Advanced Search

Education

Teaching and learning about math, Maple and MapleSim
Create a new Post in this category

Computer Algebra for Theoretical Physics

by: ecterrab 14540
physics
October 15 2014
12  4

Last week the Physics package was presented in a talk at the Perimeter Institute for Theoretical Physics and in a combined Applied Mathematics and Physics Seminar at the University of Waterloo. The presentation at the Perimeter Institute got recorded. It was a nice opportunity to surprise people with the recent advances in the package. It follows the presentation with sections closed, and at the end there is a link to a pdf with the sections open and to the related worksheet, used to run the computations in real time during the presentation.

COMPUTER ALGEBRA FOR THEORETICAL PHYSICS

 

  

Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system implemented most of the mathematical objects and mathematics used in theoretical physics computations, and dramatically approximated the notation used in the computer to the one used in paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum. In connection, in this talk the Physics project at Maplesoft is presented and the resulting Physics package illustrated tackling problems in classical and quantum mechanics, general relativity and field theory. In addition to the 10 a.m lecture, there will be a hands-on workshop at 1pm in the Alice Room.

 

... Why computers?
 

 

We can concentrate more on the ideas instead of on the algebraic manipulations

 

We can extend results with ease

 

We can explore the mathematics surrounding a problem

 

We can share results in a reproducible way

 

Representation issues that were preventing the use of computer algebra in Physics
 

 

Notation and related mathematical methods that were missing:


coordinate free representations for vectors and vectorial differential operators,

covariant tensors distinguished from contravariant tensors,

functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indices

Bras, Kets, projectors and all related to Dirac's notation in Quantum Mechanics

 

Inert representations of operations, mathematical functions, and related typesetting were missing:

 

inert versus active representations for mathematical operations

ability to move from inert to active representations of computations and viceversa as necessary

hand-like style for entering computations and texbook-like notation for displaying results

 

Key elements of the computational domain of theoretical physics were missing:

 

ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functions

ability to perform computations taking into account custom-defined algebra rules of different kinds

(problem related commutator, anticommutator, bracket, etc. rules)

Vector and tensor notation in mechanics, electrodynamics and relativity
   

Dirac's notation in quantum mechanics
   

 

• 

Computer algebra systems were not originally designed to work with this compact notation, having attached so dense mathematical contents, active and inert representations of operations, not commutative and customizable algebraic computational domain, and the related mathematical methods, all this typically present in computations in theoretical physics.

• 

This situation has changed. The notation and related mathematical methods are now implemented.

 

Tackling examples with the Physics package
 

Classical Mechanics
 

Inertia tensor for a triatomic molecule
 

 

Problem: Determine the Inertia tensor of a triatomic molecule that has the form of an isosceles triangle with two masses m[1] in the extremes of the base and mass m[2] in the third vertex. The distance between the two masses m[1] is equal to a, and the height of the triangle is equal to h.

Solution
   

Quantum mechanics
 

Quantization of the energy of a particle in a magnetic field
 


Show that the energy of a particle in a constant magnetic field oriented along the z axis can be written as

H = `ℏ`*`ω__c`*(`#msup(mi("a",mathcolor = "olive"),mo("†"))`*a+1/2)

where `#msup(mi("a",mathcolor = "olive"),mo("†"))`and a are creation and anihilation operators.

Solution
   

The quantum operator components of `#mover(mi("L",mathcolor = "olive"),mo("→",fontstyle = "italic"))` satisfy "[L[j],L[k]][-]=i `ε`[j,k,m] L[m]"
   

Unitary Operators in Quantum Mechanics
 

(with Pascal Szriftgiser, from Laboratoire PhLAM, Université Lille 1, France)

A linear operator U is unitary if 1/U = `#msup(mi("U"),mo("†"))`, in which case, U*`#msup(mi("U"),mo("†"))` = U*`#msup(mi("U"),mo("†"))` and U*`#msup(mi("U"),mo("†"))` = 1.Unitary operators are used to change the basis inside an Hilbert space, which physically means changing the point of view of the considered problem, but not the underlying physics. Examples: translations, rotations and the parity operator.

1) Eigenvalues of an unitary operator and exponential of Hermitian operators
   

2) Properties of unitary operators
   

3) Schrödinger equation and unitary transform
   

4) Translation operators
   

Classical Field Theory
 

The field equations for a quantum system of identical particles
 

 

Problem: derive the field equation describing the ground state of a quantum system of identical particles (bosons), that is, the Gross-Pitaevskii equation (GPE). This equation is particularly useful to describe Bose-Einstein condensates (BEC).

Solution
   

The field equations for the lambda*Phi^4 model
   

Maxwell equations departing from the 4-dimensional Action for Electrodynamics
   

General Relativity
 

Given the spacetime metric,

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -exp(lambda(r)), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = exp(nu(r))}))

a) Compute the trace of

"Z[alpha]^(beta)=Phi R[alpha]^(beta)+`𝒟`[alpha]`𝒟`[]^(beta) Phi+T[alpha]^(beta)"

where `≡`(Phi, Phi(r)) is some function of the radial coordinate, R[alpha, `~beta`] is the Ricci tensor, `𝒟`[alpha] is the covariant derivative operator and T[alpha, `~beta`] is the stress-energy tensor

T[alpha, beta] = (Matrix(4, 4, {(1, 1) = 8*exp(lambda(r))*Pi, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = 8*r^2*Pi, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 8*r^2*sin(theta)^2*Pi, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 8*exp(nu(r))*Pi*epsilon}))

b) Compute the components of "W[alpha]^(beta)"" ≡"the traceless part of  "Z[alpha]^(beta)" of item a)

c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)" obtained in b)

Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.

a) The trace of " Z[alpha]^(beta)=Phi R[alpha]^(beta)+`𝒟`[alpha]`𝒟`[]^(beta) Phi+T[alpha]^(beta)"
   

b) The components of "W[alpha]^(beta)"" ≡"the traceless part of " Z[alpha]^(beta)"
   

c) An exact solution for the nonlinear system of differential equations conformed by the components of  "W[alpha]^(beta)"
   

The Physics Project
 

 

"Physics" is a software project at Maplesoft that started in 2006. The idea is to develop a computational symbolic/numeric environment specifically for Physics, targeting educational and research needs in equal footing, and resembling as much as possible the flexible style of computations used with paper and pencil. The main reference for the project is the Landau and Lifshitz Course of Theoretical Physics.

 

A first version of "Physics" with basic functionality appeared in 2007. Since then the package has been growing every year, including now, among other things, a searcheable database of solutions to Einstein equations and a new dedicated programming language for Physics.

 

Since August/2013, weekly updates of the Physics package are distributed on the web, including the new developments related to our plan as well as related to people's feedback.

 

 

Presentation_at_PI_and_UW.pdf     Presentation_at_PI_and_UW.mw

 

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

2014 Maple T.A. User Summit – Schedule

by: jzivku 640
mapleta
October 15 2014
0  0

As previously announced, Maplesoft will be hosting  the 2014 Maple T.A. User Summit this October 22 – 24 in Amsterdam, The Netherlands. You might have heard about the launch of Maple T.A. 10. The User Summit in Amsterdam is a perfect opportunity to get to know more, see the new features in action, and meet Maple T.A. users from around the world.

We are happy to announce that the schedule has been finalized! The event will feature keynote and user presentations by prominent educators from around the world, first-hand discussions by Maplesoft representatives, exciting social events, and training sessions.

As you can see, this event has shaped up to be a very exciting summit for Maple T.A. users. After seeing this schedule you may be wondering why you didn’t sign-up – don’t worry, it’s not too late! To register, please visit our website: https://webstore.maplesoft.com/taconference/register.aspx

I hope to see you there!

Jonny
Maplesoft Product Manager, Maple T.A.

2014 Maple T.A. User Summit – Schedule

by: jzivku 640
mapleta
October 15 2014
0  0

As previously announced, Maplesoft will be hosting  the 2014 Maple T.A. User Summit this October 22 – 24 in Amsterdam, The Netherlands. You might have heard about the launch of Maple T.A. 10. The User Summit in Amsterdam is a perfect opportunity to get to know more, see the new features in action, and meet Maple T.A. users from around the world.

We are happy to announce that the schedule has been finalized! The event will feature keynote and user presentations by prominent educators from around the world, first-hand discussions by Maplesoft representatives, exciting social events, and training sessions.

As you can see, this event has shaped up to be a very exciting summit for Maple T.A. users. After seeing this schedule you may be wondering why you didn’t sign-up – don’t worry, it’s not too late! To register, please visit our website: https://webstore.maplesoft.com/taconference/register.aspx

I hope to see you there!

Jonny
Maplesoft Product Manager, Maple T.A.

Solutions of Differential Equations with Embedded...

by: Lenin Araujo Castillo 1790 Maple 18
engineering education embedded-components
October 15 2014
2  0

The Embedded Components are containers that currently use industries for modeling complex systems to find viable solutions in real time and thus avoid huge wait times and overload our computer; by this paper should show you how to implement a dynamic worksheet through Embedded Components in Maple; it goes from finding solutions to ordinary differential equations partial; which interact with the researcher using different parameters.
Using graphical programming will find immediate solutions to selected problems in science and engineering criteria of variability and boundary conditions evolving development with buttons on multiple actions.

 

cimac_2014.pdf

(in spanish)

Solutions_of_Differential_Equations_with_Embedded_Components.mw

 

Lenin Araujo Castillo

Physics Pure

Computer Science

 

IT Solutions for the Next Generation of Engineers

by: Lenin Araujo Castillo 1790 Maple 18
October 09 2014
2  0

Presented at the National University of Trujillo - CUICITI 2014.

IT Solutions for the Next Generation of Engineers

 

 

 

Descarga aqui los Slides de la presentación/mw CUICITI-2014

CUICITI_09102014.pdf

Soluciones_Informáticas_para_la_siguiente_generación_de_Ingenieros.mw

Lenin Araujo Castillo

Physics Pure

Computer Science

 

October Live Webinars

by: kkoserski 230 Maple
October 08 2014
1  0

Maplesoft regularly hosts live webinars on a variety of topics. Below you will find details on some upcoming webinars we think may be of interest to the MaplePrimes community.  For the complete list of upcoming webinars, visit our website.

Maplesoft Solutions for Math Education

This webinar will demonstrate how Maplesoft’s solutions for mathematics education help teachers bring complex problems to life, allow students to focus on concepts rather than the mechanics of solutions, and offer students the necessary practice to master the concepts being taught.

Key takeaways include:

• How to quickly and painlessly place incoming students in the correct math courses

• How you can use hundreds of intuitive Clickable Math tools to demonstrate and explore up to advanced-level problems and algorithms in the classroom

• How to automate your testing and assessment needs, specifically for math courses

• How to bring your STEM courses to life in an online environment

To join us for the live presentation, please click here to register.

Introduction to Maple T.A. Placement Test Suite 10

This webinar will provide an overview and demonstration of the latest release of the Maple T.A. MAA Placement Test Suite. A result of the ongoing partnership between the Mathematical Association of America (MAA) and Maplesoft, this product gives you the ability to provide the renowned MAA placement tests in an online testing environment. Learn how the Maple T.A. MAA Placement Test Suite can greatly simplify your placement process and explore the latest additions, including a streamlined interface and new tests to determine your students’ readiness for Precalculus and Algebra courses.

To join us for the live presentation, please click here to register.

There is also a recording available from another live webinar we did earlier this month: Introduction to Maple T.A. 10.

Announcing Maple T.A. 10

by: jzivku 640
October 07 2014
6  0

After lots of hard work, vast amounts of testing, and enormous anticipation, Maple T.A. 10 is now available! Maple T.A. 10 is by far our biggest release to date - and we’re not just saying that. When we compare the list of new features and improvements in Maple T.A. 10 with that of previous releases, it’s clear that Maple T.A. 10 has the largest feature set and improvements to date.

Announcing Maple T.A. 10

by: jzivku 640
October 07 2014
6  0

After lots of hard work, vast amounts of testing, and enormous anticipation, Maple T.A. 10 is now available! Maple T.A. 10 is by far our biggest release to date - and we’re not just saying that. When we compare the list of new features and improvements in Maple T.A. 10 with that of previous releases, it’s clear that Maple T.A. 10 has the largest feature set and improvements to date.

Announcing Maple T.A. 10

by: jzivku 640
October 07 2014
6  0

After lots of hard work, vast amounts of testing, and enormous anticipation, Maple T.A. 10 is now available! Maple T.A. 10 is by far our biggest release to date - and we’re not just saying that. When we compare the list of new features and improvements in Maple T.A. 10 with that of previous releases, it’s clear that Maple T.A. 10 has the largest feature set and improvements to date.

Using the position vector

by: Lenin Araujo Castillo 1790 Maple 18
application vector
September 21 2014
2  0

This is an application of vector position to better understand the vector speed and acceleration is a well defined vector space. Fully developed with embedded components for proper use.

 

    Vector_Posición.mw                   (in spanish)

 

L. Araujo C.

Physics Pure

Computer Science

Vectors in Components Embedded full

by: Lenin Araujo Castillo 1790 Maple 18
vector education embedded-components
September 07 2014
2  3

I am sure that with this vector file with embedded components will learn how it works the vector operations. The code is free and can be modified to be improved. Forward engineers.

 

Vectores_con_Components_Embedded.mw     (in spanish)      

 

Lenin Araujo Castillo

Angles directors Embedded Components

by: Lenin Araujo Castillo 1790
September 06 2014
2  0

Using MathContainer and Button for 3D vectors.

 

Angulos_Directores_con_Componentes.mw

(In spanish)

https://www.youtube.com/watch?v=J25P_qNtQe8

 

Banque de questions MapleTA en français – plus...

by: jzivku 640
September 05 2014
1  0

Aujourd’hui, je suis ravis d’annoncer la disponibilité d’une large banque de questions françaises supportant les enseignements du secondaire et de l’enseignement supérieur. Ce contenu didactique est disponible via le MapleTA Cloud, et également grâce au lien de téléchargement ci-dessous.

Lien de téléchargement de la banque de questions françaises

Ces questions sont librement et gratuitement accessibles, et vous pouvez les utiliser directement sur vos propres évaluations et exercices dans MapleTA, ou les éditer et modifier pour les adapter à vos besoins.

Le contenu de cette banque de questions françaises traite de sujets pour les classes et enseignements pré-bac, post-bac pour en majorité les matières scientifiques.

Les matières traitées par niveaux et domaines sont:

Lycées :

  • Electricité
  • Équations Différentielles
  • Gravitation universelle
  • Langues
  • Maths I
  • Maths II
  • Physique
  • Chimie
  • Mécanique

Enseignement supérieur (Post-Bac) :

  • Astrobiologie
  • Introduction au Calcul pour la Biologie
  • Chimie
  • Déplacement d'onde
  • Electricité & Magnétisme
  • Maths pour l’économie
  • Maths Post-Bac
  • Mécanique Angulaire
  • Mécanique des Fluides
  • Mécanique linéaire
  • Physique Post-Bac
  • Electrocinétique
  • Matériau
  • Mécanique des Fluides
  • Thermodynamique

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Introductory Calculus ...

by: jzivku 640
August 27 2014
1  0

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Introductory Calculus Maple T.A.. course module developed by Keele University.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

These questions are designed to accompany the first semester of an introductory honours calculus course. The course is intended primarily for students who need or expect to pursue further studies in mathematics, physics, chemistry, engineering and computer science. With over 250 question, topics include: basic material about functions, polynomials, logs and exponentials, the concept of the derivative, and lots of practise exercises for finding derivatives and integrals, and material about series.

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

Maple T.A. course materials – Introductory Calculus...

by: jzivku 640
August 27 2014
3  4

Several Maple T.A. users have developed comprehensive sets of question content and assignments to support full courses in Maple T.A. These questions are available through the Maple T.A. Cloud, and we have decided to also post the associated course modules on Maple Primes as an alternative way of accessing this content.

Below you will find a link to the Introductory Calculus for Biological Sciences Maple T.A.. course module developed by the University of Guelph.

This testing content is freely distributed, and can be used in your own Maple T.A. tests either as-is, or with edits.

The Introductory Calculus for Biological Sciences course module is designed to cover a single-semester introductory calculus course for biological sciences students at the first-year university level. The questions are designed to span the topics listed below, allowing for practice, homework or testing throughout the semester.

Topics include:

  • Introduction to Functions
  • Composite and Inverse Functions
  • Trigonometric Functions
  • Logarithms and Exponents
  • Sequences and Finite Series
  • Limits and Continuity
  • Derivatives
  • Curve Sketching
  • Differentials
  • Linear Approximation
  • Taylor Polynomials
  • Difference Equations
  • Log-Log Graphs
  • Anti-Differentiation
  • Definite Integrals

Jonny Zivku
Maplesoft Product Manager, Maple T.A.

First 35 36 37 38 39 40 41 Last Page 37 of 59