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Automatic simplification and a new Assum...

ecterrab 14837
assuming simplification assume
December 09 2014
10 6

Hi
Two new things recently added to the latest version of Physics available on Maplesoft's R&D Physics webpage are worth mentioning outside the framework of Physics.

  • automaticsimplification. This means that after "Physics:-Setup(automaticsimplification=true)", the output corresponding to every single input (literally) gets automatically simplified in size before being returned to the screen. This is fantastically convenient for interactive work in most situations.

  • Add Physics:-Library:-Assume, to perform the same operations one typically performs with the  assume command, but without the side effect that the variables get redefined. So the variables do not get redefined, they only receive assumptions.

This new Assume implements the concept of an "extended assuming". It permits re-using expressions involving the variables being assumed, expressions that were entered before the assumptions were placed, as well as reusing all the expressions computed while the variables had assumptions, even after removing the variable's assumptions. None of this is possible when placing assumptions using the standard assume. The new routine also permits placing assumptions on global variables that have special meaning, that cannot be redefined, e.g. the cartesian, cylindrical or spherical coordinates sets, or the coordinates of a coordinate spacetime system within the Physics package, etc.

Examples:

 
> 

with(Physics):

This is Physics from today:

> 

Physics:-Version()[2]

`2014, December 9, 16:51 hours`

 (1.1)
• 

Automatic simplification is here. At this point automaticsimplification is OFF by default.
> 

Setup(automaticsimplification)

[automaticsimplification = false]

 (1.2)

Hence, for instance, if you input the following expression, the computer just echoes your input:

> 

Physics:-`*`(a, c)+Physics:-`*`(a, d)+Physics:-`*`(b, c)+Physics:-`*`(b, d)

a*c+a*d+b*c+b*d

 (1.3)

There is however some structure behind (1.3) and, in most situations, it is convenient to have these structures
apparent, in part because they frequently provide hints on how to proceed ahead, but also because a more
compact expression is, roughly speaking, simpler to understand. To see this automaticsimplification in action,
turn it ON:

> 

Setup(automaticsimplification = true)

[automaticsimplification = true]

 (1.4)

Recall this same expression (you could input it with the equation label (1.3) as well) 

> 

Physics:-`*`(a, c)+Physics:-`*`(a, d)+Physics:-`*`(b, c)+Physics:-`*`(b, d)

(c+d)*(a+b)

 (1.5)

What happened: this output, as everything else after you set automaticsimplification = true and with no
exceptions, is now further processed with simplify/size before being returned. And enjoy computing with frankly
shorter expressions all around! And no need anymore for "simplify(%, size)" every three or four input lines.

Another  example, typical in computer algebra where expressions become uncomfortably large and difficult to
read: convert the following input to 2D math input mode first, in order to compare what is being entered with the
automatically simplified output on the screen

> 

-Physics:-`*`(Physics:-`*`(Physics:-`*`(3, sin(x)^(1/2)), cos(x)^2), sin(x)^m)+Physics:-`*`(Physics:-`*`(Physics:-`*`(3, sin(x)^(1/2)), cos(x)^2), cos(x)^n)+Physics:-`*`(Physics:-`*`(Physics:-`*`(4, sin(x)^(1/2)), cos(x)^4), sin(x)^m)-Physics:-`*`(Physics:-`*`(Physics:-`*`(4, sin(x)^(1/2)), cos(x)^4), cos(x)^n)

-4*(cos(x)^n-sin(x)^m)*sin(x)^(1/2)*cos(x)^2*(cos(x)^2-3/4)

 (1.6)

You can turn automaticsimplification OFF the same way

> 

Setup(automaticsimplification = false)

[automaticsimplification = false]

 (1.7)
• 

New Library:-Assume facility; welcome to the world of "extended assuming" :)

 

Consider a generic variable, x. Nothing is known about it

> 

about(x)

x:

  nothing known about this object

  

Each variable has associated a number that depends on the session, and the computer (internally) uses this
number to refer to the variable.

> 

addressof(x)

18446744078082181054

 (1.8)

When using the assume  command to place assumptions on a variable, this number, associated to it, changes,
for example:

> 

assume(0 < x and x < Physics:-`*`(Pi, 1/2))
> 

addressof(x)

18446744078179060574

 (1.9)

Indeed, the variable x got redefined and renamed, it is not anymore the variable x referenced in (1.8).

> 

about(x)

Originally x, renamed x~:

  is assumed to be: RealRange(Open(0),Open(1/2*Pi))

  


The semantics may seem confusing but that is what happened, you enter x and the computer thinks x~, not x 
anymore.This means two things:

1) all the equations/expressions, entered before placing the assumptions on x using assume, involve a variable x 
that is different than the one that exists after placing the assumptions, and so these previous expressions cannot
be reused. They involve a different variable.

2) Also, because, after placing the assumptions using assume, x refers to a different object, programs that depend
on the x that existed before placing the assumptions will not recognize the new x redefined by assume .

 

For example, if x was part of a coordinate system and the spacetime metric g[mu, nu]depends on it, the new variable x
redefined within assume, being a different symbol, will not be recognized as part of the dependency of "g[mu,nu]." This
posed constant obstacles to working with curved spacetimes that depend on parameters or on coordinates that
have a restricted range. These problems are resolved entirely with this new Library:-Assume, because it does not
redefine the variables. It only places assumptions on them, and in this sense it works like assuming , not assume .
As another example, all the Physics:-Vectors commands look for the cartesian, cylindrical or spherical coordinates
sets [x, y, z], [rho, phi, z], [r, theta, phi] in order to determine how to proceed, but these variables disappear if you use
assume to place assumptions on them. For that reason, only assuming  was fully compatible with Physics, not assume.

 

To undo assumptions placed using the assume command one reassigns the variable x to itself:

> 

x := 'x'

x

 (1.10)

Check the numerical address: it is again equal to (1.8) 

> 

addressof(x)

18446744078082181054

 (1.11)

·All these issues get resolved with the new Library:-Assume, that uses all the implementation of the existing 
assume command but with a different approach: the variables being assumed do not get redefined, and hence:
a) you can reuse expressions/equations entered before placing the assumptions, you can also undo the
assumptions and reuse results obtained with assumptions. This is the concept of an extended assuming. Also,
commands that depend on these assumed variables will all continue to work normally, before, during or after
placing the assumption, because the variables do not get redefined.

Example:

> 

about(x)

x:

  nothing known about this object

  

So this simplification attempt accomplishes nothing

> 

simplify(arccos(cos(x)))

arccos(cos(x))

 (1.12)

Let's assume now that 0 < x and x < (1/2)*Pi

> 

Library:-Assume(0 < x and x < Physics:-`*`(Pi, 1/2))

{x::(RealRange(Open(0), Open((1/2)*Pi)))}

 (1.13)

The new command echoes the internal format representing the assumption placed.

a) The address is still the same as (1.8)

> 

addressof(x)

18446744078082181054

 (1.14)

So the variable did not get redefined. The system however knows about the assumption - all the machinery of the
assume command is being used

> 

about(x)

Originally x, renamed x:

  is assumed to be: RealRange(Open(0),Open(1/2*Pi))

  


Note that the renaming is to the variable itself - i.e. no renaming.

Hence, expressions entered before placing assumptions can be reused. For example, for (1.12), we now have

> 

simplify(arccos(cos(x)))

x

 (1.15)

To clear the assumptions on x, you can use either of Library:-Assume(x=x) or Library:-Assume(clear = {x, ...}) in
the case of many variables being cleared in one go, or in the case of a single variable being cleared:

> 

Library:-Assume(clear = x)
> 

about(x)

x:

  nothing known about this object

  


The implementation includes the additionally functionality, for that purpose add the keyword additionally 
anywhere in the calling sequence. For example:

> 

Library:-Assume(x::positive)

{x::(RealRange(Open(0), infinity))}

 (1.16)
> 

about(x)

Originally x, renamed x:

  is assumed to be: RealRange(Open(0),infinity)

  
> 

Library:-Assume(additionally, x < 1)

{x::(RealRange(Open(0), Open(1)))}

 (1.17)
> 

Library:-Assume(x = x)

In summary, the new Library:-Assume command implements the concept of an extended assuming, that can be
turned ON and OFF at will at any moment without changing the variables involved.


Download AutomaticSimplificationAndAssume.mw

 

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

Computer Algebra for Theoretical Physics...

ecterrab 14837
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 = `&hbar;`*`&omega;__c`*(`#msup(mi("a",mathcolor = "olive"),mo("&dagger;"))`*a+1/2)

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

Solution
   

The quantum operator components of `#mover(mi("L",mathcolor = "olive"),mo("&rarr;",fontstyle = "italic"))` satisfy "[L[j],L[k]][-]=i `&epsilon;`[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("&dagger;"))`, in which case, U*`#msup(mi("U"),mo("&dagger;"))` = U*`#msup(mi("U"),mo("&dagger;"))` and U*`#msup(mi("U"),mo("&dagger;"))` = 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)+`&Dscr;`[alpha]`&Dscr;`[]^(beta) Phi+T[alpha]^(beta)"

where `&equiv;`(Phi, Phi(r)) is some function of the radial coordinate, R[alpha, `~beta`] is the Ricci tensor, `&Dscr;`[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)"" &equiv;"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)+`&Dscr;`[alpha]`&Dscr;`[]^(beta) Phi+T[alpha]^(beta)"
   

b) The components of "W[alpha]^(beta)"" &equiv;"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

DEs and Mathematical Functions Updates 2...

ecterrab 14837
physics differential-equations
June 23 2014
3 13

Hi,
The FunctionAdvisor project is currently developing at full speed. During the last two months, a significant amount of new conversion routines and mathematical information for Jacobi elliptic and Jacobi Theta functions, on identities, periodicity, transformations, etc. got added to the conversion network for mathematical functions and to the FunctionAdvisor. The previous months was the turn of the set of complex components, added to the network. Developments regarding the simplification and integration of special functions (e.g SphericalY for computing spherical harmonics or Dirac), as well as fixes to the numerical evaluation of JacobiAM, `assuming` and to differential equation subroutines are also part of the update.

These developments are available to everybody as usual in the Maplesoft R&D Differential Equations and Mathematical Functions webpage. Below there is a list of the latest developments as seen in the worksheet that comes in the zip with the DEsAndMathematicalFunctions update.

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

Physics updates 2014 (Jan-Jun)...

ecterrab 14837
June 19 2014
8 0

Hi,
An interesting sequence of enhancements and new developments happened in the Physics package during this first half of the year. During the last month, improvements happened in the handling of Vectorial expressions and quantum mechanics using Dirac’s notation. During April and part of May it was the turn of general relativity enhancements.

Some of the developments are also interesting beyond Physics. For example: it is now possible to multiply equations. Suppose you have A = B   (1), and C = D   (2), multiplying as in (1) (2) now results in lhs((1)) lhs((2)) = rhs((1)) rhs((2)), saving a lot of typing. You can also perform (1)/(2) or (1)^2. Some enhancements in Physics related simplification, integration, `assuming`, and typesetting - e.g. the simplification and integration of spherical harmonics (SphericalY function) are also part of the update.

These developments are available to everybody as usual in the Maplesoft R&D Physics webpage. Below there is a list of the developments for the last month as seen in the worksheet that comes in the zip with the Physics update.

 

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

DEs and Mathematical Functions Updates...

ecterrab 14837
differential-equations
May 06 2014
2 0

This is the first presentation of updates for the DE and Mathematical Functions programs of Maple 18. It includes several improvements, all in the Mathematical Functions sector, as well as some fixes. The update and instructions for its installation are available on the Maplesoft R&D webpage for DEs and mathematical functions. Some of the items below were mentioned here in Mapleprimes - you are welcome to present suggestions or issues; if possible they will be addressed right away in the next update.

  • Filling gaps in the FunctionAdvisor regarding all the 6 complex components: abs, argument, conjugate, Im, Re, signum, as well as regarding Heaviside (step function), Dirac, min and max.
  • Fix the simplification and differentation rule for doublefactorial
  • Make convert(..., hypergeometric) work the same way as convert(blabla, hypergeom)
  • Implement integral forms for Heaviside(z) and JacobiAM(z, k) via convert(..., Int)
  • Implement appropriate display for the inert %intat function as well as its conversion to the inert Int
  • Make the FunctionAdvisor/DE return not just the PDE system satisfied by f(z, k) = JacobiAM(z, k)and also (new) the ODE satisfied by f(z) = JacobiAM(z, k)
  • Fix conversion rule from Heaviside(z) to Sum
  • Fix unexpected error interruption when differentiating min(...) and max(...) containing more than three arguments
  • Fix issue in simplify/conjugate
  • Improvement in expand/int: factors in disguise are put outside the integration sign
  • Various improvements in the case of multiple integrals involving the Dirac function
  • Make Intat fully inert (before it was evaluating its arguments)
  • Make value of inert indexed objects work

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

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