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2)*(-beta^2+1)^(1/2)-`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2-`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2+1)/(-beta^2+1)^(1/2)

(62)

 

simplify(((`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2)*(-beta^2+1)^(1/2)-`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2-`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2+1)/(-beta^2+1)^(1/2), {`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]^2 = 1}, {`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1], `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]})

(-(-beta^2+1)^(1/2)*`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]^2+(-beta^2+1)^(1/2))/(-beta^2+1)^(1/2)

(63)

This result involves only `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3], and with it the form of Lambda[`~mu`, nu] = exp(`%𝕃`[`~mu`, nu]) becomes

"subs(1/(-beta^2+1)^(1/2)*((`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2)*(-beta^2+1)^(1/2)-`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1]^2-`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2]^2+1) = (-(-beta^2+1)^(1/2)*`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]^2+`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3]^2+(-beta^2+1)^(1/2))/(-beta^2+1)^(1/2),?)"

exp(`%𝕃`[`~μ`, nu]) = Matrix(%id = 36893488151918876660)

 (64)

Replacing now the components of the unit vector `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j] by the components of the vector `#mover(mi("β",fontstyle = "normal"),mo("→"))` divided by its modulus beta

seq(ub[j] = beta[j]/beta, j = 1 .. 3)

`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1] = beta[1]/beta, `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[2] = beta[2]/beta, `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[3] = beta[3]/beta

(65)

and recalling that

exp(`%𝕃`[`~μ`, nu]) = Lambda[`~μ`, nu]

exp(`%𝕃`[`~μ`, nu]) = Lambda[`~μ`, nu]

(66)

to get equation (11.98) in Jackson's book it suffices to introduce (the customary notation)

1/sqrt(-beta^2+1) = gamma

1/(-beta^2+1)^(1/2) = gamma

(67)

"simplify(subs(`#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[1] = beta[1]/beta, exp(`%𝕃`[`~μ`,nu]) = Lambda[`~μ`,nu], 1/(-beta^2+1)^(1/2) = gamma,(1/(-beta^2+1)^(1/2) = gamma)^(-1),?))"

Lambda[`~μ`, nu] = Matrix(%id = 36893488151911556148)

(68)

 

This is equation (11.98) in Jackson's book.

 

Finally, to get the form of this general Lorentz transformation excluding 3D rotations, directly expressed in terms of the relative velocity v of the two inertial systems of references, introduce

v[i] = [v__x, v__y, v__z], beta[i] = v[i]/c

v[i] = [v__x, v__y, v__z], beta[i] = v[i]/c

(69)

At this point it suffices to Define (69) as tensors

Define(v[i] = [v__x, v__y, v__z], beta[i] = v[i]/c)

{Lambda, `𝕃`[mu, nu], Physics:-Dgamma[mu], K[i], Physics:-Psigma[mu], S[i], Zeta[i], beta[i], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-gamma_[a, b], omega[i], v[i], Physics:-LeviCivita[alpha, beta, mu, nu], `#mover(mi("β",fontstyle = "normal"),mo("ˆ"))`[j]}

(70)

and remove beta and gamma from the formulation using

(rhs = lhs)(1/(-beta^2+1)^(1/2) = gamma), beta = v/c

gamma = 1/(-beta^2+1)^(1/2), beta = v/c

(71)

"simplify(subs(gamma = 1/(-beta^2+1)^(1/2),simplify(?)),size) "

Lambda[`~μ`, nu] = Matrix(%id = 36893488153289646316)

 (72)

NULL

``

NULL

References

 

[1] J.D. Jackson, "Classical Electrodynamics", third edition, 1999.

[2] L.D. Landau, E.M. Lifshitz, "The Classical Theory of Fields", Course of Theoretical Physics V.2, 4th revised English edition, 1975.

NULL

Download Deriving_the_mathematical_form_of_Lorentz_transformations.mw

Deriving_the_mathematical_form_of_Lorentz_transformations.pdf

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Should there be a Maple language reference manual?...

by: nm 11218 Maple
syntax programming documentation
July 27 2022
5  3

According to Wikipedia

"In computing, a programming language reference or language reference manual is
part of the documentation associated with most mainstream programming languages. It is
written for users and developers, and describes the basic elements of the language and how
to use them in a program. For a command-based language, for example, this will include details
of every available command and of the syntax for using it.

The reference manual is usually separate and distinct from a more detailed
 programming language specification meant for implementors of the language rather than those who simply use it to accomplish some processing task."

 

And no, Maple's Programming guide is not a Language reference manual. This is a guide to how to program in Maple. But it is not reference manual for the language itself. i.e. description of the Language itself.

Examples of Language reference manuals are

C

Original C++ 

Python

Ada

Fortran 77

 

and so on.

Why is there no LRM for Maple? Should there be one? I find Maple semantics sometime confusing, and many times I find how things work by trial and error. Having a reference to look at will be good. 

I know for a language as large as Maple, this is not easy to do. But it will good for Maplesoft to invest into making one.

The Electromagnetic Field of Moving Charges

by: ecterrab 14455 Maple
tensor physics electromagnetic
June 18 2022
11  1

This is an interesting exercise, the computation of the Liénard–Wiechert potentials describing the classical electromagnetic field of a moving electric point charge, a problem of a 3rd year undergrad course in Electrodynamics. The calculation is nontrivial and is performed below using the Physics  package, following the presentation in [1] (Landau & Lifshitz "The classical theory of fields"). I have not seen this calculation performed on a computer algebra worksheet before. Thus, this also showcases the more advanced level of symbolic problems that can currently be tackled on a Maple worksheet. At the end, the corresponding document is linked  and with it the computation below can be reproduced. There is also a link to a corresponding PDF file with all the sections open.

Moving charges:
The retarded and Liénard-Wiechert potentials, and the fields `#mover(mi("E"),mo("→"))` and `#mover(mi("H"),mo("→"))`

Freddy Baudine(1), Edgardo S. Cheb-Terrab(2)

(1) Retired, passionate about Mathematics and Physics

(2) Physics, Differential Equations and Mathematical Functions, Maplesoft

 

Generally speaking, determining the electric and magnetic fields of a distribution of charges involves determining the potentials `ϕ` and `#mover(mi("A"),mo("→"))`, followed by determining the fields `#mover(mi("E"),mo("→"))` and `#mover(mi("H"),mo("→"))` from

`#mover(mi("E"),mo("→"))` = -(diff(`#mover(mi("A"),mo("→"))`, t))/c-%Gradient(`ϕ`(X)),        `#mover(mi("H"),mo("→"))` = `&x`(VectorCalculus[Nabla], `#mover(mi("A"),mo("→"))`)

In turn, the formulation of the equations for `ϕ` and `#mover(mi("A"),mo("→"))` is simple: they follow from the 4D second pair of Maxwell equations, in tensor notation

"`∂`[k](F[]^( i, k))=-(4 Pi)/c j^( i)"

where "F[]^( i, k)" is the electromagnetic field tensor and j^i is the 4D current. After imposing the Lorentz condition

`∂`[i](A^i) = 0,     i.e.    (diff(`ϕ`, t))/c+VectorCalculus[Nabla].`#mover(mi("A"),mo("→"))` = 0

we get

`∂`[k](`∂`[`~k`](A^i)) = 4*Pi*j^i/c

which in 3D form results in

"(∇)^2A-1/(c^2) (((∂)^2)/(∂t^2)( A))=-(4 Pi)/c j"

 

Laplacian(`ϕ`)-(diff(`ϕ`, t, t))/c^2 = -4*Pi*rho/c

where `#mover(mi("j"),mo("→"))` is the current and rho is the charge density.

 

Following the presentation shown in [1] (Landau and Lifshitz, "The classical theory of fields", sec. 62 and 63), below we solve these equations for `ϕ` and `#mover(mi("A"),mo("→"))` resulting in the so-called retarded potentials, then recompute these fields as produced by a charge moving along a given trajectory `#mover(mi("r"),mo("→"))` = r__0(t) - the so-called Liénard-Wiechert potentials - finally computing an explicit form for the corresponding `#mover(mi("E"),mo("→"))` and `#mover(mi("H"),mo("→"))`.

 

While the computation of the generic retarded potentials is, in principle, simple, obtaining their form for a charge moving along a given trajectory `#mover(mi("r"),mo("→"))` = r__0(t), and from there the form of the fields `#mover(mi("E"),mo("→"))` and `#mover(mi("H"),mo("→"))` shown in Landau's book, involves nontrivial algebraic manipulations. The presentation below thus also shows a technique to map onto the computer the manipulations typically done with paper and pencil for these problems. To reproduce the contents below, the Maplesoft Physics Updates v.1252 or newer is required.

NULL

with(Physics); Setup(coordinates = Cartesian); with(Vectors)

[coordinatesystems = {X}]

 (1)

The retarded potentials phi and `#mover(mi("A"),mo("→"))`

 

 

The equations which determine the scalar and vector potentials of an arbitrary electromagnetic field are input as

CompactDisplay((`ϕ`, rho, A_, j_)(X))

j_(x, y, z, t)*`will now be displayed as`*j_

(2)

%Laplacian(`ϕ`(X))-(diff(`ϕ`(X), t, t))/c^2 = -4*Pi*rho(X)

%Laplacian(varphi(X))-(diff(diff(varphi(X), t), t))/c^2 = -4*Pi*rho(X)

(3)

%Laplacian(A_(X))-(diff(A_(X), t, t))/c^2 = -4*Pi*j_(X)

%Laplacian(A_(X))-(diff(diff(A_(X), t), t))/c^2 = -4*Pi*j_(X)

(4)

The solutions to these inhomogeneous equations are computed as the sum of the solutions for the equations without right-hand side plus a particular solution to the equation with right-hand side.

Computing the solution to the equations for `ϕ`(X) and  `#mover(mi("A"),mo("→"))`(X)

   

The Liénard-Wiechert potentials of a charge moving along `#mover(mi("r"),mo("→"))` = r__0_(t)

 

From (13), the potential at the point X = (x, y, z, t)is determined by the charge e(t-r/c), i.e. by the position of the charge e at the earlier time

`#msup(mi("t"),mo("'",fontweight = "bold"))` = t-LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)/c

The quantityLinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)is the 3D distance from the position of the charge at the time diff(t(x), x) to the 3D point of observationx, y, z. In the previous section, the charge was located at the origin and at rest, so LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`) = r, the radial coordinate. If the charge is moving, say on a path r__0_(t), we have

`#mover(mi("R"),mo("→"))` = `#mover(mi("r"),mo("→"))`-r__0_(`#msup(mi("t"),mo("'",fontweight = "bold"))`)

From (13)≡`ϕ`(r, t) = de(t-r/c)/r and the definition of `#msup(mi("t"),mo("'",fontweight = "bold"))` above, the potential `ϕ`(r, t) of a moving charge can be written as

`ϕ`(r, t(x)) = e/LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`) and e/LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`) = e/(c*(t(x)-(diff(t(x), x))))

When the charge is at rest, in the Lorentz gauge we are working, the vector potential is `#mover(mi("A"),mo("→"))` = 0. When the charge is moving, the form of `#mover(mi("A"),mo("→"))` can be found searching for a solution to "(∇)^2A-1/(c^2) (((∂)^2)/(∂t^2)( A))=-(4 Pi)/c j" that gives `#mover(mi("A"),mo("→"))` = 0 when `#mover(mi("v"),mo("→"))` = 0. Following [1], this solution can be written as

"A( )^(alpha)=(e u( )^(alpha))/(`R__beta` u^(beta))" 

where u^mu is the four velocity of the charge, "R^(mu) = r^( mu)-`r__0`^(mu) = [(r)-(`r__`),c(t-t')]".  

 

Without showing the intermediate steps, [1] presents the three dimensional vectorial form of these potentials `ϕ` and `#mover(mi("A"),mo("→"))` as

 

`ϕ` = e/(R-`#mover(mi("v"),mo("→"))`/c.`#mover(mi("R"),mo("→"))`),   `#mover(mi("A"),mo("→"))` = e*`#mover(mi("v"),mo("→"))`/(c*(R-`#mover(mi("v"),mo("→"))`/c.`#mover(mi("R"),mo("→"))`))

Computing the vectorial form of the Liénard-Wiechert potentials

   

The electric and magnetic fields `#mover(mi("E"),mo("→"))` and `#mover(mi("H"),mo("→"))` of a charge moving along `#mover(mi("r"),mo("→"))` = r__0_(t)

 

The electric and magnetic fields at a point x, y, z, t are calculated from the potentials `ϕ` and `#mover(mi("A"),mo("→"))` through the formulas

 

`#mover(mi("E"),mo("→"))`(x, y, z, t) = -(diff(`#mover(mi("A"),mo("→"))`(x, y, z, t), t))/c-(%Gradient(`ϕ`(X)))(x, y, z, t),        `#mover(mi("H"),mo("→"))`(x, y, z, t) = `&x`(VectorCalculus[Nabla], `#mover(mi("A"),mo("→"))`(x, y, z, t))

where, for the case of a charge moving on a path r__0_(t), these potentials were calculated in the previous section as (24) and (18)

`ϕ`(x, y, z, t) = e/(LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)-`#mover(mi("R"),mo("→"))`.(`#mover(mi("v"),mo("→"))`/c))

`#mover(mi("A"),mo("→"))`(x, y, z, t) = e*`#mover(mi("v"),mo("→"))`/(c*(LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)-`#mover(mi("R"),mo("→"))`.(`#mover(mi("v"),mo("→"))`/c)))

These two expressions, however, depend on the time only through the retarded time t__0. This dependence is within `#mover(mi("R"),mo("→"))` = `#mover(mi("r"),mo("→"))`(x, y, z)-r__0_(t__0(x, y, z, t)) and through the velocity of the charge `#mover(mi("v"),mo("→"))`(t__0(x, y, z, t)). So, before performing the differentiations, this dependence on t__0(x, y, z, t) must be taken into account.

CompactDisplay(r_(x, y, z), (E_, H_, t__0)(x, y, z, t))

t__0(x, y, z, t)*`will now be displayed as`*t__0

(29)

R_ = r_(x, y, z)-r__0_(t__0(x, y, z, t)), v_ = v_(t__0(x, y, z, t))

R_ = r_(x, y, z)-r__0_(t__0(X)), v_ = v_(t__0(X))

(30)

The Electric field `#mover(mi("E"),mo("→"))` = -(diff(`#mover(mi("A"),mo("→"))`, t))/c-%Gradient(`ϕ`)

 

Computation of Gradient(`ϕ`(X)) 

Computation of "(∂A)/(∂t)"

   

 Collecting the results of the two previous subsections, we have for the electric field

`#mover(mi("E"),mo("→"))`(X) = -(diff(`#mover(mi("A"),mo("→"))`(X), t))/c-%Gradient(`ϕ`(X))

E_(X) = -(diff(A_(X), t))/c-%Gradient(varphi(X))

(60)

subs(%Gradient(varphi(X)) = -c*e*(-Physics[Vectors][Norm](v_)^2*R_-Physics[Vectors][Norm](R_)*c*v_+R_*c^2+Physics[Vectors][`.`](R_, a_)*R_+Physics[Vectors][`.`](R_, v_)*v_)/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3, Physics[Vectors]:-diff(A_(X), t) = e*(Physics[Vectors][Norm](R_)^2*a_*c-v_*Physics[Vectors][Norm](v_)^2*Physics[Vectors][Norm](R_)-Physics[Vectors][Norm](R_)*Physics[Vectors][`.`](R_, v_)*a_+v_*Physics[Vectors][`.`](R_, a_)*Physics[Vectors][Norm](R_)+c*v_*Physics[Vectors][`.`](R_, v_))/((1-Physics[Vectors][`.`](R_, v_)/(Physics[Vectors][Norm](R_)*c))*(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2*Physics[Vectors][Norm](R_)), E_(X) = -(diff(A_(X), t))/c-%Gradient(varphi(X)))

E_(X) = -e*(Physics:-Vectors:-Norm(R_)^2*a_*c-v_*Physics:-Vectors:-Norm(v_)^2*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-`.`(R_, v_)*a_+v_*Physics:-Vectors:-`.`(R_, a_)*Physics:-Vectors:-Norm(R_)+c*v_*Physics:-Vectors:-`.`(R_, v_))/(c*(1-Physics:-Vectors:-`.`(R_, v_)/(Physics:-Vectors:-Norm(R_)*c))*(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^2*Physics:-Vectors:-Norm(R_))+c*e*(-Physics:-Vectors:-Norm(v_)^2*R_-Physics:-Vectors:-Norm(R_)*c*v_+R_*c^2+Physics:-Vectors:-`.`(R_, a_)*R_+Physics:-Vectors:-`.`(R_, v_)*v_)/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3

 (61)

The book, presents this result as equation (63.8):

`#mover(mi("E"),mo("→"))` = e*(1-v^2/c^2)*(`#mover(mi("R"),mo("→"))`-`#mover(mi("v"),mo("→"))`*R/c)/(R-(`#mover(mi("v"),mo("→"))`.`#mover(mi("R"),mo("→"))`)/c)^3+`&x`(e*`#mover(mi("R"),mo("→"))`/c(R-(`#mover(mi("v"),mo("→"))`.`#mover(mi("R"),mo("→"))`)/c)^6, `&x`(`#mover(mi("R"),mo("→"))`-`#mover(mi("v"),mo("→"))`*R/c, `#mover(mi("a"),mo("→"))`))

where `≡`(R, LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)) and `≡`(v, LinearAlgebra[Norm](`#mover(mi("v"),mo("→"))`)). To rewrite (61) as in the above, introduce the two triple vector products

`&x`(R_, `&x`(v_, a_)); expand(%) = %

v_*Physics:-Vectors:-`.`(R_, a_)-Physics:-Vectors:-`.`(R_, v_)*a_ = Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))

(62)

simplify(E_(X) = -e*(Physics[Vectors][Norm](R_)^2*a_*c-v_*Physics[Vectors][Norm](v_)^2*Physics[Vectors][Norm](R_)-Physics[Vectors][Norm](R_)*Physics[Vectors][`.`](R_, v_)*a_+v_*Physics[Vectors][`.`](R_, a_)*Physics[Vectors][Norm](R_)+c*v_*Physics[Vectors][`.`](R_, v_))/(c*(1-Physics[Vectors][`.`](R_, v_)/(Physics[Vectors][Norm](R_)*c))*(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2*Physics[Vectors][Norm](R_))+c*e*(-Physics[Vectors][Norm](v_)^2*R_-Physics[Vectors][Norm](R_)*c*v_+R_*c^2+Physics[Vectors][`.`](R_, a_)*R_+Physics[Vectors][`.`](R_, v_)*v_)/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3, {v_*Physics[Vectors][`.`](R_, a_)-Physics[Vectors][`.`](R_, v_)*a_ = Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))})

E_(X) = e*(-Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))+R_*c*Physics:-Vectors:-`.`(R_, a_)-Physics:-Vectors:-Norm(R_)^2*a_*c+(-c^2*v_+v_*Physics:-Vectors:-Norm(v_)^2)*Physics:-Vectors:-Norm(R_)+R_*c^3-R_*c*Physics:-Vectors:-Norm(v_)^2)/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3

(63)

`&x`(R_, `&x`(R_, a_)); expand(%) = %

Physics:-Vectors:-`.`(R_, a_)*R_-Physics:-Vectors:-Norm(R_)^2*a_ = Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(R_, a_))

(64)

simplify(E_(X) = e*(-Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))+R_*c*Physics[Vectors][`.`](R_, a_)-Physics[Vectors][Norm](R_)^2*a_*c+(-c^2*v_+v_*Physics[Vectors][Norm](v_)^2)*Physics[Vectors][Norm](R_)+R_*c^3-R_*c*Physics[Vectors][Norm](v_)^2)/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3, {Physics[Vectors][`.`](R_, a_)*R_-Physics[Vectors][Norm](R_)^2*a_ = Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](R_, a_))})

E_(X) = (c*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(R_, a_))-Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))+(c-Physics:-Vectors:-Norm(v_))*(c+Physics:-Vectors:-Norm(v_))*(R_*c-Physics:-Vectors:-Norm(R_)*v_))*e/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3

(65)

Split now this result into two terms, one of them involving the acceleration `#mover(mi("a"),mo("→"))`. For that purpose first expand the expression without expanding the cross products

lhs(E_(X) = (c*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](R_, a_))-Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))+(c-Physics[Vectors][Norm](v_))*(c+Physics[Vectors][Norm](v_))*(R_*c-Physics[Vectors][Norm](R_)*v_))*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3) = frontend(expand, [rhs(E_(X) = (c*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](R_, a_))-Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))+(c-Physics[Vectors][Norm](v_))*(c+Physics[Vectors][Norm](v_))*(R_*c-Physics[Vectors][Norm](R_)*v_))*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3)])

E_(X) = e*Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-Norm(v_)^2*v_/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3-e*Physics:-Vectors:-Norm(R_)*c^2*v_/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3-e*Physics:-Vectors:-Norm(v_)^2*R_*c/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3+e*R_*c^3/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3+e*c*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(R_, a_))/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3-e*Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3

 (66)

Introduce the notation used in the textbook, `≡`(R, LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)) and `≡`(v, LinearAlgebra[Norm](`#mover(mi("v"),mo("→"))`)) and proceed with the splitting

lhs(E_(X) = (c*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](R_, a_))-Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))+(c-Physics[Vectors][Norm](v_))*(c+Physics[Vectors][Norm](v_))*(R_*c-Physics[Vectors][Norm](R_)*v_))*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3) = subs(Norm(R_) = R, Norm(v_) = v, add(normal([selectremove(`not`(has), rhs(E_(X) = e*Physics[Vectors][Norm](R_)*Physics[Vectors][Norm](v_)^2*v_/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3-e*Physics[Vectors][Norm](R_)*c^2*v_/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3-e*Physics[Vectors][Norm](v_)^2*R_*c/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3+e*R_*c^3/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3+e*c*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](R_, a_))/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3-e*Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3), `#mover(mi("a"),mo("→"))`)])))

E_(X) = e*(-R*c^2*v_+R*v^2*v_+R_*c^3-R_*c*v^2)/(c*R-Physics:-Vectors:-`.`(R_, v_))^3-e*(R*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))-c*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(R_, a_)))/(c*R-Physics:-Vectors:-`.`(R_, v_))^3

(67)

Rearrange only the first term using simplify; that can be done in different ways, perhaps the simplest is using subsop

subsop([2, 1] = simplify(op([2, 1], E_(X) = e*(-R*c^2*v_+R*v^2*v_+R_*c^3-R_*c*v^2)/(c*R-Physics[Vectors][`.`](R_, v_))^3-e*(R*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))-c*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](R_, a_)))/(c*R-Physics[Vectors][`.`](R_, v_))^3)), E_(X) = e*(-R*c^2*v_+R*v^2*v_+R_*c^3-R_*c*v^2)/(c*R-Physics[Vectors][`.`](R_, v_))^3-e*(R*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](v_, a_))-c*Physics[Vectors][`&x`](R_, Physics[Vectors][`&x`](R_, a_)))/(c*R-Physics[Vectors][`.`](R_, v_))^3)

E_(X) = e*(c-v)*(c+v)*(-R*v_+R_*c)/(c*R-Physics:-Vectors:-`.`(R_, v_))^3-e*(R*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(v_, a_))-c*Physics:-Vectors:-`&x`(R_, Physics:-Vectors:-`&x`(R_, a_)))/(c*R-Physics:-Vectors:-`.`(R_, v_))^3

(68)

NULL

By eye this result is mathematically equal to equation (63.8) of the textbook, shown here above before (62) .

 

Algebraic manipulation rewriting (68) as the textbook equation (63.8)

   

The magnetic field  `#mover(mi("H"),mo("→"))` = `&x`(VectorCalculus[Nabla], `#mover(mi("A"),mo("→"))`)

 

 

The book does not show an explicit form for `#mover(mi("H"),mo("→"))`, it only indicates that it is related to the electric field by the formula

 

`#mover(mi("H"),mo("→"))` = `&x`(`#mover(mi("R"),mo("→"))`, `#mover(mi("E"),mo("→"))`)/LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)

 

Thus in this section we compute the explicit form of `#mover(mi("H"),mo("→"))` and show that this relationship mentioned in the book holds. To compute `#mover(mi("H"),mo("→"))` = `&x`(VectorCalculus[Nabla], `#mover(mi("A"),mo("→"))`) we proceed as done in the previous sections, the right-hand side should be taken at the previous (retarded) time t__0. For clarity, turn OFF the compact display of functions.

OFF

 

We need to calculate

H_(X) = Curl(A_(x, y, z, t__0(x, y, z, t)))

H_(X) = Physics:-Vectors:-Curl(A_(x, y, z, t__0(X)))

(75)

Deriving the chain rule `&x`(VectorCalculus[Nabla], `#mover(mi("A"),mo("→"))`(t__0(x, y, z, t))) = %Curl(A_(x, y, z, `#msub(mi("t"),mi("0"))`))+`&x`(%Gradient(`#msub(mi("t"),mi("0"))`(X)), diff(`#mover(mi("A"),mo("→"))`(t__0), t__0))

   

So applying to (75)  the chain rule derived in the previous subsection we have

H_(X) = %Curl(A_(x, y, z, t__0))+`&x`(%Gradient(t__0(X)), diff(A_(x, y, z, t__0), t__0))

H_(X) = %Curl(A_(x, y, z, t__0))+Physics:-Vectors:-`&x`(%Gradient(t__0(X)), diff(A_(x, y, z, t__0), t__0))

(87)

where t__0 is taken as a function of x, y, z, t only in %Gradient(`#msub(mi("t"),mi("0"))`(X)). Now that the functionality is understood, turning ON the compact display of functions and displaying the fields by their names,

CompactDisplay(H_(X) = %Curl(A_(x, y, z, t__0))+Physics[Vectors][`&x`](%Gradient(t__0(X)), diff(A_(x, y, z, t__0), t__0)), E_(X))

E_(x, y, z, t)*`will now be displayed as`*E_

(88)

The value of %Gradient(`#msub(mi("t"),mi("0"))`(X)) is computed lines above as (48)

%Gradient(t__0(X)) = R_/(-c*Physics[Vectors][Norm](R_)+Physics[Vectors][`.`](R_, v_))

%Gradient(t__0(X)) = R_/(-c*Physics:-Vectors:-Norm(R_)+Physics:-Vectors:-`.`(R_, v_))

(89)

The expression for `#mover(mi("A"),mo("→"))` with no dependency is computed lines above, as (28),

subs(A_ = A_(x, y, z, t__0), A_ = e*v_/((Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_/c))*c))

A_(x, y, z, t__0) = e*v_/((Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_)/c)*c)

(90)

The expressions for `#mover(mi("R"),mo("→"))` and the velocity in terms of t__0 with no dependency are

R_ = r_(x, y, z)-r__0_(t__0), v_ = v_(t__0)

R_ = r_(x, y, z)-r__0_(t__0), v_ = v_(t__0)

(91)

CompactDisplay(r_(x, y, z))

r_(x, y, z)*`will now be displayed as`*r_

(92)

subs(R_ = r_(x, y, z)-r__0_(t__0), v_ = v_(t__0), [%Gradient(t__0(X)) = R_/(-c*Physics[Vectors][Norm](R_)+Physics[Vectors][`.`](R_, v_)), A_(x, y, z, t__0) = e*v_/((Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_)/c)*c)])

[%Gradient(t__0(X)) = (r_(x, y, z)-r__0_(t__0))/(-c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))+Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), v_(t__0))), A_(x, y, z, t__0) = e*v_(t__0)/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), v_(t__0))/c)*c)]

(93)

Introducing this into "H(X)=`%Curl`(A_(x,y,z,t[`0`]))+(`%Gradient`(t[`0`](X)))*((∂A)/(∂`t__0`))",

eval(H_(X) = %Curl(A_(x, y, z, t__0))+Physics[Vectors][`&x`](%Gradient(t__0(X)), diff(A_(x, y, z, t__0), t__0)), [%Gradient(t__0(X)) = (r_(x, y, z)-r__0_(t__0))/(-c*Physics[Vectors][Norm](r_(x, y, z)-r__0_(t__0))+Physics[Vectors][`.`](r_(x, y, z)-r__0_(t__0), v_(t__0))), A_(x, y, z, t__0) = e*v_(t__0)/((Physics[Vectors][Norm](r_(x, y, z)-r__0_(t__0))-Physics[Vectors][`.`](r_(x, y, z)-r__0_(t__0), v_(t__0))/c)*c)])

H_(X) = %Curl(e*v_(t__0)/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), v_(t__0))/c)*c))+Physics:-Vectors:-`&x`(r_(x, y, z)-r__0_(t__0), -e*v_(t__0)*(-Physics:-Vectors:-`.`(diff(r__0_(t__0), t__0), r_(x, y, z)-r__0_(t__0))/Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))-(-Physics:-Vectors:-`.`(diff(r__0_(t__0), t__0), v_(t__0))+Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), diff(v_(t__0), t__0)))/c)/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), v_(t__0))/c)^2*c)+e*(diff(v_(t__0), t__0))/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), v_(t__0))/c)*c))/(-c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_(t__0))+Physics:-Vectors:-`.`(r_(x, y, z)-r__0_(t__0), v_(t__0)))

 (94)

Before computing the first term `&x`(VectorCalculus[Nabla], () .. ()), for readability, re-introduce the velocity diff(`#msub(mi("r"),mi("0_"))`(t__0), t__0) = `#mover(mi("v"),mo("→"))`, the acceleration diff(`#mover(mi("v"),mo("→"))`(t__0), t__0) = `#mover(mi("a"),mo("→"))`, then remove the dependency of these functions on t__0, not relevant anymore since there are no more derivatives with respect to t__0. Performing these substitutions in sequence,

diff(`#msub(mi("r"),mi("0_"))`(t__0), t__0) = `#mover(mi("v"),mo("→"))`, diff(`#mover(mi("v"),mo("→"))`(t__0), t__0) = `#mover(mi("a"),mo("→"))`, `#mover(mi("v"),mo("→"))`(t__0) = `#mover(mi("v"),mo("→"))`, `#msub(mi("r"),mi("0_"))`(t__0) = `#msub(mi("r"),mi("0_"))`

diff(r__0_(t__0), t__0) = v_, diff(v_(t__0), t__0) = a_, v_(t__0) = v_, r__0_(t__0) = r__0_

(95)

subs(diff(r__0_(t__0), t__0) = v_, diff(v_(t__0), t__0) = a_, v_(t__0) = v_, r__0_(t__0) = r__0_, H_(X) = %Curl(e*v_(t__0)/((Physics[Vectors][Norm](r_(x, y, z)-r__0_(t__0))-Physics[Vectors][`.`](r_(x, y, z)-r__0_(t__0), v_(t__0))/c)*c))+Physics[Vectors][`&x`](r_(x, y, z)-r__0_(t__0), -e*v_(t__0)*(-Physics[Vectors][`.`](diff(r__0_(t__0), t__0), r_(x, y, z)-r__0_(t__0))/Physics[Vectors][Norm](r_(x, y, z)-r__0_(t__0))-(-Physics[Vectors][`.`](diff(r__0_(t__0), t__0), v_(t__0))+Physics[Vectors][`.`](r_(x, y, z)-r__0_(t__0), diff(v_(t__0), t__0)))/c)/((Physics[Vectors][Norm](r_(x, y, z)-r__0_(t__0))-Physics[Vectors][`.`](r_(x, y, z)-r__0_(t__0), v_(t__0))/c)^2*c)+e*(diff(v_(t__0), t__0))/((Physics[Vectors][Norm](r_(x, y, z)-r__0_(t__0))-Physics[Vectors][`.`](r_(x, y, z)-r__0_(t__0), v_(t__0))/c)*c))/(-c*Physics[Vectors][Norm](r_(x, y, z)-r__0_(t__0))+Physics[Vectors][`.`](r_(x, y, z)-r__0_(t__0), v_(t__0))))

H_(X) = %Curl(e*v_/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_)/c)*c))+Physics:-Vectors:-`&x`(r_(x, y, z)-r__0_, -e*v_*(-Physics:-Vectors:-`.`(v_, r_(x, y, z)-r__0_)/Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-(-Physics:-Vectors:-`.`(v_, v_)+Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, a_))/c)/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_)/c)^2*c)+e*a_/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_)/c)*c))/(-c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)+Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_))

 (96)

Activate now the inert curl `&x`(VectorCalculus[Nabla], () .. ())

value(H_(X) = %Curl(e*v_/((Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_)/c)*c))+Physics[Vectors][`&x`](r_(x, y, z)-r__0_, -e*v_*(-Physics[Vectors][`.`](v_, r_(x, y, z)-r__0_)/Physics[Vectors][Norm](r_(x, y, z)-r__0_)-(-Physics[Vectors][`.`](v_, v_)+Physics[Vectors][`.`](r_(x, y, z)-r__0_, a_))/c)/((Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_)/c)^2*c)+e*a_/((Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_)/c)*c))/(-c*Physics[Vectors][Norm](r_(x, y, z)-r__0_)+Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_)))

H_(X) = e*Physics:-Vectors:-`&x`(-c^2*_i*Physics:-Vectors:-`.`(diff(r_(x, y, z), x), r_(x, y, z)-r__0_)/((c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_))^2*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_))+c*_i*Physics:-Vectors:-`.`(diff(r_(x, y, z), x), v_)/(c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_))^2-c^2*_j*Physics:-Vectors:-`.`(diff(r_(x, y, z), y), r_(x, y, z)-r__0_)/((c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_))^2*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_))+c*_j*Physics:-Vectors:-`.`(diff(r_(x, y, z), y), v_)/(c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_))^2-c^2*_k*Physics:-Vectors:-`.`(diff(r_(x, y, z), z), r_(x, y, z)-r__0_)/((c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_))^2*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_))+c*_k*Physics:-Vectors:-`.`(diff(r_(x, y, z), z), v_)/(c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_))^2, v_)/c+Physics:-Vectors:-`&x`(r_(x, y, z)-r__0_, -e*v_*(-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_)/Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-(-Physics:-Vectors:-Norm(v_)^2+Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, a_))/c)/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_)/c)^2*c)+e*a_/((Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)-Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_)/c)*c))/(-c*Physics:-Vectors:-Norm(r_(x, y, z)-r__0_)+Physics:-Vectors:-`.`(r_(x, y, z)-r__0_, v_))

 (97)

From (34)≡diff(`#mover(mi("r"),mo("→"))`, x) = `#mover(mi("i"),mo("∧"))`, diff(`#mover(mi("r"),mo("→"))`, y) = `#mover(mi("j"),mo("∧"))`, diff(`#mover(mi("r"),mo("→"))`, z) = `#mover(mi("k"),mo("∧"))`, and reintroducing `#mover(mi("r"),mo("→"))`(x, y, z)-r__0_ = `#mover(mi("R"),mo("→"))`

subs(diff(r_(x, y, z), x) = _i, diff(r_(x, y, z), y) = _j, diff(r_(x, y, z), z) = _k, `#mover(mi("r"),mo("→"))`(x, y, z)-r__0_ = `#mover(mi("R"),mo("→"))`, H_(X) = e*Physics[Vectors][`&x`](-c^2*_i*Physics[Vectors][`.`](diff(r_(x, y, z), x), r_(x, y, z)-r__0_)/((c*Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_))^2*Physics[Vectors][Norm](r_(x, y, z)-r__0_))+c*_i*Physics[Vectors][`.`](diff(r_(x, y, z), x), v_)/(c*Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_))^2-c^2*_j*Physics[Vectors][`.`](diff(r_(x, y, z), y), r_(x, y, z)-r__0_)/((c*Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_))^2*Physics[Vectors][Norm](r_(x, y, z)-r__0_))+c*_j*Physics[Vectors][`.`](diff(r_(x, y, z), y), v_)/(c*Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_))^2-c^2*_k*Physics[Vectors][`.`](diff(r_(x, y, z), z), r_(x, y, z)-r__0_)/((c*Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_))^2*Physics[Vectors][Norm](r_(x, y, z)-r__0_))+c*_k*Physics[Vectors][`.`](diff(r_(x, y, z), z), v_)/(c*Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_))^2, v_)/c+Physics[Vectors][`&x`](r_(x, y, z)-r__0_, -e*v_*(-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_)/Physics[Vectors][Norm](r_(x, y, z)-r__0_)-(-Physics[Vectors][Norm](v_)^2+Physics[Vectors][`.`](r_(x, y, z)-r__0_, a_))/c)/((Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_)/c)^2*c)+e*a_/((Physics[Vectors][Norm](r_(x, y, z)-r__0_)-Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_)/c)*c))/(-c*Physics[Vectors][Norm](r_(x, y, z)-r__0_)+Physics[Vectors][`.`](r_(x, y, z)-r__0_, v_)))

H_(X) = e*Physics:-Vectors:-`&x`(-c^2*_i*Physics:-Vectors:-`.`(_i, R_)/((c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^2*Physics:-Vectors:-Norm(R_))+c*_i*Physics:-Vectors:-`.`(_i, v_)/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^2-c^2*_j*Physics:-Vectors:-`.`(_j, R_)/((c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^2*Physics:-Vectors:-Norm(R_))+c*_j*Physics:-Vectors:-`.`(_j, v_)/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^2-c^2*_k*Physics:-Vectors:-`.`(_k, R_)/((c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^2*Physics:-Vectors:-Norm(R_))+c*_k*Physics:-Vectors:-`.`(_k, v_)/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^2, v_)/c+Physics:-Vectors:-`&x`(R_, -e*v_*(-Physics:-Vectors:-`.`(R_, v_)/Physics:-Vectors:-Norm(R_)-(-Physics:-Vectors:-Norm(v_)^2+Physics:-Vectors:-`.`(R_, a_))/c)/((Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_)/c)^2*c)+e*a_/((Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_)/c)*c))/(-c*Physics:-Vectors:-Norm(R_)+Physics:-Vectors:-`.`(R_, v_))

 (98)

Simplify(H_(X) = e*Physics[Vectors][`&x`](-c^2*_i*Physics[Vectors][`.`](_i, R_)/((c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2*Physics[Vectors][Norm](R_))+c*_i*Physics[Vectors][`.`](_i, v_)/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2-c^2*_j*Physics[Vectors][`.`](_j, R_)/((c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2*Physics[Vectors][Norm](R_))+c*_j*Physics[Vectors][`.`](_j, v_)/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2-c^2*_k*Physics[Vectors][`.`](_k, R_)/((c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2*Physics[Vectors][Norm](R_))+c*_k*Physics[Vectors][`.`](_k, v_)/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^2, v_)/c+Physics[Vectors][`&x`](R_, -e*v_*(-Physics[Vectors][`.`](R_, v_)/Physics[Vectors][Norm](R_)-(-Physics[Vectors][Norm](v_)^2+Physics[Vectors][`.`](R_, a_))/c)/((Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_)/c)^2*c)+e*a_/((Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_)/c)*c))/(-c*Physics[Vectors][Norm](R_)+Physics[Vectors][`.`](R_, v_)))

H_(X) = (-e*c*Physics:-Vectors:-`&x`(R_, v_)*(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))+e*(-c*Physics:-Vectors:-`.`(R_, v_)+(Physics:-Vectors:-Norm(v_)^2-Physics:-Vectors:-`.`(R_, a_))*Physics:-Vectors:-Norm(R_))*Physics:-Vectors:-`&x`(R_, v_)-e*(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))*Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-`&x`(R_, a_))/((c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3*Physics:-Vectors:-Norm(R_))

 (99)

To conclude, rearrange this expression as done with the one for the electric field `#mover(mi("E"),mo("→"))` at (65), so first expand  (99) without expanding the cross products

lhs(H_(X) = (-e*c*Physics[Vectors][`&x`](R_, v_)*(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))+e*(-c*Physics[Vectors][`.`](R_, v_)+(Physics[Vectors][Norm](v_)^2-Physics[Vectors][`.`](R_, a_))*Physics[Vectors][Norm](R_))*Physics[Vectors][`&x`](R_, v_)-e*(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))*Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, a_))/((c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3*Physics[Vectors][Norm](R_))) = frontend(expand, [rhs(H_(X) = (-e*c*Physics[Vectors][`&x`](R_, v_)*(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))+e*(-c*Physics[Vectors][`.`](R_, v_)+(Physics[Vectors][Norm](v_)^2-Physics[Vectors][`.`](R_, a_))*Physics[Vectors][Norm](R_))*Physics[Vectors][`&x`](R_, v_)-e*(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))*Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, a_))/((c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3*Physics[Vectors][Norm](R_)))])

H_(X) = -Physics:-Vectors:-Norm(R_)*Physics:-Vectors:-`&x`(R_, a_)*c*e/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3+Physics:-Vectors:-`&x`(R_, v_)*Physics:-Vectors:-Norm(v_)^2*e/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3-Physics:-Vectors:-`&x`(R_, v_)*c^2*e/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3+Physics:-Vectors:-`.`(R_, v_)*Physics:-Vectors:-`&x`(R_, a_)*e/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3-Physics:-Vectors:-`&x`(R_, v_)*Physics:-Vectors:-`.`(R_, a_)*e/(c*Physics:-Vectors:-Norm(R_)-Physics:-Vectors:-`.`(R_, v_))^3

 (100)

Then introduce the notation used in the textbook, `≡`(R, LinearAlgebra[Norm](`#mover(mi("R"),mo("→"))`)) and `≡`(v, LinearAlgebra[Norm](`#mover(mi("v"),mo("→"))`)) and split into two terms, one of which contains the acceleration `#mover(mi("a"),mo("→"))`

lhs(H_(X) = -Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, a_)*c*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3+Physics[Vectors][`&x`](R_, v_)*Physics[Vectors][Norm](v_)^2*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3-Physics[Vectors][`&x`](R_, v_)*c^2*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3+Physics[Vectors][`.`](R_, v_)*Physics[Vectors][`&x`](R_, a_)*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3-Physics[Vectors][`&x`](R_, v_)*Physics[Vectors][`.`](R_, a_)*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3) = subs(Norm(R_) = R, Norm(v_) = v, add(normal([selectremove(`not`(has), rhs(H_(X) = -Physics[Vectors][Norm](R_)*Physics[Vectors][`&x`](R_, a_)*c*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3+Physics[Vectors][`&x`](R_, v_)*Physics[Vectors][Norm](v_)^2*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3-Physics[Vectors][`&x`](R_, v_)*c^2*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3+Physics[Vectors][`.`](R_, v_)*Physics[Vectors][`&x`](R_, a_)*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3-Physics[Vectors][`&x`](R_, v_)*Physics[Vectors][`.`](R_, a_)*e/(c*Physics[Vectors][Norm](R_)-Physics[Vectors][`.`](R_, v_))^3), `#mover(mi("a"),mo("→"))`)])))

H_(X) = Physics:-Vectors:-`&x`(R_, v_)*e*(-c^2+v^2)/(c*R-Physics:-Vectors:-`.`(R_, v_))^3-e*(Physics:-Vectors:-`&x`(R_, a_)*R*c-Physics:-Vectors:-`&x`(R_, a_)*Physics:-Vectors:-`.`(R_, v_)+Physics:-Vectors:-`.`(R_, a_)*Physics:-Vectors:-`&x`(R_, v_))/(c*R-Physics:-Vectors:-`.`(R_, v_))^3

(101)

Verifying `#mover(mi("H"),mo("→"))` = `&x`(`#mover(mi("R"),mo("→"))`, `#mover(mi("E"),mo("→"))`)/R

   

References

 

NULL

[1] Landau, L.D., and Lifshitz, E.M. Course of Theoretical Physics Vol 2, The Classical Theory of Fields. Elsevier, 1975.

NULL

 

Download document: The_field_of_moving_charges.mw

Download PDF with sections open: The_field_of_moving_charges.pdf

 

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

What Are Your Favourite Maple Learn Tricks?

by: pstaadecker 125 Maple Learn
tip education beginners
January 19 2022
2  0

I’m looking for users’ favourite tips and tricks in Maple Learn. Specifically, small pieces of advice that most people don’t know about, but that helped you create better Maple Learn documents. For instance,

  • A favorite feature that you think is hard to discover;
  • Common techniques you use when creating documents;
  • Things about Maple Learn you wish you knew when you started.

These tricks could be for newbies or for experienced users.

To start off the discussion, let me share three of my own favorite tricks in Maple Learn.

1. Using Documents from the Document Gallery

Writing a Maple Learn document from scratch can seem overwhelming, especially for beginners. A much easier way to create documents is to start with a template from the Document Gallery.

There are hundreds of Maple Learn documents in the Document Gallery, available here. Instead of writing Maple Learn documents from scratch, I like to search the gallery for documents relating to my topic. I then select a document, and just modify it slightly to get what I want.

2. Toggling from Math Mode to Text Mode

If you want to write text in a group element, it’s best to toggle to text mode (otherwise Maple Learn will treat your text as math).

While this can be done using the toolbar, there is a nifty keyboard shortcut to toggle to text mode: place your cursor at the beginning of the group element, and press the space key.

3. Using Double Arrows in Plots to Show Distance

Here’s one for the advanced users. The Vector Command lets you draw arrows on a Maple Learn plot. Combine two such arrows of the same colour going in opposite directions, and you get a double arrow (see below), which I like to use to represent distances in my Maple Learn documents.

Indeed, here is an example document where I use double arrows to provide a visualization of the product rule in calculus (plot pictured below). Notice how the double arrows (created using the vector command) represent distances in the plot.

Comment your favourite tips and tricks down below!

Suggestion: improvement of help pages in future...

by: mmcdara 7711 Maple
documentation
November 13 2021
3  9

A simple suggestion...

I would appreciate being able to open multiple help pages simultaneously instead of just one.
This seems to me particularly interesting when you have to browse back and forth between several related items.

Maple Learn - Now with Images!

by: ecarette 490 Maple Learn
image education circle
November 11 2021
1  0

Yes, that’s right! You can now add images to your Maple Learn documents! Whether you’re adding a diagram to help visualize a physics concept, inserting the logo or your school or organization, or just adding a cute selfie so that everyone knows how great you looked while making this document, you can add any image you’d like using the image icon on the toolbar. You’ll need to be logged in to access this new feature, but luckily making an account is completely free!

To insert the image, just click the image icon and select the image you want from your computer or tablet. To resize it, highlight the image and click the image icon again. You can also turn the image into a hyperlink by highlight the image and clicking the link button! Now, not only will your document look snazzy, but it can take you anywhere you’d like.

Images aren’t the only exciting new feature in Maple Learn. If you were excited by all the circles in the last set of updates, then you’re going to love this one, because we’ve introduced the Circle command! Just plug in the centre of the circle and the radius, and bam, circle. What’s more, you can easily turn your circle into an arc by adding the angle measures of the two endpoints of the arc. Infinitely customizable round objects, right at your fingertips. To learn more, check out the How-To documents Using the Circle Command and Plotting Arcs.

Ancient Greek mathematicians thought that there was nothing that couldn’t be constructed with only a compass and a straightedge. A wise math professor once tasked my class with using these same tools to draw a pretty picture. With Maple Learn’s Circle function and ability to graph straight lines, you have all the tools you need to complete this same task! We look forward to seeing the results.

 

Comparing Units

by: Anthrazit 885 Maple
units
November 03 2021
1  13

Guys, this is still the most painful thing i Maple for me, and I hope this gets a high priority for future development.

It is still not possible to compare variables, when one of them could become zero.

with(Units[Standard])

[`*`, `+`, `-`, `/`, `<`, `<=`, `<>`, `=`, Im, Re, `^`, abs, add, arccos, arccosh, arccot, arccoth, arccsc, arccsch, arcsec, arcsech, arcsin, arcsinh, arctan, arctanh, argument, ceil, collect, combine, conjugate, cos, cosh, cot, coth, csc, csch, csgn, diff, eval, evalc, evalr, exp, expand, factor, floor, frac, int, ln, log, log10, log2, max, min, mul, normal, polar, root, round, sec, sech, seq, shake, signum, simplify, sin, sinh, sqrt, surd, tan, tanh, trunc, type, verify]

 (1)

a := 15*Unit('kN')

15*Units:-Unit(kN)

 (2)

b := 0*Unit('kN')

0

 (3)

NULL``

> 

if a < b then "True" else "False" end if

Error, cannot determine if this expression is true or false: 15*Units:-Unit(kN) < 0

  

NULL

Download CompareUnits.mw

New Application Center - Beta

by: Bryon 1728 MaplePrimes
November 01 2021
8  14

As many of you are aware, the Maple Application Center is a very important resource for Maple users. It is a place for authors to share their Maple work, and for users to have access to a rich collection of over 2,500 curated Maple documents covering a wide array of topics and disciplines.

I am very pleased to announce that we have been hard at work on a new version of the Application Center, and it’s at a state where we’re ready to open it up to the public for testing. You can access the new site here: https://www.maplesoft.com/applications_beta . We are looking for feedback, so please give it a try, and let us know what you think!

Here are a few of my favorite features of the new site:

Updated Look & Feel
The interface of the current version of the Application Center has not changed in many years, and it was time for a new paint job. I think you’ll find that the new site is cleaner, modern, and more enjoyable to use.

Easier to Find the Documents you Want
The updated Application Center provides multiple new ways to find content that is relevant for you. Browse user-made collections of documents or use tags (the same tags used in MaplePrimes) to find documents for the topics you are interested in. Alternatively, you can use the search bar to quickly find documents, tags or authors.

Personalize your Experience
If you are logged-in when using the Application Center, you will be able to customize what you see by pinning your favorite collections, authors or tags to your home page.

Community Moderation & Reputation
As with MaplePrimes, the strength of the Application Center comes from the amazing community of individuals who contribute to it. In addition to submitting your own content to the Application Center, users can now edit tags and create collections of content that others can use. Similar to MaplePrimes, community moderation is restricted to members who have a sufficient reputation score. Speaking of reputation, quality contributions to MaplePrimes will now be reflected in your reputation score. When someone likes one of your submissions, your reputation will increase by 5.

 

There are many other great new features as well, and we have a roadmap of future updates planned that will make it even better.

I invite you to take a look at the new site and play with it. Browse some content, search, look through tags, and create some collections. Most importantly, I’m really hopeful that you will then use the comments section below to let us know what you think. Did you discover any bugs or issues? What do you like? What do you dislike? What other features would you like to see?

We are hoping to run the Beta for a period of a few weeks, and I’m looking forward to hearing and reading your thoughts. Hope you enjoy it!

https://www.maplesoft.com/applications_beta

Bryon

Concerning Chromebooks and Calculations

by: ecarette 490 Maple Learn
October 27 2021
4  0

Do you have a Chromebook?  Are you a student or a teacher looking for the mighty power of Maple, but find yourself limited by your web-only computer? Well, have no fear, because Maple Learn is here!

As a web-based application, Maple Learn is fully supported by Chromebooks. You can create graphs, perform and check calculations, and share documents all within the comfort of your own browser. No need to download any kind of software—just go to learn.maplesoft.com to get started!

Students, if you’re looking for some use for your school-provided Chromebook and wondering how it can help you learn instead of just weighing down your backpack, Maple Learn can help. It’s the perfect, all-inclusive tool to help you learn, visualize, and check your math. And, if you’re looking to brush up on all that math you forgot over the summer, you can check out the Maple Learn Example Gallery, home to hundreds of examples and explanations of a wide variety of math concepts. And it’s all accessible on your Chromebook!

Check out Engineering Matters Podcast #127 - Tools...

by: alex_b 75 Maple , Maple Flow
October 08 2021
2  0

Here's a podcast that covers a few topics that get discussed on MaplePrimes.
 

We all like finding the right tool for the job. In the Sep 2021 episode of the Engineering Matters Podcast “#127 – Tools for Thinking” you can discover how far engineers have come in their quest for better tools.

It features contributions from several members of the Maplesoft team as they discuss how the user experience shapes the adoption of engineering software tools.

The hosts have fun describing some early calculation hacks - from early Sumerian farmers using their fingers as tally counters, to the paper calculus notebooks of the 1850s used by historical engineering figures like Isambard Kingdom Brunel. What starts as a necessity gets improved over time to save them mental effort – all driven by the way users interact with the tool.

This episode gives a behind-the-scenes look at some of the decisions that shaped the engineering product that is now Maple Flow from its roots in Maple. Maplesoft CEO Laurent Bernardin describes the spark of innovation in the late 1970s, when two professors at the University of Waterloo developed Maple. “The two professors got together, realising that there was a need in math education for a tool to help with calculations and setting out to create that tool. And Maple was born quickly, was adopted across universities around the globe.”

As engineers typically work in ways far removed from the regular academic setting, Product Manager Samir Khan weighs in on the shift that comes from a different user base: “Different tools have different design intents,” says Khan. “Some tools are designed for programmers such as code development environments, like Visual Studio. Some environments are aimed at mathematicians, people who need precise control over the mathematical structure of their equations, and some environments are designed for engineers who simply want to throw down a few equations on a virtual whiteboard and manipulate them and get results.”

The conversation also touches on the design of the GUI itself. Margaret Hinchcliffe, Maple’s Senior GUI Developer expresses the importance of smoothing the user experience - drilling down and taking “the typical tasks that people want to do the most, and make those the most immediate. So really focusing on how many keystrokes do they need to do this task?”.

Ironically the idea of the paper notebook still has features that are desirable. Khan muses on the idea that Maplesoft has “taken the first step with having a virtual whiteboard, but Maple Flow still relies on keyboard and mouse input”. He offers suggestions for what may be next in the industry: “It’d be interesting to see if we can take advantage of modern advances in deep learning and AI to imitate what humans are doing and interpreting handwritten mathematics.”

You can listen to the entire podcast (~30 min) here: https://engineeringmatters.reby.media/2021/09/30/the-evolution-of-tools-for-thinking/

A Chebyshev Flower

by: rcorless 710 Maple
chebyshev polarplot plotting
October 07 2021
9  1

From a tweet by Tamás Görbe : plotting Chebyshev polynomials in polar coordinates leads to some interesting pictures.  Screenshot here, link to the worksheet (and some perhaps interesting puzzles) at the end.

 

ChebyshevRose.mw

Reversion of series: Looking for "ambiverts"

by: rcorless 710 Maple
series
September 23 2021
1  5

Dear all,

Reversion of series---computing a series for the functional inverse of a function---has been in Maple since forever, but many people are not aware of how easy it is.  Here's an example, where we are looking for "self-reverting" series---which I called "ambiverts".  Anyway have fun.

 

https://maple.cloud/app/5974582695821312/Series+Reversion%3A+Looking+for+ambiverts

PS There looks to be some "code rot" in the branch point series for Lambert W in Maple, which we encounter in that worksheet.  Or, I may simply have not coded it very well in the first place (yeah, that was mine, once upon a time).  Checking now.  But there is a workaround (albeit an ugly one) shown in that worksheet.

 

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