Can sympy do tensor based Euler-Lagrange?

[Joseph Smidt]

Hey everyone,

     From documentation I see sympy has a tensor package (awesome) and I see some discussion on being able to calculate Eular-Lagrange equations. However, I was wondering if sympy can calculate tensor based Euler-Lagrange equations of motion. 

    For, example, let's take the example of electricity and magnitism [1].  Here our Lagrangian = 1/4 F_uv F^uv - A_a J^a. Here F_uv is the maxwell tensor, A_a is the E&M vector potential and J^a is the source of that potential.  Varying this action with respect A_a gives:

d(F^uv)dx_u = J^v, the standard E&M equation of motion.   See the wikipedia link below for a better description.

   Does anyone know if sympy is equipped to handle tensor based Eular-Lagrange equations equations such as these? If so, how would I go about solving for this E&M example? Thanks.

[1] http://en.wikipedia.org/wiki/Covariant_formulation_of_classical_electromagnetism#Lagrangian_for_classical_electrodynamics

[Alan Bromborsky]

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For an alternate approach to tensors see sections 8.2.4 - 8.2.6 in attached pdf.  Also see chapter 7 in attached for alternate treatment of tensors.

[Joseph Smidt]

Brombo,

    I appreciate that there is an alternative formulation of E&M but I am wondering is sympy has the capability to do these calculations? If so, is there any relevant documentation or examples in sympy you could point to or provide so that I can see?  Thanks.

[F. B.]

On Monday, March 10, 2014 7:16:41 PM UTC+1, Joseph Smidt wrote:

Brombo,

    I appreciate that there is an alternative formulation of E&M but I am wondering is sympy has the capability to do these calculations? If so, is there any relevant documentation or examples in sympy you could point to or provide so that I can see?  Thanks.

The current approach to symbolic tensors, encoded in sympy.tensor.tensor, does not support that, nor does it support partial derivatives or operators on tensors.

[Aaron Meurer]

How similar is that to the idea from
https://github.com/sympy/sympy/issues/5858? Maybe one could be
implemented and use to make the other work?

This would all be great work for a GSoC project, by the way.

Aaron Meurer

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[Joseph Smidt]

> This would all be great work for a GSoC project, by the way. 

This would be a great project if it could be done because in the theoretical physics world deriving equations of motian from tensorial actions (Like this E&M example)  is very common and can lead to pages of algebra.

Thanks.

[Sergey Kirpichev]

On Monday, March 10, 2014 9:24:27 PM UTC+4, Joseph Smidt wrote:

   Does anyone know if sympy is equipped to handle tensor based Eular-Lagrange equations equations such as these?

Yes, with euler_equations from calculus module.  But it doesn't support tensor notation directly.

[Alan Bromborsky]

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If you are talking about Lagrangian field theory the question is how do you take a derivative with respect to the field and the gradient of the field and not with respect to the position vector the field is a function of.

[Sergey Kirpichev]

On Thursday, March 13, 2014 7:29:40 PM UTC+4, brombo wrote:

If you are talking about Lagrangian field theory the question is how do you take a derivative with respect to the field and the gradient of the field and not with respect to the position vector the field is a function of.

The docstring of euler_equation function has a trivial example of scalar field.  There should not be any problems for more complex field (e.g. for electrodynamics).