Varying an action in components xCoba

[Sergi Sirera-Lahoz]

Hi all,

I'm having problems using VarD and VarAction (defined in the notebook ActionVariation_Metric_Fields.nb by Cyril Pitrou) when the lagrangian is defined in terms of the components from xCoba.

I think the problem reduces to applying a variation to an expression with different scalar functions and their derivatives.

VarD and VarAction seem to recognize only tensors. I've tried making a rule to write the scalar functions to "scalar tensors" (i.e. Tensor with no indices). This works for the scalar functions but not for the derivatives of the scalar functions. I haven't been able to write a rule that converts derivatives of a function in a specific basis to derivatives of a tensor that works with VarD or VarAction. I think both use ToCanonical which doesn't work well with partial derivatives.

I'm attaching a simplified notebook that shows the issue but I'm happy to share the notebook with the full problem if this is too simplified. Please let me know if there's something very basic that I'm missing.

Best wishes,

Sergi

[Sergi Sirera-Lahoz]

Hello again,

I found a promising solution that seems to work. It involves defining a normalized vector in the r-direction and substituting derivatives wrt r for a contraction of this vector and a PD. This way, VarD knows how to deal with the derivative. I attach here the notebook. A simplified example can be found in the section Toy Example. You can ignore the rest.

Best wishes,

Sergi

[Jose]

Hi,

Yes, by construction VarD only works on scalars. Indeed, the simplest way to bypass this is to contract anything else with some arbitrary vector field(s) and then remove (somehow) this vector field (or fields) at the end. Of course, you need to decide how to do this removal. I don't know of a general way to do it that is valid for all cases. Derivatives and symmetry arguments may play a role.

Cheers,

Jose.