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by: ecterrab 14702 Product: Maple

April 29 2016

13 0

Below is the worksheet with the whole material presented yesterday in the webinar, “Applying the power of computer algebra to theoretical physics”, broadcasted by the “Institute of Physics” (IOP, England). The material was very well received, rated 4.5 out of 5 (around 30 voters among the more than 300 attendants), and generated a lot of feedback. The webinar was recorded so that it is possible to watch it (for free, of course, click the link above, it will ask you for registration, though, that’s how IOP works).

Anyway, you can reproduce the presentation with the worksheet below (mw file linked at the end, or the corresponding pdf also linked with all the input lines executed). As usual, to reproduce the input/output you need to have installed the latest version of Physics, available in the Maplesoft R&D Physics webpage.

Why computer algebra?

... and why computer algebra?

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 textbook-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

(commutator, anticommutator and bracket rules, etc.)

Examples

	The Maple computer algebra environment

Classical Mechanics

	Inertia tensor for a triatomic molecule

Classical Field Theory

	*The field equations for the model

	*Maxwell equations departing from the 4-dimensional Action for Electrodynamics

	*The Gross-Pitaevskii field equations for a quantum system of identical particles

Quantum mechanics

	*The quantum operator components of satisfy

	Quantization of the energy of a particle in a magnetic field

Unitary Operators in Quantum Mechanics

	*Eigenvalues of an unitary operator and exponential of Hermitian operators

Properties of unitary operators

Consider two set of kets and , each of them constituting a complete orthonormal basis of the same space.

One can always build an unitary operator that maps one basis to the other, i.e.:

	*Verify that implies on

	*Show that is unitary

	*Show that the matrix elements of in the and basis are equal

	Show that and have the same spectrum

Schrödinger equation and unitary transform

Consider a ket that solves the time-dependant Schrödinger equation:

and consider

,

where is a unitary operator.

Does evolves according a Schrödinger equation

and if yes, which is the expression of ?

	Solution

Translation operators using Dirac notation

In this section, we focus on the operator

	Settings

	The Action (translation) of the operator on a ket

	Action of on an operator

General Relativity

	*Exact Solutions to Einstein's Equations

*"Physical Review D" 87, 044053 (2013)

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 Ricci and Weyl scalars

b) Compute the trace of

where is some function of the radial coordinate, is the Ricci tensor, is the covariant derivative operator and is the stress-energy tensor

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

c) Compute the components of the traceless part of of item b)

d) Compute an exact solution to the nonlinear system of differential equations conformed by the components of obtained in c)

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 Ricci and Weyl scalars

	b) The trace of

	b) The components of the traceless part of

	c) An exact solution for the nonlinear system of differential equations conformed by the components of

*The Equivalence problem between two metrics

From the "What is new in Physics in Maple 2016" page:

In the Maple PDEtools package, you have the mathematical tools - including a complete symmetry approach - to work with the underlying [Einstein’s] partial differential equations. [By combining that functionality with the one in the Physics and Physics:-Tetrads package] you can also formulate and, depending on the metrics also resolve, the equivalence problem; that is: to answer whether or not, given two metrics, they can be obtained from each other by a transformation of coordinates, as well as compute the transformation.

	Example from: A. Karlhede, "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation, Vol. 12, No. 9, 1980

	*Equivalence for Schwarzschild metric (spherical and Krustal coordinates)

Tetrads and Weyl scalars in canonical form

Generally speaking a canonical form is obtained using transformations that leave invariant the tetrad metric in a tetrad system of references, so that theWeyl scalars are fixed as much as possible (conventionally, either equal to 0 or to 1).

Bringing a tetrad in canonical form is a relevant step in the tackling of the equivalence problem between two spacetime metrics.

The implementation is as in "General Relativity, an Einstein century survey", edited by S.W. Hawking (Cambridge) and W. Israel (U. Alberta, Canada), specifically Chapter 7 written by S. Chandrasekhar, page 388:

The transformations (rotations of the tetrad system of references) used are of Class I, II and III as defined in Chandrasekar's chapter - equations (7.79) in page 384, (7.83) and (7.84) in page 385. Transformations of Class I can be performed with the command Physics:-Tetrads:-TransformTetrad using the optional argument nullrotationwithfixedl_, of Class II using nullrotationwithfixedn_ and of Class III by calling TransformTetrad(spatialrotationsm_mb_plan, boostsn_l_plane), so with the two optional arguments simultaneously.

The determination of appropriate transformation parameters to be used in these rotations, as well as the sequence of transformations happens all automatically by using the optional argument, canonicalform of TransformTetrad .

>

restart; with(Physics); with(Tetrads)

0, "%1 is not a command in the %2 package", Tetrads, Physics

(7.4.1)

	Petrov type I

	Petrov type II

	Petrov type III

	Petrov type N

	Petrov type D

Physics_2016_IOP_webinar.mw Physics_2016_IOP_webinar.pdf

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

Asked by: rdpdo 15

April 29 2016

1 2

Hello,

I have a complex set of non linear diff eqns in the form :

y1'' = f(y1',y1,y2'',y2',y2,y3'',y3',y3,y4'',....,y6'',y6',y6,u1,u2,u3,u4) ;

y2'' = f(y1'',y1',y1,y2',y2,y3'',y3',y3,y4'',....,y6'',y6',y6,u1,u2,u3,u4)

and so on ... y6''=(...)

As I want to resolve this coupled systeme in matlab using @ODE45... I wanted the equations in the form : y1''=f(y1',y1,y2',y2,....) and so on ... => X'[] = f(X[],U[])

How can I force maple to rearrange a system of coupled eqns with only the variables i want ?

I know this is possible beacause it is a nonlinear state space model but maple do not work with nonlinear state space model... It give me error when I tried to create statespace model with my non linear diff eqns.

Thanks a lot !

Asked by: sand15 1613

April 29 2016

1 4

----------------------------------------------------------------------------------------------------------------------
Introduction

I have a matrix (named DC on the piece of code below) all the elements of which are complex numbers a+b*I with a and b floating point numbers.
I want to obtain the real part of DC.

Obviously, if you do something like :
DC := Matrix(2,2, [1.0+1.0*I, 1.0-1.0*I, -1.0+1.0*I, -1.0-1.0*I];
Re~(DC);

the result corresponds to the desired matrix

----------------------------------------------------------------------------------------------------------------------
Context

In fact this matrix DC comes from some computations described in the piece of code below

# Purpose :
# Given N points in a plane (here in [0,1]X[0,1]), compute the matrix DX of distances between these points
#
# Example : if Pi and Pj are two such points, DX[i,j]=DX[j,i] denotes the Euclidian distance between Pi and Pj
#
# As I did not be able to find any single function in MAPLE that would construct DX , I proceed that way :
#   1/ let X the (N,2) matrix that contains the coordinates of the N points
#   2/ I represent these N points as N complex numbers (vector C)
#   3/ I construct the (N,N) matrix MC = <C | C …..| C>
#   4/ I put MC = C – Transpose(C) :
#   5/ I take the norm DX of each elements of DC : DX := abs~(DC)
#       At this point, DX should contain the desired distances
#       But, due to floating point arithmetics, each element of DX writes a+0.*I where a is some floating point number)
#   6/ Last stage : execute Re~(DX)
#

with(Statistics):
with(LinearAlgebra):

N := 4:
X := Matrix(N,2, convert(Sample(Uniform(0,1), 2*N), list)): #just an example

C   := X[..,1] +~ X[..,2] *~I;
MC := Multiply(C, Vector[row](N, 1));
DC := MC - Transpose(MC);
DX := abs~(DC);
Re~(DX)

----------------------------------------------------------------------------------------------------------------------
My observations :

1/ Maple 2015, Windows XP, 64 bytes
Re~(DX) returns DX and does not remove the imaginary (0.*I) part
But Matrix(N, N, Re~(convert(DC, list))) does (which is a satisfactory, even if not clever, stopgap)

Why (it is just a question to help me to understand correctly how MAPLE proceeds) Re~(DC) does not (seem) to work here ?

2/ Maple 2015.2, Mac OS X El Capitan
DX := abs~(DC) gives me this strange result :
If (for instance) DC[i, j] = -1 – 2*I, DX[i, j] = +1 + 2*I
According to the compatibility problems between Maple 2015.1 and “El Capitan” (fixed from February), could it remain a few other problems ?

Last but not lesat : Did I do any syntax error ?

I look forward to your responses

Asked by: Bendesarts 450

April 29 2016

1 2

Hello,

I have a little question about the property "local" or not of a index in a for loop.

I notice that when I make "for loop" the index i is not local. In the small example, when I ask to evaluate i after having done this loop, the answer is 5 and not i.

Example :

for i from 1 to 4 do
i^2
od;

i; --> 5

How can I do to ask Maple to use index as local ?

Thanks a lot for your help.

Asked by: baustamm1 35

April 29 2016

0 8

Dear all,

I have a question: how to compute the roots of exp(z) = -1 with z in C?

I tried:

fsolve( exp(z) = -1, z, complex );

But it only gives one root (0.1671148658e-3+4.934802220*10^9*I) which does not even seem to be correct. I would prefere smth like z_n = I*(2*n-1)*pi or at least multiple roots...

By using

solve(exp(x) = -1, x);

it returns I*Pi.

MATLAB MuPAD gives the desired result:

solve(exp(x) = -1, x)

(PI*I + 2*PI*k*I, k in Z)

Thanks!

Asked by: jiggity 5

April 28 2016

0 4

solarsysem.mw Sorry for the repost but this is my newest document.

I have to create a solar system model on maple by defining a force equation then using the seq function to create a diffeq and then solving those differential equations using the initial conditions with the sun at (0, 0, 0) in xyz coordinates.

It works until my last "ic1" entry and I get an error in dsolve/numeric/process_input

I'm pretty desperate, I'll appreciate any help I can get

Asked by: bsexton 10

April 28 2016

1 1

I am a new user. I have a worksheet and need to see how the the procs are being accessed as it does not look like they are being called and executed as I would expect. What is the most efficient way to debug and see the calling order of my Procs?

Thanks in advance,

Bonnie

Asked by: roiweber 10

April 28 2016

1 2

Hi,

I am a college student that is aspiring to become a theoretical theorist. I have postulates that are not in mathematical form yet, and I would like to use a program that can help with this process. I am new to the usage of Maple products, and I have questions. Is it possible to use Maple in this way, before getting to the difficult calculus steps? Can MapleSim or Maple Physics, perform simulations, or models of physics calculations?

Thank you,

Roi

Asked by: John Fredsted 2253

April 28 2016

0 7

Wanting to edit a post of mine, I accidentally clicked the 'More'-button and chose 'Edit' on anothers post. I immediately saw that it was not my post, and did of course not alter it. But it left me somewhat shocked, for it would seem to imply that anyone can edit (and delete, as well, have just investigated that) anyones post(s). That I think is something that really needs to be remedied as soon as possible.

Asked by: joyceljy9990 50

April 28 2016

1 2

Here is my command and i will use to make the user interface.

Orginal:

for i to l do

for j from x to y do

block[i][j] := 1-block[i][j]

end do;

c_block[i] := block[i];

c_block[i] := parse(cat(c_block[i][]))

end do

Can someone help me to edit my command so that i can write it in Click Edit Action for a button? thank you.

Do(for i from 1 to %l do)
Do(for j from x to y do)
Do(block[i][j] = 1-block[i][j]);
end do);
Do(c_block[i] := block[i]);
Do( c_block[i] := parse(cat(c_block[i][])));
end do);

Asked by: rostam 5

April 28 2016

1 15

Hey

I have a very simple function and I need to plot it. The plot is however not smooth and there are annoying cuts and steps which don't go away no matter how I set the options. This problem has been puzzling me for a while and I really appreciate any help.

The followings are all the possible options I could find, none seems to work. You can take a look at this file: ploting_problem.mw

Thanks

Asked by: mskalsi 290

April 28 2016

0 6

Download triplet.mw

Regards

Asked by: Zeineb 540

April 28 2016

2 1

Dear all

I need a help to compute the integral from -1 to 1 of product of derivative of legendre polynomial

> restart;
> with(orthopoly);

P(n, x); # Designe the legendre polynomial of degree n
I would like to compute

int(diff(P(n,x),x)*diff(P(m,x),x),x=-1..1) ;

In this previous formula when I put n=2 and m=3 I can get the result but in general case i cannot,

Many thanks

Asked by: nrebman1 15

April 27 2016

1 5

Howdy all,

I am trying to create a solar system model by defining a force equation then using the sequence function to create a differential equation and then solving those differential equations using the initial conditions (in X,Y,Z coordinates). So far I have the code below.

>

$mass := Vector(4, {(1) = ` 15 x 1 `*Matrix, (2) = `Data Type: `*anything, (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

(1)

>

sqrt(sum(x[i](t)^2, i = 1 .. 3));

proc (i, j) options operator, arrow; sqrt(sum((x[i, k](t)-x[j, k](t))^2, k = 1 .. 3)) end proc

(2)

>

diff(x[0, 1](t), t, t) = force(0)[1]:

>

$initialPositions := Vector(4, {(1) = ` 15 x 3 `*Matrix, (2) = `Data Type: `*anything, (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

(3)

>

$initialVelocities := Vector(4, {(1) = ` 15 x 3 `*Matrix, (2) = `Data Type: `*anything, (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

(4)

>

ic1 := seq(seq(x[i, k](0) = initialPositions[i+1, k], k = 1 .. 3), i = 0 .. N-1); ic2 := seq(seq((D(x[i, k]))(0) = initialVelocities[i+1, k], k = 1 .. 3), i = 0 .. N-1); equations := {ic1, ic2, ode}; sol := dsolve(equations, numeric)

Error, (in dsolve/numeric/process_input) invalid specification of initial conditions, got {x[0, 1](0) = 0, x[0, 2](0) = 0, x[0, 3](0) = 0, x[1, 1](0) = -0.210e8, x[1, 2](0) = 0.426e8, x[1, 3](0) = 0.541e7, x[2, 1](0) = 0.106e9, x[2, 2](0) = -0.244e8, x[2, 3](0) = -0.644e7, x[3, 1](0) = -0.139e9, x[3, 2](0) = -0.569e8, x[3, 3](0) = 0.316e4, x[4, 1](0) = -0.177e9, x[4, 2](0) = -0.155e9, x[4, 3](0) = 0.111e7, x[5, 1](0) = -0.802e9, x[5, 2](0) = 0.131e9, x[5, 3](0) = 0.174e8, x[6, 1](0) = -0.480e9, x[6, 2](0) = -0.142e10, x[6, 3](0) = 0.438e8, x[7, 1](0) = 0.280e10, x[7, 2](0) = 0.103e10, x[7, 3](0) = -0.324e8, x[8, 1](0) = 0.420e10, x[8, 2](0) ...

>

Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

>

Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

>

Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

>

Download ass4.mw

Everything works fine until I try to execute the last line. When I do that I get an error that says "Error, in dsolve/numeric/process_input. invalid specifications of initial conditions.

At this point I am not sure if the problem lies in how I have defined my initial conditions or the way i've defined the force equation but I am open to any suggestions or ideas on where I should go from here.

Thanks in advance!

Nick

Asked by: zhuxian 60

April 27 2016

1 3

i use the pdsolve to find the solutions of a system of partial differential equations,

but the result contains some indefinite integrals, how to simplify it further?

thank you

code:

eq1 := {6*(diff(_xi[t](x, t, u), u))-3*(diff(_xi[x](x, t, u), u)), 12*(diff(_xi[t](x, t, u), u, u))-6*(diff(_xi[x](x, t, u), u, u)), 2*(diff(_xi[t](x, t, u), u, u, u))-(diff(_xi[x](x, t, u), u, u, u)), diff(_eta[u](x, t, u), t)+diff(_eta[u](x, t, u), x, x, x)+(diff(_eta[u](x, t, u), x))*u, 18*(diff(_xi[t](x, t, u), x, u))+3*(diff(_eta[u](x, t, u), u, u))-9*(diff(_xi[x](x, t, u), x, u)), 6*(diff(_xi[t](x, t, u), x, x))+3*(diff(_eta[u](x, t, u), x, u))-3*(diff(_xi[x](x, t, u), x, x)), 6*(diff(_xi[t](x, t, u), x, u, u))+diff(_eta[u](x, t, u), u, u, u)-3*(diff(_xi[x](x, t, u), x, u, u)), 12*(diff(_xi[t](x, t, u), u))-6*(diff(_xi[x](x, t, u), u))+6*(diff(_xi[t](x, t, u), x, x, u))-6*(diff(_xi[t](x, t, u), u))*u+3*u*(diff(_xi[x](x, t, u), u))-3*(diff(_xi[x](x, t, u), x, x, u))+3*(diff(_eta[u](x, t, u), x, u, u)), 12*(diff(_xi[t](x, t, u), x))-6*(diff(_xi[x](x, t, u), x))+2*(diff(_xi[t](x, t, u), t))+2*(diff(_xi[t](x, t, u), x, x, x))-4*(diff(_xi[t](x, t, u), x))*u+2*(diff(_xi[x](x, t, u), x))*u+_eta[u](x, t, u)-(diff(_xi[x](x, t, u), t))+3*(diff(_eta[u](x, t, u), x, x, u))-(diff(_xi[x](x, t, u), x, x, x))};

simplify(pdsolve(eq1))

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Residual invariance

Petrov type I

Petrov type II

Petrov type III

Petrov type D

remains invariant under rotations of Class III

Petrov type N

remains invariant under rotations of Class II