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MaplePrimes Posts are for sharing your experiences, techniques and opinions about Maple, MapleSim and related products, as well as general interests in math and computing.

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  • Hi, 

    In a recent post  (Monte Carlo Integration) Radaar shared its work about the numerical integration, with the Monte Carlo method, of a function defined in polar coordinates.
    Radaar used a raw strategy based on a sampling in cartesian coordinates plus an ad hoc transformation.
    Radaar obtained reasonably good results, but I posted a comment to show how Monte Carlo summation in polar coordinates can be done in a much simpler way. Behind this is the choice of a "good" sampling distribution which makes the integration problem as simple as Monte Carlo integration over a 2D rectangle with sides parallel to the co-ordinate axis.

    This comment I sent pushed me to share the present work on Monte Carlo integration over simple polygons ("simple" means that two sides do not intersect).
    Here again one can use raw Monte Carlo integration on the rectangle this polygon is inscribed in. But as in Radaar's post, a specific sampling distribution can be used that makes the summation method more elegant.

    This work relies on three main ingredients:

    1. The Dirichlet distribution, whose one form enables sampling the 2D simplex in a uniform way.
    2. The construction of a 1-to-1 mapping from this simplex into any non degenerated triangle (a mapping whose jacobian is a constant equal to the ratio of the areas of the two triangles).
    3. A tesselation into triangles of the polygon to integrate over.


    This work has been carried out in Maple 2015, which required the development of a module to do the tesselation. Maybe more recent Maple's versions contain internal procedures to do that.
     

    Monte_Carlo_Integration.mw

     

    This is an animation of the spread of the COVID-19 over the U.S. in the first 150 days.  It was created in Maple 2020, making extensive use of DataFrames. 

     

    https://www.youtube.com/watch?v=XHXeJKTeoRw

     

    The animation of 150 Day history includes COVID-19 data published by the NY Times and geographic data assembled from other sources. Each cylinder represents a county or in two special cases New York City and Kansas City. The cross-sectional area of each cylinder is the area in square miles of the corresponding county. The height of each cylinder is on a logarithmic scale (in particular it is 100*log base 2 of the case count for the county. The argument of the logarithm function is the number of cases per county divided by the are in square miles-so an areal density.  Using a logarithmic scale facilitates showing super high density areas (e.g., NYC) along with lower density areas.  The heights are scaled by a prefactor of 100 for visualization.

    Hi. My name is Eugenio and I’m a Professor at the Departamento de Didáctica de las Ciencias Experimentales, Sociales y Matemáticas at the Facultad de Educación of the Universidad Complutense de Madrid (UCM) and a member of the Instituto de Matemática Interdisciplinar (IMI) of the UCM.

    I have a 14-year-old son. In the beginning of the pandemic, a confinement was ordered in Spain. It is not easy to make a kid understand that we shouldn't meet our friends and relatives for some time and that we should all stay at home in the first stage. So, I developed a simplified explanation of virus propagation for kids, firstly in Scratch and later in Maple, the latter using an implementation of turtle geometry that we developed long ago and has a much better graphic resolution (E. Roanes-Lozano and E. Roanes-Macías. An Implementation of “Turtle Graphics” in Maple V. MapleTech. Special Issue, 1994, 82-85). A video (in Spanish) of the Scratch version is available from the Instituto de Matemática Interdisciplinar (IMI) web page: https://www.ucm.es/imi/other-activities

    Introduction

    Surely you are uncomfortable being locked up at home, so I will try to justify that, although we are all looking forward going out, it is good not to meet your friends and family with whom you do not live.

    I firstly need to mention a fractal is. A fractal is a geometric object whose structure is repeated at any scale. An example in nature is Romanesco broccoli, that you perhaps have eaten (you can search for images on the Internet). You can find a simple fractal in the following image (drawn with Maple):

    Notice that each branch is divided into two branches, always forming the same angle and decreasing in size in the same proportion.

    We can say that the tree in the previous image is of “depth 7” because there are 7 levels of branches.

    It is quite easy to create this kind of drawing with the so called “turtle geometry” (with a recursive procedure, that is, a procedure that calls itself). Perhaps you have used Scratch programming language at school (or Logo, if you are older), which graphics are based in turtle geometry.

    All drawings along these pages have been created with Maple. We can easily reform the code that generated the previous tree so that it has three, four, five,… branches at each level, instead of two.

    But let’s begin with a tale that explains in a much simplified way how the spread of a disease works.

    - o O o -

    Let's suppose that a cat returns sick to Catland suffering from a very contagious disease and he meets his friends and family, since he has missed them so much.

    We do not know very well how many cats each sick cat infects in average (before the order to STAY AT HOME arrives, as cats in Catland are very obedient and obey right away). Therefore, we’ll analyze different scenarios:

    1. Each sick cat infects two other cats.
    2. Each sick cat infects three other cats.
    3. Each sick cat infects five other cats

     

    1. Each Sick Cat Infects Two Cats

    In all the figures that follow, the cat initially sick is in the center of the image. The infected cats are represented by a red square.

    · Before everyone gets confined at home, it only takes time for that first sick cat to see his friends, but then confinement is ordered (depth 1)

    As you can see, with the cat meeting his friends and family, we already have 3 sick cats.

    · Before all cats confine themselves at home, the first cat meets his friends, and these in turn have time to meet their friends (depth 2)

    In this case, the number of sick cats is 7.

    · Before every cat is confined at home, there is time for the initially sick cat to meet his friends, for these to meet their friends, and for the latter (friends of the friends of the first sick cat) to meet their friends (depth 3).

    There are already 15 sick cats...

    · Depth 4: 31 sick cats.

    · Depth 5: 63 sick cats.

    Next we’ll see what would happen if each sick cat infected three cats, instead of two.

     

    2. Every Sick Cat Infects Three Cats

    · Now we speed up, as you’ve got the idea.

    The first sick cat has infected three friends or family before confining himself at home. So there are 4 infected cats.

    · If each of the recently infected cats in the previous figure have in turn contact with their friends and family, we move on to the following situation, with 13 sick cats:

    · And if each of these 13 infected cats lives a normal life, the disease spreads even more, and we already have 40!

    · At the next step we have 121 sick cats:

    · And, if they keep seeing friends and family, there will be 364 sick cats (the image reminds of what is called a Sierpinski triangle):

     

    4. Every Sick Cat Infects Five Cats

    · In this case already at depth 2 we already have 31 sick cats.

     

    5. Conclusion

    This is an example of exponential growth. And the higher the number of cats infected by each sick cat, the worse the situation is.

    Therefore, avoiding meeting friends and relatives that do not live with you is hard, but good for stopping the infection. So, it is hard, but I stay at home at the first stage too!

    We have just released an update to Maple, Maple 2020.1.

    Maple 2020.1 includes corrections and improvement to the mathematics engine, export to PDF, MATLAB connectivity, support for Ubuntu 20.04, and more.  We recommend that all Maple 2020 users install these updates.

    This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2020.1 download page, where you can also find more details.

    In particular, please note that this update includes a fix to the SMTLIB problem reported on MaplePrimes. Thanks for the feedback!

    Monte Carlo integration uses random sampling unlike classical techniques like the trapezoidal or Simpson's rule in evaluating the integration numerically.

    restart; ff := proc (rho, phi) return exp(rho*cos(phi))*rho end proc; aa := 0; bb := 1; cc := 0; dd := 2*Pi; alfa := 5; nrun := 15000; sum1 := 0; sum2 := 0; X := Statistics:-RandomVariable(Uniform(0, 1)); SX := Statistics:-Sample(X); for ii to nrun do u1 := SX(1)[1]; u2 := SX(1)[1]; xx1 := aa+(bb-aa)*u1; xx2 := cc+(dd-cc)*u2; xx3 := (bb-aa)*(1-u1); xx4 := (dd-cc)*(1-u2); sum1 := sum1+evalf(ff(xx1, xx2)); sum2 := sum2+evalf(ff(xx1, xx2))+evalf(ff(xx1, xx4))+evalf(ff(xx3, xx2))+evalf(ff(xx3, xx4)) end do; area1 := (bb-aa)*(dd-cc)*sum1/nrun; area2 := (bb-aa)*(dd-cc)*sum2/(4*nrun); area2

    HFloat(3.5442090450428574)

    (1)

    evalf(Int(exp(rho*cos(phi))*rho, rho = 0 .. 1, phi = 0 .. 2*Pi))

    3.550999378

    (2)

    NULL


     

    Download MONTE_CARLO_INTEGRATION1.mw

     

     

    The purpose of this document is:

    a) to correct the physics that was used in the document "Minimal Road Radius for Highway Superelevation" recently submitted to the Maple Applications Center;

    b) to confirm the values found in the manual for the American Association of State Highway and Transportation Officials (AASHTO) that engineers use to design and build these banked curves are physically sound. 

    c) to highlight the pedagogical value inherent in the Maple language to distinguish between assignment ( := )  and equivalence (  =  );

    d) but most importantly, to demonstrate the pedagogical value Maple has in thinking about solving a problem involving a physical process. Given Maple's symbolic mathematics capabilities, one can implement a top-down approach to the physics and the mathematics, working from the general principle to the specific example. This allows one to avoid the types of errors that occur when translating the problem into a bottom up approach, from specific values of the example to the general principle, an approach that is required by most other computational systems.

    I hope that others are willing to continue to engage in discussions related to the pedagogical value of Maple beyond mathematics.

    I was asked to post this document to both here and the Maple Applications Center

    [Document edited for typos.]

    Minimum_Road_Radius.mw

     

    We’re excited to announce a new version of MapleSim! The MapleSim 2020 family of products lets you build and test models faster than ever, including faster simulations, powerful new tools for machine builders, and expanded modeling capabilities. Improvements include:

    • Faster results, with more efficient models, faster simulations, and more powerful design tools.
    • Powerful new features for machine builders, with new components, improved visualizations, and automation-focused connectivity tools that make it faster than ever to build and test digital twins.
    • Improved modeling capabilities, with an extensive collection of updates to components, libraries, and analysis tools.
    • More realistic machine visualizations with an expanded Kinematic Cam Generation App.
    • New product: MapleSim Insight, giving machine builders powerful, simulation-based debugging and 3-D visualization capabilities that connect directly to common automation platforms.
    • New add-on library:  MapleSim Ropes and Pulleys Library for the easy creation of winch and pulley systems as part of your machine development. 

    See What’s New in MapleSim 2020 for more information about these and other improvements!
     

    Maple's pdsolve() is quite capable of solving the PDE that describes the motion of a single-span Euler beam.  As far as I have been able to ascertain, there is no obvious way of applying pdsolve() to solve multi-span beams.  The worksheet attached to this post provides tools for solving multi-span Euler beams.  Shown below are a few demos.  The worksheet contains more demos.

     

    A module for solving the Euler beam with the method of lines

    beamsolve

     

    The beamsolve proc solves a (possibly multi-span) Euler beam equation:``

    "rho ((∂)^2u)/((∂)^( )t^2)+ ((∂)^2)/((∂)^( )x^2)(EI ((∂)^(2)u)/((∂)^( )x^(2)))=f"

    subject to initial and boundary conditions.  The solution u = u(x, t) is the

    transverse deflection of the beam at point x at time t, subject to the load
    density (i.e., load per unit length) given by f = f(x, t). The coefficient rho 

    is the beam's mass density (mass per unit length), E is the Young's modulus of

    the beam's material, and I is the beam's cross-sectional moment of inertia

    about the neutral axis.  The figure below illustrates a 3-span beam (drawn in green)
    supported on four supports, and loaded by a variable density load (drawn in gray)
    which may vary in time.  The objective is to determine the deformed shape of the
    beam as a function of time.


    The number of spans, their lengths, and the nature of the supports may be specified

    as arguments to beamsolve.

     

    In this worksheet we assume that rho, E, I are constants for simplicity. Since only
    the product of the coefficients E and I enters the calculations, we lump the two

    together into a single variable which we indicate with the two-letter symbol EI.

    Commonly, EI is referred to as the beam's rigidity.

     

    The PDE needs to be supplied with boundary conditions, two at each end, each

    condition prescribing a value (possibly time-dependent) for one of u, u__x, u__xx, u__xxx 
    (that's 36 possible combinations!) where I have used subscripts to indicate

    derivatives.  Thus, for a single-span beam of length L, the following is an admissible

    set of boundary conditions:
    u(0, t) = 0, u__xx(0, t) = 0, u__xx(L, t) = 0, u__xxx(t) = sin*t.   (Oops, coorection, that last
    condition was meant to be uxxx(L,t) = sin t.)

    Additionally, the PDE needs to be supplied with initial conditions which express

    the initial displacement and the initial velocity:
    "u(x,0)=phi(x),   `u__t`(x,0)=psi(x)."

     

    The PDE is solved through the Method of Lines.  Thus, each span is subdivided into

    subintervals and the PDE's spatial derivatives are approximated through finite differences.

    The time derivatives, however, are not discretized.  This reduces the PDE into a set of

    ODEs which are solved with Maple's dsolve().  

     

    Calling sequence:

            beamsolve(L, n, options)

     

    Parameters:

            L:  List of span lengths, in order from left to right, as in [L__1, L__2 .. (), `L__ν`].

            n The number of subintervals in the shortest span (for the finite difference approximation)

     

    Notes:

    • 

    It is assumed that the spans are laid back-to-back along the x axis, with the left end
    of the overall beam at x = 0.

    • 

    The interior supports, that is, those supports where any two spans meet, are assumed
    to be of the so-called simple type.  A simple support is immobile and it doesn't exert
    a bending moment on the beam.  Supports at the far left and far right of the beam can
    be of general type; see the BC_left and BC_right options below.

    • 

    If the beam consists of a single span, then the argument L may be entered as a number
    rather than as a list. That is, L__1 is equivalent to [L__1].

     

    Options:

            All options are of the form option_name=value, and have built-in default values.

            Only options that are other than the defaults need be specified.

     

            rho: the beam's (constant) mass density per unit length (default: rho = 1)

            EI: the beam's (constant) rigidity (default: EI = 1)

            T: solve the PDE over the time interval 0 < t and t < T (default: T = 1)

            F: an expression in x and t that describes the applied force f(x, t)  (default: F = 0)
            IC: the list [u(x, 0), u__t(x, 0)]of the initial displacement and velocity,  as
                    expressions in x (default: IC = [0,0])

            BC_left: a list consisting of a pair of boundary conditions at the left end of
                    the overall (possibly multi-span beam.  These can be any two of
                    u = alpha(t), u_x = beta(t), u_xx = gamma(t), u_xxx = delta(t). The right-hand sides of these equations

                    can be any expression in t.  The left-hand sides should be entered literally as indicated.

                    If a right-hand side is omitted, it is taken to be zero.   (default: BC_left = [u, u_xx] which

                    corresponds to a simple support).

            BC_right: like BC_left, but for the right end of the overall beam (default: BC_right = "[u,u_xx])"

     

    The returned module:

            A call to beamsolve returns a module which presents three methods.  The methods are:

     

            plot (t, refine=val, options)

                    plots the solution u(x, t) at time t.  If the discretization in the x direction

                    is too coarse and the graph looks non-smooth, the refine option

                    (default: refine=1) may be specified to smooth out the graph by introducing

                    val number of intermediate points within each discretized subinterval.

                    All other options are assumed to be plot options and are passed to plots:-display.

     

            plot3d (m=val, options)

                    plots the surface u(x, t).  The optional m = val specification requests

                    a grid consisting of val subintervals in the time direction (default: "m=25)"

                    Note that this grid is for plotting purposes only; the solution is computed

                    as a continuous (not discrete) function of time. All other options are assumed

                    to be plot3d options and are passed to plots:-display.

     

            animate (frames=val, refine=val, options)

                    produces an animation of the beam's motion.  The frames option (default = 50)

                    specifies the number of animation frames.  The refine option is passed to plot
                    (see the description above. All other options are assumed to be plot options and
                    are passed to plots:-display.

    Note:

            In specifying the boundary conditions, the following reminder can be helpful.  If the beam

            is considered to be horizontal, then u is the vertical displacement, `u__x ` is the slope,  EI*u__xx

            is the bending moment, and EI*u__xxx is the transverse shear force.

     

    A single-span simply-supported beam with initial velocity

     

    The function u(x, t) = sin(Pi*x)*sin(Pi^2*t) is an exact solution of a simply supported beam with

    "u(x,0)=0,   `u__t`(x,0)=Pi^(2)sin(Pi x)."  The solution is periodic in time with period 2/Pi.

    sol := beamsolve(1, 25, 'T'=2/Pi, 'IC'=[0, Pi^2*sin(Pi*x)]):
    sol:-animate(size=[600,250]);

    The initial condition u(x, 0) = 0, u__t(x, 0) = 1  does not lead to a separable form, and

    therefore the motion is more complex.

    sol := beamsolve(1, 25, 'T'=2/Pi, 'IC'=[0, 1]):
    sol:-animate(frames=200, size=[600,250]);


     

    A single-span cantilever beam

     

    A cantilever beam with initial condition "u(x,0)=g(x),  `u__t`(x,0)=0," where g(x) is the
    first eigenmode of its free vibration (calculated in another spreadsheet).  The motion is
    periodic in time, with period "1.787018777."

    g := 0.5*cos(1.875104069*x) - 0.5*cosh(1.875104069*x) - 0.3670477570*sin(1.875104069*x) + 0.3670477570*sinh(1.875104069*x):
    sol := beamsolve(1, 25, 'T'=1.787018777, 'BC_left'=[u,u_x], 'BC_right'=[u_xx,u_xxx], 'IC'=[g, 0]):
    sol:-animate(size=[600,250]);

    If the initial condition is not an eigenmode, then the solution is rather chaotic.

    sol := beamsolve(1, 25, 'T'=3.57, 'BC_left'=[u,u_x], 'BC_right'=[u_xx,u_xxx], 'IC'=[-x^2, 0]):
    sol:-animate(size=[600,250], frames=100);


     

    A single-span cantilever beam with a weight hanging from its free end

     

    sol := beamsolve(1, 25, 'T'=3.57, 'BC_left'=[u,u_x], 'BC_right'=[u_xx,u_xxx=1]):
    sol:-animate(size=[600,250], frames=100);


     

    A single-span cantilever beam with oscillating support

     

    sol := beamsolve(1, 25, 'T'=Pi, 'BC_left'=[u=0.1*sin(10*t),u_x], 'BC_right'=[u_xx,u_xxx]):
    sol:-animate(size=[600,250], frames=100);


     

    A dual-span simply-supported beam with moving load

     

    Load moves across a dual-span beam.

    The beam continues oscillating after the load leaves.

    d := 0.4:  T := 4:  nframes := 100:
    myload := - max(0, -6*(x - t)*(d + x - t)/d^3):
    sol := beamsolve([1,1], 20, 'T'=T, 'F'=myload):
    sol:-animate(frames=nframes):
    plots:-animate(plot, [2e-3*myload(x,t), x=0..2, thickness=1, filled=[color="Green"]], t=0..T, frames=nframes):
    plots:-display([%%,%], size=[600,250]);


     

    A triple-span simply-supported beam with moving load

     

    Load moves across a triple-span beam.

    The beam continues oscillating after the load leaves.

    d := 0.4:  T := 6: nframes := 100:
    myload := - max(0, -6*(x - t)*(d + x - t)/d^3):
    sol := beamsolve([1,1,1], 20, 'T'=T, 'F'=myload):
    sol:-plot3d(m=50);
    sol:-animate(frames=nframes):
    plots:-animate(plot, [2e-3*myload(x,t), x=0..3, thickness=1, filled=[color="Green"]], t=0..T, frames=nframes):
    plots:-display([%%,%], size=[600,250]);

    z3d;


     

    A triple-span beam, moving load falling off the cantilever end

     

    In this demo the load move across a multi-span beam with a cantilever section at the right.

    As it skips past the cantilever end, the beam snaps back violently.

    d := 0.4:  T := 8: nframes := 200:
    myload := - max(0, -6*(x - t/2)*(d + x - t/2)/d^3):
    sol := beamsolve([1,1,1/2], 10, 'T'=T, 'F'=myload, BC_right=[u_xx, u_xxx]):
    sol:-animate(frames=nframes):
    plots:-animate(plot, [1e-2*myload(x,t), x=0..3, thickness=1, filled=[color="Green"]], t=0..T, frames=nframes):
    plots:-display([%%,%], size=[600,250]);


     


    Download worksheet: euler-beam-with-method-of-lines.mw

     

    This is my attempt to produce a subplot within a larger plot for magnifying data in a small region, and putting that subplot into the white space of the figure.
    Based on the questions: How to insert a plot into another plot? and Inset figure in Maple, I wrote a couple of procedures that create sub-plots and allow the user to place the subplot window as he/she chooses. This avoids the graininess issues mentioned by acer in the second link (and experienced by me).

    So far, I only have this completed for point plots, but using acer's method of piecewise functions posted in the plotin2b.mw of the second article, with the subplot function being defined only if it satisfies your conditions, would allow the subplot generating procedure to be generalized easily enough. But the data I'm working with all point plots, so that's the example here.

    The basic idea  is to use one procedure to create boxes, make tickmarks on the expanded region, and make tickmark labels, combine all of those into one graph. Then create scaled and shifted versions of the data series, then make graphs of those. Lastly, combine them all into one picture.

    Hope this helps someone who has to do the same.

    Mapleprimes isn't inserting the contents, but here is the download of the file: SubPlotBoxesandVectorDataSeries.mw

     

    Hello all ,

    It's been years and years that I have been using Maple (since Maple V.2 in 1992 if my memory is still

    correct. Now we are at Maple 2020 and the brilliant scientific library GMP is still in Maple at version

    5.1.1.

    kernelopts(gmpversion);
                                "5.1.1"

    The relatively new GMP is 6.2.0 : https://gmplib.org/

    I wonder if in a not too far version of Maple we could have GMP updated. It is really an impressive mathematical library alongside with GSL.

    Voilà !

     

    Kind regards to all,

     

    Jean-Michel

     

    Here is a little cute demo that shows how a cube may be paritioned into three congruent pyramids.  This was inspired by a Mathematica demo that I found in the web but I think this one's better :-)

    A Cube as a union of three right pyramids

    Here is an animated demo of the well-known fact that a cube may be partitioned

    into three congruent right pyramids.

     

    2020-05-21

    restart;

    with(plots):

    with(plottools):

    A proc to plot a general polyhedron.
    V = [[x, y, z], [x, y, z], () .. (), [x, y, z]]                list of vertices
    F = [[n__1, n__2, () .. ()], [n__1, n__2, () .. ()], () .. (), [n__1, n__2, () .. ()]]  list of faces

    An entry "[`n__1`,`n__2`. ...]" in Fdescribes a face made of the vertices "V[`n__1`], V[`n__2`], ...," etc.

    polyhedron := proc(V::list, F::list)
      seq(plottools:-polygon([seq( V[F[i][j]], j=1..nops(F[i]))]), i=1..nops(F));
      plots:-display(%);
    end proc:

    Define the vertices and faces of a pyramid:

    v := [[0,0,0],[1,0,0],[1,1,0],[0,1,0],[0,0,1]];
    f := [ [1,2,3,4], [5,2,3], [5,3,4], [1,5,4], [1,2,5] ];

    [[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 0, 1]]

    [[1, 2, 3, 4], [5, 2, 3], [5, 3, 4], [1, 5, 4], [1, 2, 5]]

    Build three such pyramids:

    P1 := polyhedron(v, f):
    P2 := reflect(P1, [[1,0,0],[1,1,0],[1,0,1]]):
    P3 := reflect(P1, [[0,1,0],[1,1,0],[0,1,1]]):

    This is what we have so far:

    display(P1,P2,P3, scaling=constrained);

    Define an animation frame.  The parameter t goes from 0 to 1.

    Any extra options are assumed to be plot3d options and are

    passed to plots:-display.

    frame := proc(t)
      plots:-display(
        P1,

        rotate(P2, Pi/2*t, [[1,1,0],[1,0,0]]),
        rotate(P3, Pi/2*t, [[0,1,0],[1,1,0]]),
        color=["Red", "Green", "Blue"], _rest);
    end proc:

    Animate:

    display(frame(0) $40, seq(frame(t), t=0..1, 0.01), frame(1) $40,
      insequence, scaling=constrained, axes=none,
      orientation=[45,0,120], viewpoint=circleleft);

     

    Download square-partitioned-into-pyramids.mw

     

     

    The equations of motion in curvilinear coordinates, tensor notation and Coriolis force

    ``

     

    The formulation of the equations of motion of a particle is simple in Cartesian coordinates using vector notation. However, depending on the problem, for example when describing the motion of a particle as seen from a non-inertial system of references (e.g. a rotating planet, like earth), there is advantage in using curvilinear coordinates and also tensor notation. When the particle's movement is observed from such a rotating referential, we also see "acceleration" that is not due to any force but to the fact that the referential itself is accelerated. The material below discusses and formulates these topics, and derives the expression for the Coriolis and centripetal force in cylindrical coordinates as seen from a rotating system of references.

     

    The computation below is reproducible in Maple 2020 using the Maplesoft Physics Updates v.681 or newer.

     

    Vector notation

     

    Generally speaking the equations of motion of a particle are easy to formulate: the position vector is a function of time, the velocity is its first derivative and the acceleration is its second derivative. For instance, in Cartesian coordinates

    with(Physics); with(Vectors)

    r_(t) = x(t)*_i+y(t)*_j+z(t)*_k

    r_(t) = x(t)*_i+y(t)*_j+z(t)*_k

    (1)

    diff(r_(t) = x(t)*_i+y(t)*_j+z(t)*_k, t)

    diff(r_(t), t) = (diff(x(t), t))*_i+(diff(y(t), t))*_j+(diff(z(t), t))*_k

    (2)

    diff(diff(r_(t), t) = (diff(x(t), t))*_i+(diff(y(t), t))*_j+(diff(z(t), t))*_k, t)

    diff(diff(r_(t), t), t) = (diff(diff(x(t), t), t))*_i+(diff(diff(y(t), t), t))*_j+(diff(diff(z(t), t), t))*_k

    (3)

    Newton's 2nd law, that in an inertial system of references when there is force there is acceleration and viceversa, is then given by

    F_(t) = m*lhs(diff(diff(r_(t), t), t) = (diff(diff(x(t), t), t))*_i+(diff(diff(y(t), t), t))*_j+(diff(diff(z(t), t), t))*_k)

    F_(t) = m*(diff(diff(r_(t), t), t))

    (4)

    where `#mover(mi("F"),mo("&rarr;"))`(t) = F__x(t)*`#mover(mi("i"),mo("&and;"))`+F__y(t)*`#mover(mi("j"),mo("&and;"))`+F__z(t)*`#mover(mi("k"),mo("&and;"))` represents the total force acting on the particle. This vectorial equation represents three second order differential equations which, for given initial conditions, can be integrated to arrive at a closed form expression for `#mover(mi("r"),mo("&rarr;"))`(t) as a function of t.

     

    Tensor notation

     

    In Cartesian coordinates, the tensorial form of the equations (4) is also straightforward. In a flat spacetime - Galilean system of references - the three space coordinates x, y, z form a 3D tensor, and so does its first derivate and the second one. Set the spacetime to be 3-dimensional and Euclidean and use lowercaselatin indices for the corresponding tensors

    Setup(coordinates = cartesian, metric = Euclidean, dimension = 3, spacetimeindices = lowercaselatin)

    `The dimension and signature of the tensor space are set to `[3, `+ + +`]

     

    `Systems of spacetime coordinates are:`*{X = (x, y, z)}

     

    _______________________________________________________

     

    `The Euclidean metric in coordinates `*[x, y, z]

     

    _______________________________________________________

     

    Physics:-g_[mu, nu] = Matrix(%id = 18446744078329083054)

     

    _______________________________________________________

    (5)

    The position, velocity and acceleration vectors are expressed in tensor notation as done in (1), (2) and (3)

    X[j](t)

    (X)[j](t)

    (6)

    diff((X)[j](t), t)

    Physics:-Vectors:-diff((Physics:-SpaceTimeVector[j](X))(t), t)

    (7)

    diff(Physics[Vectors]:-diff((Physics[SpaceTimeVector][j](X))(t), t), t)

    Physics:-Vectors:-diff(Physics:-Vectors:-diff((Physics:-SpaceTimeVector[j](X))(t), t), t)

    (8)

    Setting a tensor F__j(t) to represent the three Cartesian components of the force

    Define(F[j] = [F__x(t), F__y(t), F__z(t)])

    `Defined objects with tensor properties`

     

    {Physics:-Dgamma[a], F[j], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}

    (9)

    Newton's 2nd law (4), now expressed in tensorial notation, is given by

    F[j] = m*Physics[Vectors]:-diff(Physics[Vectors]:-diff((Physics[SpaceTimeVector][j](X))(t), t), t)

    F[j] = m*(diff(diff((Physics:-SpaceTimeVector[j](X))(t), t), t))

    (10)

    The three differential equations behind this tensorial form of (4) are as expected

    TensorArray(F[j] = m*(diff(diff((Physics[SpaceTimeVector][j](X))(t), t), t)), output = setofequations)

    {F__x(t) = m*(diff(diff(x(t), t), t)), F__y(t) = m*(diff(diff(y(t), t), t)), F__z(t) = m*(diff(diff(z(t), t), t))}

    (11)

    Things are straightforward in Cartesian coordinates because the components of the line element `#mover(mi("dr"),mo("&rarr;"))` = dx*`#mover(mi("i"),mo("&and;"))`+dy*`#mover(mi("j"),mo("&and;"))`+dz*`#mover(mi("k"),mo("&and;"))` are exactly the components of the tensor d(X[j])

    TensorArray(d_(X[j]))

    Array(%id = 18446744078354237310)

    (12)

    and so, the form factors (see related Mapleprimes post) are all equal to 1.

     

    In the case of no external forces, `#mover(mi("F"),mo("&rarr;"))`(t) = 0 and 0 = F[j] and the equations of motion, whose solution are the trajectory, can be formulated as the path of minimal length between two points, a geodesic. In the case under consideration, because the spacetime is flat (Galilean) those two points lie on a plane, we are talking about Euclidean geometry, that information is encoded in the metric (the 3x3 identity matrix (5)), and the geodesic is a straight line. The differential equations of this geodesic are thus the equations of motion (11) with  `#mover(mi("F"),mo("&rarr;"))`(t) = 0, and can be computed using Geodesics

     

    Geodesics(t)

    [diff(diff(z(t), t), t) = 0, diff(diff(y(t), t), t) = 0, diff(diff(x(t), t), t) = 0]

    (13)

     

    Curvilinear coordinates

     

    Vector notation

     

    The form of these equations in the case of curvilinear coordinates, for example in cylindrical or spherical variables, is obtained performing a change of coordinates.

    tr := `~`[`=`]([X], ChangeCoordinates([X], cylindrical))

    [x = rho*cos(phi), y = rho*sin(phi), z = z]

    (14)

    This change keeps the z axis unchanged, so the corresponding unit vector `#mover(mi("k"),mo("&and;"))` remains unchanged.

    Changing the basis and coordinates used to represent the position vector `#mover(mi("r"),mo("&rarr;"))`(t) = x(t)*`#mover(mi("i"),mo("&and;"))`+y(t)*`#mover(mi("j"),mo("&and;"))`+z(t)*`#mover(mi("k"),mo("&and;"))`, it becomes

    r_(t) = ChangeBasis(rhs(r_(t) = x(t)*_i+y(t)*_j+z(t)*_k), cylindrical, alsocomponents)

    r_(t) = z(t)*_k+rho(t)*_rho(t)

    (15)

    where since in (1) the coordinates (x, y, z) depend on t, in (15), not just rho(t) and z(t) but also the unit vector `#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`(t)depends on t. The velocity is computed as usual, differentiating

    diff(r_(t) = z(t)*_k+rho(t)*_rho(t), t)

    diff(r_(t), t) = (diff(z(t), t))*_k+(diff(rho(t), t))*_rho(t)+rho(t)*(diff(phi(t), t))*_phi(t)

    (16)

    The second term is due to the dependency of `#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))` on the coordinate phi together with the chain rule diff(`#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`(t), t) = (diff(`#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`, phi))*(diff(phi(t), t)) and (diff(`#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`, phi))*(diff(phi(t), t)) = `#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))`(t)*(diff(phi(t), t)). The dependency of curvilinear unit vectors on the coordinates is automatically taken into account when differentiating due to the Setup setting geometricdifferentiation = true.

     

    For the acceleration,

    diff(diff(r_(t), t) = (diff(z(t), t))*_k+(diff(rho(t), t))*_rho(t)+rho(t)*(diff(phi(t), t))*_phi(t), t)

    diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k

    (17)

    where the term involving (diff(phi(t), t))^2 comes from differentiating `#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))`(t) in (16) taking into account the dependency of `#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))` on the coordinate "phi." This result can also be obtained by directly changing variables in the acceleration diff(`#mover(mi("r"),mo("&rarr;"))`(t), t, t), in equation (3)

    lhs(diff(diff(r_(t), t), t) = (diff(diff(x(t), t), t))*_i+(diff(diff(y(t), t), t))*_j+(diff(diff(z(t), t), t))*_k) = ChangeBasis(rhs(diff(diff(r_(t), t), t) = (diff(diff(x(t), t), t))*_i+(diff(diff(y(t), t), t))*_j+(diff(diff(z(t), t), t))*_k), cylindrical, alsocomponents)

    diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k

    (18)

     

    Newton's 2nd law becomes

    F_(t) = m*rhs(diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k)

    F_(t) = m*(_rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k)

    (19)

    In the absence of external forces, equating to 0 the vector components (coefficients of the unit vectors) of the acceleration diff(`#mover(mi("r"),mo("&rarr;"))`(t), t, t)we get the system of differential equations in cylindrical coordinates whose solution is the trajectory of the particle expressed in the ("rho(t),phi(t),z(t))"

    `~`[`=`]({coeffs(rhs(diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k), [`#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`(t), `#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))`(t), `#mover(mi("k"),mo("&and;"))`])}, 0)

    {2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)) = 0, diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2 = 0, diff(diff(z(t), t), t) = 0}

    (20)

    solve({2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)) = 0, diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2 = 0, diff(diff(z(t), t), t) = 0}, {diff(phi(t), t, t), diff(rho(t), t, t), diff(z(t), t, t)})

    {diff(diff(phi(t), t), t) = -2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t), diff(diff(rho(t), t), t) = rho(t)*(diff(phi(t), t))^2, diff(diff(z(t), t), t) = 0}

    (21)

    In this formulation (21) with `#mover(mi("F"),mo("&rarr;"))`(t) = 0, although diff(z(t), t, t) = 0, no acceleration in the `#mover(mi("k"),mo("&and;"))` direction, is naturally expected, the same cannot be said about the other two equations for diff(phi(t), t, t) and diff(rho(t), t, t). Those two equations are discussed below under Coriolis and Centripetal forces. The key observation at this point, however, is that the right-hand sides of both unexpected equations involve diff(phi(t), t), rotation around the z axis.

     

    Tensor notation

     

    The same equations (19) and (21) result when using tensor notation. For that purpose, one can transform the position, velocity and acceleration tensors (6), (7), (8), but since they are expressed as functions of a parameter (the time), it is simpler to transform only the underlying metric using TransformCoordinates. That has the advantage that all the geometrical subtleties of curvilinear coordinates, like scale factors and dependency of unit vectors on curvilinear coordinates, get automatically, very succinctly, encoded in the metric:

    TransformCoordinates(tr, g_[j, k], [rho, phi, z], setmetric)

    _______________________________________________________

     

    `Coordinates: `[rho, phi, z]*`. Signature: `(`+ + +`)

     

    _______________________________________________________

     

    Physics:-g_[a, b] = Matrix(%id = 18446744078263848958)

     

    _______________________________________________________

    (22)

    The computation of geodesics assumes that the coordinates (rho, phi, z) depend on a parameter. That parameter is passed as the first argument to Geodesics

    Geodesics(t)

    [diff(diff(rho(t), t), t) = rho(t)*(diff(phi(t), t))^2, diff(diff(phi(t), t), t) = -2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t), diff(diff(z(t), t), t) = 0]

    (23)

    These equations of motion (23) are the same as the equations (21) computed using standard vector notation, differentiating and taking into account the dependency of curvilinear unit vectors on the curvilinear coordinates in (16) and (17).  One of the interesting features of computing with tensors is, as said, that all those geometrical algebraic subtleties of curvilinear coordinates are automatically encoded in the metric (22).

     

    To understand how are the geodesic equations computed in one go in (23), one can perform the calculation in steps:

    1. 

    Make rho be a function of t directly in the metric

    2. 

    Compute - not the final form of the equations (23) - but the intermediate form expressing the geodesic equation using tensor notation, in terms of Christoffel symbols

    3. 

    Compute the components of that tensorial equation for the geodesic (using TensorArray)

     

    For step 1, we have

    subs(rho = rho(t), g_[])

    Physics:-g_[a, b] = Matrix(%id = 18446744078354237910)

    (24)

    Set this metric where `&equiv;`(rho, rho(t))

    "Setup(?):"

    _______________________________________________________

     

    `Coordinates: `[rho, phi, z]*`. Signature: `(`+ + +`)

     

    _______________________________________________________

     

    Physics:-g_[a, b] = Matrix(%id = 18446744078342604430)

     

    _______________________________________________________

    (25)

    Step 2, the geodesic equations in tensor notation with the coordinates depending on the time t are computed passing the optional argument tensornotation

    Geodesics(t, tensornotation)

    diff(diff((Physics:-SpaceTimeVector[`~a`](X))(t), t), t)+Physics:-Christoffel[`~a`, b, c]*(diff((Physics:-SpaceTimeVector[`~b`](X))(t), t))*(diff((Physics:-SpaceTimeVector[`~c`](X))(t), t)) = 0

    (26)

    Step 3: compute the components of this tensorial equation

    TensorArray(diff(diff((Physics[SpaceTimeVector][`~a`](X))(t), t), t)+Physics[Christoffel][`~a`, b, c]*(diff((Physics[SpaceTimeVector][`~b`](X))(t), t))*(diff((Physics[SpaceTimeVector][`~c`](X))(t), t)) = 0, output = listofequations)

    [diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2 = 0, diff(diff(phi(t), t), t)+2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t) = 0, diff(diff(z(t), t), t) = 0]

    (27)

    These are the same equations (23).

     

    Having the tensorial equation (26) is also useful to formulate the equations of motion in tensorial form in the presence of force. For that purpose, redefine the contravariant tensor F^j to represent the force in the cylindrical basis

    Define(F[`~j`] = [`F__&rho;`(t), `F__&phi;`(t), F__z(t)])

    `Defined objects with tensor properties`

     

    {Physics:-D_[a], Physics:-Dgamma[a], F[j], Physics:-Psigma[a], Physics:-Ricci[a, b], Physics:-Riemann[a, b, c, d], Physics:-Weyl[a, b, c, d], Physics:-d_[a], Physics:-g_[a, b], Physics:-Christoffel[a, b, c], Physics:-Einstein[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}

    (28)

     

    Newton's 2nd law (19)

    F_(t) = m*(_rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k)

    F_(t) = m*(_rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k)

    (29)

    now using tensorial notation, becomes

    F[`~a`] = m*lhs(diff(diff((Physics[SpaceTimeVector][`~a`](X))(t), t), t)+Physics[Christoffel][`~a`, b, c]*(diff((Physics[SpaceTimeVector][`~b`](X))(t), t))*(diff((Physics[SpaceTimeVector][`~c`](X))(t), t)) = 0)

    F[`~a`] = m*(diff(diff((Physics:-SpaceTimeVector[`~a`](X))(t), t), t)+Physics:-Christoffel[`~a`, b, c]*(diff((Physics:-SpaceTimeVector[`~b`](X))(t), t))*(diff((Physics:-SpaceTimeVector[`~c`](X))(t), t)))

    (30)

    TensorArray(F[`~a`] = m*(diff(diff((Physics[SpaceTimeVector][`~a`](X))(t), t), t)+Physics[Christoffel][`~a`, b, c]*(diff((Physics[SpaceTimeVector][`~b`](X))(t), t))*(diff((Physics[SpaceTimeVector][`~c`](X))(t), t))))

    Array(%id = 18446744078329063774)

    (31)

    where we recall (see related Mapleprimes post) that to obtain the vector components entering `#mover(mi("F"),mo("&rarr;"))`(t) from these tensor components F[`~a`]we need to multiply the latter by the scale factors (1, rho(t), 1), the component of `#mover(mi("F"),mo("&rarr;"))`(t) in the direction of `#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))` is given by rho(t)*m*(diff(phi(t), t, t)+2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t)).

     

    Coriolis force and centripetal force

     

    After changing variables the position vector of the particle got expressed in (15) as

     

    `#mover(mi("r"),mo("&rarr;"))`(t) = z(t)*`#mover(mi("k"),mo("&and;"))`+`#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`(t)*rho(t)

     

    A distinction needs to be made here, according to whether the unit vector `#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))` depends or not on the time t, the former being the general case. When `#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))` is a constant, the value of the coordinate phi - the angle between `#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))` and the x axis - does not change, there is no rotation around the z axis. On the other hand, when `&equiv;`(`#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`, `#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`(t)), the orientation of `#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))` and so the coordinate phi changes with time, so either the force `#mover(mi("F"),mo("&rarr;"))`(t)acting on the particle has a component in the `#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))` direction that produces rotation around the z axis, or the system of references - itself - is rotating around the z axis.

     

    Likewise, the expression (15)  can represent the position vector measured in the original Galilean (inertial) system of references, where a force `#mover(mi("F"),mo("&rarr;"))`(t)is producing rotation around the z axis, or it can represent the position of the particle measured in a rotating, non-inertial system references. Hence the transformation (14) can also be interpreted in two different ways, as representing a choice of different functions (generalized coordinates) to represent the position of the particle in the original inertial system of references, or it can represent a transformation from an inertial to another rotating, non-inertial, system of references.

     

    This equivalence between the trajectory of a particle subject to an external force, as observed in an inertial system of references, and the same trajectory observed from a non-inertial accelerated system of references where there is no external force, actually at the root of the formulation of general relativity, is also well known in classical mechanics. The (apparent) forces observed in the rotating non-inertial system of references, due only to its acceleration, are called Coriolis and centripetal forces.

     

    To see that the equations

     

    diff(rho(t), t, t) = (diff(phi(t), t))^2*rho(t), diff(phi(t), t, t) = -2*(diff(phi(t), t))*(diff(rho(t), t))/rho(t)

     

    that appeared in (27) when in the inertial system of references `#mover(mi("F"),mo("&rarr;"))`(t) = m*(diff(`#mover(mi("r"),mo("&rarr;"))`(t), t, t)) and m*(diff(`#mover(mi("r"),mo("&rarr;"))`(t), t, t)) = 0, are related to the Coriolis and centripetal forces in the non-inertial referencial, following [1] introduce a vector `#mover(mi("&omega;",fontstyle = "normal"),mo("&rarr;"))`representing the rotation of that referencial around the z axis (when, in the inertial system of references, the non-inertial system rotates clockwise, in the non-inertial system phi increases in value in the anti-clockwise direction)

    `#mover(mi("&omega;",fontstyle = "normal"),mo("&rarr;"))` = -(diff(phi(t), t))*`#mover(mi("k"),mo("&and;"))`

    omega_ = -(diff(phi(t), t))*_k

    (32)

    According to [1], (39.7), the acceleration diff(`#mover(mi("r"),mo("&rarr;"))`(t), t, t)in the inertial system is expressed in terms of the quantities in the non-inertial rotating system by the sum of the following three vectorial terms.

    First, the components of the acceleration `#mover(mi("a"),mo("&rarr;"))`(t)measured in the non-inertial system are given by the second derivatives of the coordinates (rho(t), phi(t), z(t)) multiplied by the scale factors, which in this case are given by (1, rho(t), 1) (see this post in Mapleprimes)

    `#mover(mi("a"),mo("&rarr;"))`(t) = (diff(rho(t), t, t))*`#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`(t)+rho(t)*(diff(phi(t), t, t))*`#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))`(t)+(diff(z(t), t, t))*`#mover(mi("k"),mo("&and;"))`

    a_(t) = (diff(diff(rho(t), t), t))*_rho(t)+rho(t)*(diff(diff(phi(t), t), t))*_phi(t)+(diff(diff(z(t), t), t))*_k

    (33)

    Second, the Coriolis force divided by the mass, by definition given by

    2*`&x`(diff(r_(t), t) = (diff(z(t), t))*_k+(diff(rho(t), t))*_rho(t)+rho(t)*(diff(phi(t), t))*_phi(t), omega_ = -(diff(phi(t), t))*_k)

    2*Physics:-Vectors:-`&x`(diff(r_(t), t), omega_) = -2*rho(t)*(diff(phi(t), t))^2*_rho(t)+2*(diff(rho(t), t))*(diff(phi(t), t))*_phi(t)

    (34)

    Third the centripetal force divided by the mass, defined by

    `&x`(omega_ = -(diff(phi(t), t))*_k, `&x`(r_(t) = z(t)*_k+rho(t)*_rho(t), omega_ = -(diff(phi(t), t))*_k))

    Physics:-Vectors:-`&x`(omega_, Physics:-Vectors:-`&x`(r_(t), omega_)) = rho(t)*(diff(phi(t), t))^2*_rho(t)

    (35)

    Adding these three terms,

    (a_(t) = (diff(diff(rho(t), t), t))*_rho(t)+rho(t)*(diff(diff(phi(t), t), t))*_phi(t)+(diff(diff(z(t), t), t))*_k)+(2*Physics[Vectors][`&x`](diff(r_(t), t), omega_) = -2*rho(t)*(diff(phi(t), t))^2*_rho(t)+2*(diff(rho(t), t))*(diff(phi(t), t))*_phi(t))+(Physics[Vectors][`&x`](omega_, Physics[Vectors][`&x`](r_(t), omega_)) = rho(t)*(diff(phi(t), t))^2*_rho(t))

    a_(t)+2*Physics:-Vectors:-`&x`(diff(r_(t), t), omega_)+Physics:-Vectors:-`&x`(omega_, Physics:-Vectors:-`&x`(r_(t), omega_)) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k

    (36)

    So that

    diff(`#mover(mi("r"),mo("&rarr;"))`(t), t, t) = lhs(a_(t)+2*Physics[Vectors][`&x`](diff(r_(t), t), omega_)+Physics[Vectors][`&x`](omega_, Physics[Vectors][`&x`](r_(t), omega_)) = _rho(t)*(diff(diff(rho(t), t), t)-(diff(phi(t), t))^2*rho(t))+_phi(t)*(2*(diff(phi(t), t))*(diff(rho(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k)

    diff(diff(r_(t), t), t) = a_(t)+2*Physics:-Vectors:-`&x`(diff(r_(t), t), omega_)+Physics:-Vectors:-`&x`(omega_, Physics:-Vectors:-`&x`(r_(t), omega_))

    (37)

    and where the right-hand side of (36) is, actually, the result computed lines above in (18)

    diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-(diff(phi(t), t))^2*rho(t))+_phi(t)*(2*(diff(phi(t), t))*(diff(rho(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k

    diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k

    (38)

    rhs(a_(t)+2*Physics[Vectors][`&x`](diff(r_(t), t), omega_)+Physics[Vectors][`&x`](omega_, Physics[Vectors][`&x`](r_(t), omega_)) = _rho(t)*(diff(diff(rho(t), t), t)-(diff(phi(t), t))^2*rho(t))+_phi(t)*(2*(diff(phi(t), t))*(diff(rho(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k)-rhs(diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-(diff(phi(t), t))^2*rho(t))+_phi(t)*(2*(diff(phi(t), t))*(diff(rho(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k)

    0

    (39)

    From (37), in the absence of external forces diff(`#mover(mi("r"),mo("&rarr;"))`(t), t, t) = 0 and so the acceleration `#mover(mi("a"),mo("&rarr;"))`(t) measured in the rotating system is given by the sum of the Coriolis and centripetal accelerations

    isolate(rhs(diff(diff(r_(t), t), t) = a_(t)+2*Physics[Vectors][`&x`](diff(r_(t), t), omega_)+Physics[Vectors][`&x`](omega_, Physics[Vectors][`&x`](r_(t), omega_))), `#mover(mi("a"),mo("&rarr;"))`(t))

    a_(t) = -2*Physics:-Vectors:-`&x`(diff(r_(t), t), omega_)-Physics:-Vectors:-`&x`(omega_, Physics:-Vectors:-`&x`(r_(t), omega_))

    (40)

    In other words: in the absence of external forces, the acceleration of a particle observed in a rotating (non-inertial) system of references is not zero.

     

    Expressing this equation (40) in terms of (rho(t), phi(t), z(t)) we get

    subs(a_(t) = (diff(diff(rho(t), t), t))*_rho(t)+rho(t)*(diff(diff(phi(t), t), t))*_phi(t)+(diff(diff(z(t), t), t))*_k, -(2*Physics[Vectors][`&x`](diff(r_(t), t), omega_) = -2*rho(t)*(diff(phi(t), t))^2*_rho(t)+2*(diff(rho(t), t))*(diff(phi(t), t))*_phi(t)), Physics[Vectors][`&x`](omega_, Physics[Vectors][`&x`](r_(t), omega_)) = rho(t)*(diff(phi(t), t))^2*_rho(t), a_(t) = -2*Physics[Vectors][`&x`](diff(r_(t), t), omega_)-Physics[Vectors][`&x`](omega_, Physics[Vectors][`&x`](r_(t), omega_)))

    (diff(diff(rho(t), t), t))*_rho(t)+rho(t)*(diff(diff(phi(t), t), t))*_phi(t)+(diff(diff(z(t), t), t))*_k = rho(t)*(diff(phi(t), t))^2*_rho(t)-2*(diff(rho(t), t))*(diff(phi(t), t))*_phi(t)

    (41)

    resulting in the three equations

    ((diff(diff(rho(t), t), t))*_rho(t)+rho(t)*(diff(diff(phi(t), t), t))*_phi(t)+(diff(diff(z(t), t), t))*_k = rho(t)*(diff(phi(t), t))^2*_rho(t)-2*(diff(rho(t), t))*(diff(phi(t), t))*_phi(t)).`#mover(mi("&rho;",fontstyle = "normal"),mo("&and;"))`(t)

    diff(diff(rho(t), t), t) = rho(t)*(diff(phi(t), t))^2

    (42)

    ((diff(diff(rho(t), t), t))*_rho(t)+rho(t)*(diff(diff(phi(t), t), t))*_phi(t)+(diff(diff(z(t), t), t))*_k = rho(t)*(diff(phi(t), t))^2*_rho(t)-2*(diff(rho(t), t))*(diff(phi(t), t))*_phi(t)).`#mover(mi("&phi;",fontstyle = "normal"),mo("&and;"))`(t)

    rho(t)*(diff(diff(phi(t), t), t)) = -2*(diff(rho(t), t))*(diff(phi(t), t))

    (43)

    ((diff(diff(rho(t), t), t))*_rho(t)+rho(t)*(diff(diff(phi(t), t), t))*_phi(t)+(diff(diff(z(t), t), t))*_k = rho(t)*(diff(phi(t), t))^2*_rho(t)-2*(diff(rho(t), t))*(diff(phi(t), t))*_phi(t)).`#mover(mi("k"),mo("&and;"))`

    diff(diff(z(t), t), t) = 0

    (44)

    which are the equations returned by Geodesics in (23)

    [diff(diff(rho(t), t), t) = rho(t)*(diff(phi(t), t))^2, diff(diff(phi(t), t), t) = -2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t), diff(diff(z(t), t), t) = 0]

    [diff(diff(rho(t), t), t) = rho(t)*(diff(phi(t), t))^2, diff(diff(phi(t), t), t) = -2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t), diff(diff(z(t), t), t) = 0]

    (45)

    ``

    References

    [1] L.D. Landau, E.M. Lifchitz, Mechanics, Course of Theoretical Physics, Volume 1, third edition, Elsevier.


     

    Download The_equations_of_motion_in_curvilinear_coordinates.mw

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

     

    POSSIBILITIES OF USING OF COMPUTER IN MATHEMATICS

    AND OTHER APPLICATIONS IN INCLUSIVE EDUCATION

     

    Alsu Gibadullina, math teacher, math teacher

    Secondary and high school # 57, Kazan, Russia

     

    In recent years Russia actively promoted and implemented the so-called inclusive education (IE). According to the materials of Alliance of human rights organizations “Save the children”: "Inclusive or included education is the term used to describe the process of teaching children with special needs in General (mass) schools. In the base of inclusive education is the ideology that ensures equal treatment for everybody, but creates special conditions for children with special educational needs. Experience shows that any of the rigid educational system some part of the children is eliminated because the system is not ready to meet the individual needs of these children in education. This ratio is 15 % of the total number of children in schools and so retired children become separated and excluded from the overall system. You need to understand that children do not fail but the system excludes children. Inclusive approaches can support such children in learning and achieving success. Inclusive education seeks to develop a methodology that recognizes that all children have different learning needs tries to develop a more flexible approach to teaching. If teaching and learning will become more effective as a result of the changes that introduces Inclusive Education, all children will win (not only children with special needs)."

    There are many examples of schools that have developed their strategy implementation of IE, published many theoretical and practical benefits of inclusion today. All of them have common, immaterial character. There is no description of specific techniques implementing the principles of IO in the teaching of certain disciplines, particularly mathematics. In this paper we propose a methodology that can be successfully used as in “mathematical education for everyone", also for the development of scientific creativity of children at all age levels of the school in any discipline.

    According to the author, one of the most effective methodological tools for education is a computer mathematics (SCM, SSM). Despite the fact that the SCM were created for solving problems of higher mathematics, their ability can successfully implement them in the school system. This opinion is confirmed by more than 10–year-old author's experience of using the package Maple in teaching mathematics. At first it was just learning the system and primitive using its. Then author’s interactive demonstrations, e-books, programs of analytical testing were created by the tools packages. The experience of using the system Maple in teaching inevitably led to the necessity to teach children to work with it. At first worked a club who has studied  the principles of the package’s work, which eventually turned into a research laboratory for the use of computer technology. Later on its basis there was created the scientific student society (SSS) “GEODROMhic" which operates to this day. The main idea and the ultimate goal of SSS – individual research activities on their interests with the creation of the author electronic scientific journals through the use of computer mathematics Maple. The field of application of the package was very diverse: from mathematics to psychology and cultural phenomena. SSS’s activity is very high: they are constantly and successfully participate in intellectual high-level activities (up to international). Obviously, not every SSS’s member reaches high end result. However, even basic experience in scientific analysis, modeling, intelligent  using of the computer teaches the critical thinking skills, evokes interest to new knowledge, allows you to experience their practical value, gives rise to the development of creative abilities. As a result, the research activity improves intellectual culture, self-esteem and confidence, resistance to external negative influence. It should be noted, however, that members of the scientific societies are not largely the so-called "gifted", than ordinary teenagers with different level of intellectual development and mathematical training. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models, which contributes to the development of mathematical thinking.

    From 2007 to 2012. our school (№. 57 of Kazan) was the experimental platform of the Republican study SKM (Maple) and other application software in the system of school mathematical education under the scientific management of Professor Yu. G. Ignatyev of Kazan state University (KF(P)U).

    Practical adaptation of computer mathematics and other useful information technologies to the educational process of secondary schools passed and continues to work in the following areas:

    1. The creation of a demonstration support of different types of the lessons;
    2. The embedding of computing to the structure of practical trainings;
    3. In the form of additional courses - studying of computer applications through which you can conduct a research of the mathematical model and create animated presentation videos, web-pages, auto-run menu;
    4. Students’ working on individual creative projects:
      • construction of computer mathematical models;
      • creating author's programs with elements of scientific researches;
      • students create interactive computer-based tutorials;
      • creation of an electronic library of creative projects;
    5. The participation of students  in the annual competitions and scientific conferences for students;
    6. The accumulation and dissemination of new methodological experience.

    Traditionally, the training system has the structure: explanation of a new →  the solution of tasks→ check, self-test and control → planning of the new unit  with using analysis. However, the main task types: 1) elementary, 2) basic, 3) combined, 4) integrated, 5) custom. With the increasing the level of training a number of basic tasks are growing and some integrated tasks become a class of basic. Thus, the library for basic operations is generated. The decision of the educational task occurs on the way of mastering the theoretical knowledge of mathematical modeling: 1) analysis of conditions (and construction drawing), 2) the search for methods of solution, 3) computation, 4) the researching.

    To introduce computer mathematics in this training system, you can:

    • At demonstrations. For example, with Maplе facilities you can create a step-by-step interactive and animated images, which are essentially the exact graphic interpretation of mathematical models.
    • If we have centralized collective control.
    • If students have individual self-control.
    • In the analysis of the conditions of the problem, for the construction and visualization of its model, the study of this model.
    • In the computations.
    • In practical training of different forms.
    • In individual projects with elements of research.

    In the learning process with the use of computer mathematics in the school a library of themed demonstrations, tasks of different levels and purpose, programs, analytical testing, research projects is generated. With all this especially valuable is that the student is dealing with mathematical signs and mathematical models. Addiction to them processed in the course of working with them it’s unobtrusive, naturally, organically.

    Mathematical modelling (MM) is increasingly becoming an important component of scientific research. Today's powerful engineering tools allow to carry out numerous computer experiments, deep and full enough of exploring the object, without significant cost painless. Thus provided the advantages of theoretical approach, and experiment. The integration of information technology and MM method is effective, safe and economical. This explains its wide distribution and makes unavoidable component of scientific and technical progress.

    Modeling is a natural process for people, it is present in any activity. The introduction with nature by man occurs through constant  modeling of situation, comparing with the basic models and past experience by them. Method for modeling, abstraction as a method of understanding the world is therefore  an effective method of learning. Training activities associated with the creative transformation of the subject. The main feature of educational training activities is the systematic solution of the educational problems. The connection of the principles of developmental education, mathematical modeling, neurophysiological mechanisms of the brain and experience with Maple leads to the following conclusions: the method of mathematical modeling is not only scientific research but also the way of development of thinking in general; computer and mathematical environment (Maple), which is a powerful tool for scientific simulation can be considered as the elementary analogue of the brain. These qualities of computer mathematics led to the idea of using it not only as an effective methodological tool but as a means of nurturing the thinking and development of mental functions of the brain. To study this effect the school psychologist conducted a test, which confirm the observations: the dynamics of intellectual options students  who working with Maple compares favorably with peers. In the process of doing computer math, in particular Maple, are involved in complex different mental functions. It is in the inclusion of all mental functions is the essence of integration of learning, its educational character. And this, in turn, contributes to the solution of moral problems.

    Long-term work with computer mathematics led to the idea to use it as a tool for psychological testing. One of the projects focuses on the psychology and contains authors Maple–tests to identify the degree of development of different mental functions. Interactive mathematical environment  gives a wide variability and creative testing capabilities. Moreover, Maple–test can be used not only as diagnostic but also as educational, and corrective. This technology was tested in one of psycho neurological dispensaries a few years ago.

    Currently, one of the author's students, the so-called "homeworkers", the second year is a young man with a diagnosis, categories F20, who does not speak and does not write independently. It was impossible to get feedback from him and have basic training until then author have begun to apply computer-based tools, including system Maple. Working with the computer tests and mathematical objects helps to see not only the mental and even the simplest thinking movement, but also emotional movement!

    In general, the effectiveness of the implementation in the structure of educational process of secondary school of new organizational forms of the use of computers, based on the application of the symbolic mathematics package Maple, computer modeling and information technology, has many aspects, here are some of them:

    • goals of education and math in particular;
    • additional education;
    • methodical and professional opportunities;
    • theoretical education;
    • modeling;
    • scientific creativity;
    • logical language;
    • spatial thinking, the development of the imagination;
    • programming skills;
    • the specificity of technical translation;
    • differentiation and individualization of educational process;
    • prospective teaching, the continuity of higher and secondary mathematics education;
    • development of creative abilities, research skills;
    • analytical thinking;
    • mathematical thinking;
    • mental diagnosis;
    • mental correction.

           According to the author, the unique experience of the Kazan 57–th school suggests that computer mathematics (Maple) is the most effective also universal tool of new methods of inclusion. In recent decades, there are more children with a specific behavior, with a specific perception, not able to focus, with a poor memory, poor thinking processes. There are children, emotionally and intellectually healthy, or even ahead of their peers in one team together with them. High school should provide all the common core learning standards. It needed a variety of programs and techniques, as well as specialists who use them. Due to its remarkable features, computer mathematics, in particular Maple, can be used or be the basis of the variation of methods of physico-mathematical disciplines of inclusive education.

     

    Research work

    The fractal structure’s researching.

    Modeling of the fractal sets in the Maple program.

    Municipal Budget Educational Establishment “School # 57” of Kirov district of Kazan

        Author:  Ibragimova Evelina

        Scientific advisor:  Alsu Gibadullina - mathematics teacher

        Translator:  Aigul Gibadullina

    In Russian

    ИбрагимоваЭ_Фракталы.docx

    In English

    Fractals_researching.doc

     

         ( Images - in attached files )

    Table of contents:

    Introduction

    I. Studying of principles of fractals construction

    II. Applied meaning of fractals

    III. Researching of computer programs of fractals construction

    Conclusion

    Introduction

    We don’t usually think about main point of things, which we have to do with every day. Environmental systems are many-sided, ever–changing and compli­cated, but they are formed by a little number of rules. Fractals are apt example of this – they are complicated, but based on simple regulations. Self – similarity is the main attribute of them.  Just one fractal element contains genetically information about all system.  This information have a forming role for all system. But some­times self – similarity is partial.

    Hypothesis of the research. Fractals and various elements of the Universe have general principles of structural organization. It is a reason why the theory of frac­tals is instrument for cognition of the world.

    Purpose of the research. Studying  of genetic analogy  between fractals and alive and non-living Universe systems with computer-based mathematical mod­eling in the Mapel’s computer space.

    Problems of the research. 

    1. studying of principles of fractal’s construction;
    2. Detection of  general fractal content of physical, biological and artificial sys­tems;
    3. Researching of applied meaning of fractals;
    4. Searching of computer programs which can generate all of known fractals;
    5. Researching of fractals witch was assigned by complex variables;
    6. Formation of innovative ideas of using of fractals in different spaces;

    The object of research.  Fractal structures, nature and society objects.

    The subject of research. Manifestation of fractality in different objects of the Universe.

    Methods of the researching.

    1. Studying and analysis of literature of research’s problem;
    2. Searching of computer programs which can generate fractals and experimenta­tion with them;
    3. Comparative analysis of principles of generating of fractals and structural or­ganizations of physical, biological and artificial systems;
    4. Generation and formulation of innovation ways to applied significance of fractals.

    Applied significance.

    Researching of universality of fractals gives general academic way of cognition of nature and society.

     

    I. Studying of principles of fractal construction

    We can see fractal constructions everywhere – from crystals and different accu­mulations (clouds, rivers, mountains, stars etc.) to complex ecosystems and bio­logical objects like fern leafs or human brain. Actually, the idea that frac­tal principles are genetic code of our Universe has been discussed for about fifteen years. The first attempt of modeling of the process of the Universe construction was done by A.D. Linde. We also know that young Andrey Saharov had solved “fractal” calculation problem – it was already half a century ago.

    Now therefore, fractal picture of the world was intuitively anticipated by human genius and it inevitably manifested in its activity.

    Fractals are divided into four groups in the traditional way: geometric (constructive), algebraical (dynamical), stochastical and natural.

    The first group of fractals is geometric. It is the most demonstrative type of fractals, because we can instantly observe the self-similarity in it. This type of fractals is constructed in the basis of original figure by her fragmentation and real­izing of different transformations. Geometrical fractals ensue on repeating of this procedure. They are using in computer-generated graphics for generating the pic­tures of leafs, bush, dimensional structures, etc.

    The second large group of fractals -  algebraical. This fractals are constructed by iteration of nonlinear displays, which set by simple formulas. There are two types of algebraical fractals – linear and nonlinear. The first of them are determined by first order equates (linear equates), and the second by nonlinear equates, their na­ture significantly brighter, richer and more diverse than first order equates.

    The third known group of fractals – stochastical. It is generated by method of random modification of options in iterative process. Therefore, we get an objects which is similar to nature fractals – asymmetrical trees, rugged coasts, mountain scenery etc. Such fractals are useful in modeling of land topography, sea–surface and electrolysis process etc.

    The fourth group of fractals is nature, they are dominate in our life. The main difference of such fractals is that they can’t demonstrate infinite self-similarity. There is “physical fractals” term in the classification concept for nature fractals, this term notes their naturalness. These fractals are created with two simple opera­tions: copy and scaling. We can indefinitely list examples of nature fractals: hu­man’s circulatory system, crowns and leafs of trees, lungs, etc.  It is impossible to show all diversity of nature fractals.

     

    II. Applied meaning of fractals

    Fractals are having incredibly widespread application nowadays.

    In the medicine. Human’s organism is consists from fractal structures: circulatory system, bronchus, muscle, neuron system, etc. So it’s naturally that fractal algorithms are useful in the medicine. For example, assessment of rhythm of fractal dimension while electric diagrams analyzing allows to make more infor­mative and accurate view on the beginning of specific illnesses. Also fractals are using for high–quality processing of  X–ray images (in the experimental way). There are designing of new methods in the gastroenterology which allows to ex­plore gastrointestinal tract organs qualitative and painlessly. Actually, there are discoveries of application of fractal methods for diagnosis and treatment of cancer.

    In the science. There are no scientific and technical areas without fractal calcu­lations nowadays. It happens due to the fact that majority of nature objects have fractal structures and dimension: coasts of the continents; natural resources alloca­tion; magnetic field anomaly; dissemination of surges and vibrations in an elastic environments; porous, solid and fungal bodies; crystals; turbulence; dynamic of complicated systems in general, etc. Fractals are useful in geology, geophysics, in the oil sciences… It’s impossible to list all the spaces of adaptability.

    Modeling of chaotic processes, particularly, in description of population models.

    In telecommunications. It’s naturally that fractals are popular in this area too. Natan Coen is person, who had started to use fractal antennas. Fractal antenna has very compact form which provides high productivity. Due to this, such antennas are used in marine and air transport, in personal devises. The theory of fractal an­tennas has become an independent, well-developed apparatus of synthesis and analysis of electric small antenna (ESA) nowadays. There are developments of possibility of fractal compression of the traffic which is transmitted through the web. The goal of this is more effective transfer of information.

    In the visual effects. The theory of fractals has penetrated area of formation of different kinds of visualizations and creation of special effects in the computer graphics soon. This theory are very useful in modeling of nature landscapes in computer games. The film industry also has not been without fractal geometry. All the special effects are based in fractal structure: mountain landscape, lava, flame, fog, large flows and the same. In the modern level of the cinema creation of the special effects is impossible without modeling of fractals.

    In the economics. The Veirshtrass’s function is famous example of stochastic fractals. Analysis of graph of the function in interactive mathematic environment Maple allows to make sure in fractal structure of function by way of entry of dif­ferent ranges of graphic visualization. In any indefinitely small area of the part graph of the function absolutely looks like area of this part in the all . The property of function is used in analysis of stock markets.

    In the architect. Notably, fractal structures have become useful in the architect more earlier than B. Mandelbrode had discovered them. S.B. Pomorov, Doctor of Architecture, Professor, member of Russian Architect Union, talks about applica­tion of fractal theory in the architect in his article. Let’s see on the part of this arti­cle:

    “Fractal structures were found in configuration of African tribal villages, in an­cient Vavilon’s ziggurats, in iconic buildings of ancient India and China, in gothic temples of ancient Russia .

    We can see the high fractal level in Malevich’s Architectons. But they were cre­ated long before emergence of the notion of fractals in the architect. People started to use fractal algorithms on the architect morphogenesis consciously after Mandel­brot’s publications. It was made possible to use fractal geometry for analyzing of architectural forms.

    Fractals had become available to the majority of specialists due to the comput­erization.  They had been incredibly attractive for architectors, designers and town planners in aesthetic, philosophical and psychological way. Fractal theory was per­ceived on emotional, sensual level in the first phase. The constant repression lead­ing to loss of sensuality.

    Application of fractal structures is effective on the microenvironment designing level: interior, household items and their elements. Fractal structures introduction allows creating a new surroundings for people with fractal properties on all levels. It corresponds to nesting spaces.

    Fractal formations are not a panacea or a new era in the architect history. But it’s a new way to design architect forms which enriches the architectural theory and practice language. The understanding of na­ture fractal impacts on architectural view of urban environment. An attempt to de­velop the method of architectural designing which will base in an in-depth fractal forms is especially interesting. Will this method base only on mathematics? Will it be different methods and features symbiosis? The practice experiments and re­searches will show us. It’s safe to say that modern fractal approach can be useful not only for analysis, but also for harmonic order and nature’s chaos, architect which may be semantic dominant in nature and historic context.”

    Computer systems. Fractal data compression is the most useful fractal applica­tion in the computer science. This kind of compression is based on the fact that it’s easy to describe the real world by fractal geometry. Nevertheless, pictures are compressed better than by other methods (like jpeg or gif). Another one advantage is that picture isn’t pixelateing while compressing. Often picture looks better after increase in fractal compressing.

    Basic concept for fractal computer graphics is “Fractal triangle”. Also there are “Fractal figure”, “Fractal object”, “Fractal line”, “Fractal composition”, “Parent object” and “Heir object”. However, it should be noted that fractal computer graphics has recently received as a kind of computer graphics of 21th century.

     The opportunities of fractal computer graphics cannot be overemphasized. It allows creating abstract compositions where we can realize a lot of moves: hori­zontal and vertical, diagonal directions, symmetry and asymmetry etc. Only a few programmers from all over the world know about fractal graphics today.  To what can we compare fractal picture? For example, with complex structure of crystal or with snowflake, the elements of which line up in the one complex composition. This property of fractal object can be useful in ornament creating or designing of decorative composition. Algorithms of synthesis of fractal rates which allows to reproduce copy of any picture too close to the original are developed today.

     

    III. Researching of computer programs of fractal construction

    Strict algorithms of fractals are really good for programming. There are a lot of computer programs which introduce fractals nowadays. Computer mathematic systems are stand out from over programs, especially, Maple. Computer mathe­matics is mathematic modeling tool. So programming represents genetic structure of fractal in these systems and we can see precise submission of fractal structure in the picture while we enter a number of iterations . This is the reason why mathematic fractals should be studied with computer mathematics.  The last dis­covery in fractal geometry has been made possible by powerful, modern com­puters. Fractal property researching is almost completely based on computer cal­culations. It allows making computer experiments which reproduce processes and phenomenon which we can’t experiment in the real world with.

    Our school has been worked with computer mathematics Maple package more than 10 years. So we have unique opportunity to experiment with mathematic fractals, thanks to that we can understand how initial values impact on outcome   (it is stochastic fractal). For example, we have understood the meaning of the fact that color is the fourth dimension: color changing leads to changing of physical char­acteristics. That is what astrophysics mean talking about “multicolored” of the Universe. While fractal constructing in interactive mathematic environment we re­ceived graphic models which was like A. D. Linde’s model of the Universe. Perhaps, it demonstrates that Universe has fractal structure.

     

    Conclusion

    Scientists and philosophers argue, can we talk about universality of fractals in recent years. There are two groups of two opposite positions. We agree with the fact that fractals are universal. Due to the fact that movement is inherent property of material also we always have fractals wherever we have movement.  

    We are convinced that fractal is genetic property of the Universe, but it is not mean that all the Universe elements to the one fractal organization. In deployment process fractal structure is undergoing a lot of fluctuations (deviations) and a lot of points of bifurcation (branching) lead grate number of fractal development varie­ties.  

    Therefore, we think that fractals are general academic method of real world re­searching. Fractals give the methodology of nature and community researching.

    In transitional, chaotic period of society development social life become harder. Different social systems clash. Ancient values are exchanged for new values literally in all spaces. So it’s vitally important for science to develop behavior strategies which allow to avoid tragic mistakes. We think that fractals play important role in developing of such technologies. And – synergy is theory of evolving systems self- organization. But evolution happens on fractal principles, as we know now.

     

    P.S.  Images - in attached files

     

    This is still a work in progress, but might be of use to anybody interested in Maxwell's Equations :-)

    Examining_Maxwells_Equations.mw

     

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