The diffgeom Module – Status Report

I have written already a few posts about the theory behind the module, the structure of the module, etc. However, besides some rare examples, I have not described in much details how the work progresses. So here is a short summary (check the git log for more details):

  • The basics about coordinate systems and fields are already in. There are numerous issues with all the simplify-like algorithms inside SymPy, however they are slowly ironed out.
  • Some simplistic methods for work with integral curves are implemented.
  • The basics of tensor/wedge products are in. Many simplification routines can be added. Contraction between tensor products and vectors is possible (special case of “lowering of an index”).
  • Over-a-map, pushforwards and pullbacks are not implemented yet.
  • Instead of them I have focused my work on derivatives and curvature tensors. For the moment work on these can be done in a limited coordinate-dependent way. A longer post explaining the theory is coming and with it an implementation slightly less dependent on coordinates (working with Christoffel symbols is a pain).
  • Hodge star operator – still not implemented.

An example that I want to implement is a theorem that in irrotational cosmology isotropy implies homogeneity. Doing this will be the first non-trivial example in this module.

A serendipitous detour from the project was my work on the differential equations solver. Aaron had implemented a very thorough solver for single equations. I had tried to extend it in a few simple ways in order to work with systems of ODEs and initial conditions. However this led me to Jordan forms of matrices, generalized eigenvectors and a bunch of interesting details on which I work in my free time (especially this week).

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2 Comments

  1. Thank you for your great work. Could you give some examples of what you aim to make possible with the solver? I am especially interested in what surfaces you will be able to generate.

    Reply
    • Most of the work is focused on application to general relativity. However if you have ideas about what can be implemented feel free to point me to any references. For the moment I have not thought much about any helper functions related to generating equations of surfaces.

      Reply

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