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ecterrab

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Maplesoft Physics Updates v.1167...

ecterrab 14560
March 08 2022
3 0

Hi
The error interruption is unexpected, and a fix to that for Maple 2021 is distributed within the Maplesoft Physics Updates v.1167; to install it suffices to input Physics:-Version(latest) and follow instructions.

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

Physics...

ecterrab 14560
February 22 2022
3 0

See the Physics package, the spacetime metric help page ?g_, and more general ?Physics,Tensors. The answers to your questions are all there.

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

Explanation...

ecterrab 14560
February 13 2022
2 0

restart

with(Physics)

Setup(spacetime)

[spacetimeindices = greek]

 (1)

Physics:-Version()

`The "Physics Updates" version in the MapleCloud is 1142 and is the same as the version installed in this computer, created 2022, February 12, 11:16 hours Pacific Time.`

 (2)

Define(t[mu])

{Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], t[mu], Physics:-LeviCivita[alpha, beta, mu, nu]}

 (3)

NULL

SumOverRepeatedIndices(t[mu]*t[`~mu`])

t[1]*t[`~1`]+t[2]*t[`~2`]+t[3]*t[`~3`]+t[4]*t[`~4`]

 (4)

NULL

SumOverRepeatedIndices(t[mu]*t[`~mu`])

t[1]*t[`~1`]+t[2]*t[`~2`]+t[3]*t[`~3`]+t[4]*t[`~4`]

 (5)

NULL

SumOverRepeatedIndices(t[mu]*t[`~μ`])

t[mu]*t[`~μ`]

 (6)

When you see things like this, try lprint

> 

lprint(%)

t[mu]*t[`~μ`]

  

The following works, because, if the covariant index is A, the contravariant index is ~A (here, A = `μ`)

> 

t[`μ`]*t[`~μ`]

t[`μ`]*t[`~μ`]

 (7)
> 

SumOverRepeatedIndices(%)

t[1]*t[`~1`]+t[2]*t[`~2`]+t[3]*t[`~3`]+t[4]*t[`~4`]

 (8)

For the same reason, this works

> 

t[`mu`]*t[`~mu`]

t[mu]*t[`~mu`]

 (9)
> 

SumOverRepeatedIndices(%)

t[1]*t[`~1`]+t[2]*t[`~2`]+t[3]*t[`~3`]+t[4]*t[`~4`]

 (10)

And yet for the same reason this does not work

> 

t[`μ`]*t[`~mu`]

t[`μ`]*t[`~mu`]

 (11)
> 

SumOverRepeatedIndices(%)

t[`μ`]*t[`~mu`]

 (12)

Likewise neither t[mu] * t[`~μ`], your example, could work. You need to be aware: that shortcut to enter indices that you are using, depending on how you use it for contravariant indices, it results in a different thing, displayed the same way (~mu and `~μ`)

NULL

 

Download greek-index_(reviewed).mw

Edgardo S. Cheb-Terrab

 

Physics, Differential Equations and Mathematical Functions, Maplesoft

FunctionAdvisor(plot, Zeta(z)) and plots...

ecterrab 14560
January 21 2022
3 2

I think FunctionAdvisor(plot, Zeta) is the simplest and most comprehensive way of plotting the Zeta function. The plots you will see are not random but follow the NIST (Digital Library of Mathematical Functions), a somehow modern version of the Abramowitz and Stegun book.

Another very convenient way to visualize specific cases of the Zeta function is using plots:-plotcompare with the expression_plot optional argument, e.g. plotcompare(Zeta(z^2/3), expression_plot).

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

Physics:-Fundiff and ?Physics,Examples...

ecterrab 14560
January 11 2022
2 0

Hi
The command that works in general to compute equations of motion (or field equations in the case of field theory, quantum or not) is Physics:-Fundiff, to perform functional differentiation. Take a look, please, at its help page (?Fundiff).

Beyond Fundiff's help page, more examples of computing motion/field equations using Fundiff are found in the help page ?Physics,Examples.

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

Yes, see ?PDEtools,diff_table...

ecterrab 14560
January 10 2022
3 1

with(PDEtools) 

PDEtools:-declare(y(x), prime = x)

`derivatives with respect to`*x*`of functions of one variable will now be displayed with '`

 (1)

Now set a label, Y, to represent y(x) and its derivatives using indexed notation, using PDEtools:-diff_table

Y := diff_table(y(x))


Example:

Y[x, x]+Y[] = f(x)

diff(diff(y(x), x), x)+y(x) = f(x)

 (2)

show

diff(diff(y(x), x), x)+y(x) = f(x)

 (3)

NULL

To also input using prime notation, there is the Typesetting package, as indicated by acer. If, in addition, you have installed the lasted Maplesoft Physics Updates for Maple 2021.2, you can in addition use dot notation for time derivatives together with all the above.

Download Input_using_indexed_notation_for_derivatives.mw

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

PDEtools:-Solve is the simplest approach...

ecterrab 14560
January 07 2022
6 5
 

This problem has a rather simple solution using PDEtools:-Solve , which has a whole set of routines to handle things like this.

> 

r := x*(diff(theta(t), t))^2+y*(diff(`ϕ`(t), t))^2
> 

g := (4*(f+T))*(diff(theta(t), t))^2+u*(diff(`ϕ`(t), t))^2

All you need to do is to ask for a solution independent of certain variables, in you example: t

> 

PDEtools:-Solve(r = g, {x, y}, independentof = t)

{x = 4*f+4*T, y = u}

 (1)

This is the example you called more complicated. PDEtools:-Solve works fine as well, whether you load Physics or not

> 

with(Physics)
> 

r := x*(diff(theta(t), t))^2+y*(diff(`ϕ`(t), t))^2+z*(diff(theta(t), t, t))+w*(diff(`ϕ`(t), t, t))
> 

g := (4*(f+T))*(diff(theta(t), t))^2+u*(diff(`ϕ`(t), t))^2+(f+9)*(diff(theta(t), t, t))+4*s*(diff(`ϕ`(t), t, t))
> 

PDEtools:-Solve(r = g, {w, x, y, z}, independentof = t)

{w = 4*s, x = 4*f+4*T, y = u, z = f+9}

 (2)
> 

NULL

Download PDEtools_Solve_independentof.mw

Edgardo S. Cheb-Terrab

Physics, Differential Equations and Mathematical Functions, Maplesoft

Fixed within the Maplesoft Physics Updat...

ecterrab 14560
January 06 2022
4 0

Thanks @nm for reporting the problem; the fix is available to everybody by installing the Maplesoft Physics Updates v.1128 or newer.

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

Fixed in Maplesoft Physics Updates v.112...

ecterrab 14560
December 30 2021
4 1

This is fixed, and the fix is available for everybody using Maple 2021.2 by installing the Maplesoft Physics Updates v.1125.

Details: the solution proposed by @mmcdara is correct, but will convert to the commutative :-`*` operator, not to the generalized product operator of the Physics package, that also handles anticommutative and noncommutative variables. The fix introduced converts to the generalized `*` instead of  :-`*`. And if what you wanted when entering [a, b] is their commutator, the solution by @Pascal4QM is the way to go.

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

There is no way you can 'change'...

ecterrab 14560
December 24 2021
2 1

There is no way you can 'change' the display of these differential operators. Being able to change the display would be equivalent to being able to change the display of - basically - everything else too. Although that is possible, it requires internal programing of the Typesetting routines. Not something for everyone.

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

The problem is in solve...

ecterrab 14560
December 24 2021
3 0

The following, attempted to transform the implicit solution in explicit, hangs:

> solve(1/tan(1/2*ln(y(x)/x))-ln(x)-_C1, y(x));

I am tracking this in the bugs' database.

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

This is how you do it...

ecterrab 14560
December 22 2021
6 0

restartode := diff(w(z), z, z)+f(z)*(diff(w(z), z))+g(z)*w(z) = 0

diff(diff(w(z), z), z)+f(z)*(diff(w(z), z))+g(z)*w(z) = 0

 (1)

transformation := eta = int(exp(-(int(f(z), z))), z)

eta = int(exp(-(int(f(z), z))), z)

 (2)

If the above is the new independent variable eta, then in the terminology used in the help page PDEtools:-dchange your inverse transformation is of the form

ITR := {transformation, v(eta) = w(z)}

{eta = int(exp(-(int(f(z), z))), z), v(eta) = w(z)}

 (3)

The problem with tthe above is that you do not have a simple representation for the direct transformation; that is: you don't know how to express z = F(eta). The best algebraic representation you have for F(eta)is using RootOf, so that the complete transformation you need to use is

TR__complicated := PDEtools:-Solve(ITR, {z, w(z)})

{z = RootOf(eta-Intat(exp(-(Int(f(_b), _b))), _b = _Z)), w(z) = v(eta)}

 (4)

You can us this one, but don't expect something simple right away

PDEtools:-dchange(TR__complicated, ode, [eta, v(eta)], known = all, simplify)

g(RootOf(eta-Intat(exp(-(Int(f(_b), _b))), _b = _Z)))*v(eta)+(diff(diff(v(eta), eta), eta))*exp(-2*Intat(f(_b), _b = RootOf(eta-Intat(exp(-(Int(f(_b), _b))), _b = _Z)))) = 0

 (5)

In your post, however, you mix variables by expressing g as a function of z, not eta, and ditto for the exponential. To do that, select the first transformation equation in TR__complicated

TR__complicated[1]

z = RootOf(eta-Intat(exp(-(Int(f(_b), _b))), _b = _Z))

 (6)

subs((rhs = lhs)(z = RootOf(eta-Intat(exp(-(Int(f(_b), _b))), _b = _Z))), g(RootOf(eta-Intat(exp(-(Int(f(_b), _b))), _b = _Z)))*v(eta)+(diff(diff(v(eta), eta), eta))*exp(-2*Intat(f(_b), _b = RootOf(eta-Intat(exp(-(Int(f(_b), _b))), _b = _Z)))) = 0)

v(eta)*g(z)+(diff(diff(v(eta), eta), eta))*exp(-2*Intat(f(_b), _b = z)) = 0

 (7)

And this is the result you show. You can still remove the intat

convert(v(eta)*g(z)+(diff(diff(v(eta), eta), eta))*exp(-2*Intat(f(_b), _b = z)) = 0, Int)

v(eta)*g(z)+(diff(diff(v(eta), eta), eta))*exp(-2*(Int(f(z), z))) = 0

 (8)

 

NULL


 

Download given_inverse_transformation_of_the_independent_variable.mw

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

Physics:-Vectors:-ChangeCoordinates...

ecterrab 14560
November 18 2021
3 3
> 

with(Physics); with(Vectors)
> 

sys := {diff(x(t), t) = y(t), diff(y(t), t) = -x(t)}

{diff(x(t), t) = y(t), diff(y(t), t) = -x(t)}

 (1)

This is perhaps what you are looking for when saying more automatic? The system knows about these things. Note however that, in thie context of this package, what you call polar coordinates is the 2D (x,y) part of a 3D cylindrical system of coordinates. So,

> 

ChangeCoordinates(sys, cylindrical)

{(diff(rho(t), t))*cos(phi(t))-rho(t)*(diff(phi(t), t))*sin(phi(t)) = rho(t)*sin(phi(t)), (diff(rho(t), t))*sin(phi(t))+rho(t)*(diff(phi(t), t))*cos(phi(t)) = -rho(t)*cos(phi(t))}

 (2)
> 

solve({(diff(rho(t), t))*cos(phi(t))-rho(t)*(diff(phi(t), t))*sin(phi(t)) = rho(t)*sin(phi(t)), (diff(rho(t), t))*sin(phi(t))+rho(t)*(diff(phi(t), t))*cos(phi(t)) = -rho(t)*cos(phi(t))}, diff(({phi, rho})(t), t))

{diff(phi(t), t) = -1, diff(rho(t), t) = 0}

 (3)

By the way, the documentation of the Physics:-Vectors package looks clear and complete ... if you feel it doesn't please post some suggestions (thanks) and I will take them a look.

Download ChangeCoordinates.mw

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

Physics,Tensors...

ecterrab 14560
November 14 2021
4 0

 

with(Physics)

g_[sc]

Physics:-g_[mu, nu] = Matrix(%id = 36893488152178629316)

 (1)

Riemann[alpha, beta, mu, nu]^2

Physics:-Riemann[alpha, beta, mu, nu]*Physics:-Riemann[`~alpha`, `~beta`, `~mu`, `~nu`]

 (2)

SumOverRepeatedIndices(Physics[Riemann][alpha, beta, mu, nu]*Physics[Riemann][`~alpha`, `~beta`, `~mu`, `~nu`])

48*m^2/r^6

 (3)

Weyl[alpha, beta, mu, nu]^2

Physics:-Weyl[alpha, beta, mu, nu]*Physics:-Weyl[`~alpha`, `~beta`, `~mu`, `~nu`]

 (4)

SumOverRepeatedIndices(Physics[Weyl][alpha, beta, mu, nu]*Physics[Weyl][`~alpha`, `~beta`, `~mu`, `~nu`])

48*m^2/r^6

 (5)

For things like this, or more details, or in general related to tensors, see the help page Physics,Tensors , it is a well organized help page with sections, subsections and examples.

NULL


 

Download Riemann_and_Weyl_SumOverRepeatedIndices.mw

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

see ?simplify,siderels...

ecterrab 14560
November 14 2021
3 2
This is a portion of your worksheet, showing the operation that I think resolves your problem.
> 

restart
> 

with(Physics)
> 

Setup(mathematicalnotation = true)

[mathematicalnotation = true]

 (1)
> 

Setup(noncommutativeprefix = e)

[noncommutativeprefix = {e}]

 (2)
> 

p := Commutator(e[0], e[1])

Physics:-Commutator(e[0], e[1])

 (3)
> 

expand(p)

Physics:-`*`(e[0], e[1])-Physics:-`*`(e[1], e[0])

 (4)

Suppose you want to eliminate portions of it in favor of something else. That includes, for instance, going back the operation. Here is one somewhat general way of accomplishing that. In steps, you want to use

> 

(%Commutator = `@`(expand, Commutator))(e[0], e[1])

%Commutator(e[0], e[1]) = Physics:-`*`(e[0], e[1])-Physics:-`*`(e[1], e[0])

 (5)

to eliminate e[0]*e[1]. You can do this

> 

Physics[`*`](e[0], e[1])-Physics[`*`](e[1], e[0])

Physics:-`*`(e[0], e[1])-Physics:-`*`(e[1], e[0])

 (6)
> 

simplify(Physics[`*`](e[0], e[1])-Physics[`*`](e[1], e[0]), {%Commutator(e[0], e[1]) = Physics[`*`](e[0], e[1])-Physics[`*`](e[1], e[0])}, {e[0]*e[1]})

%Commutator(e[0], e[1])

 (7)

In that way, for example, your

> 

H[2] := a^2*(e[0]*e[1]-e[0]*e[1])*zeta[2]

a^2*zeta[2]*(Physics:-`*`(e[0], e[1])-Physics:-`*`(e[1], e[0]))

 (8)

becomes

> 

simplify(H[2], {%Commutator(e[0], e[1]) = Physics[`*`](e[0], e[1])-Physics[`*`](e[1], e[0])}, {e[0]*e[1]})

a^2*zeta[2]*%Commutator(e[0], e[1])

 (9)
> 

``


NOTE AFTER ANSWERING: You can also use the command Physics:-SortProducts  with the optional argument usecommutator; see the related help page.

Download Drinfieldstuff_display_(reviewed).mw

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

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