Best way to explore Finsler Geometry calculations with Maple?

• Maple
• strangerep
In summary, the conversation discusses the use of Finsler geometry with Cartan connection and introduces various auxiliary definitions such as the Minkowski metric, unit velocity vector, local time direction, and a constant with dimensions of inverse length. The main focus is on the computation of various tensor components, including the fundamental function and the inverse matrix. The speaker is seeking suggestions on the best programming paradigm or symbolic math package to use for their particular, more complicated case. Some potential options, such as the "New Finsler" package for Maple and a Mathematica package, are also mentioned.

strangerep

Science Advisor
TL;DR Summary
I need to perform some highly tedious calculations in Finsler-like geometry, at least an order of magnitude more complicated than Riemannian geometry. I'm proficient (A-level) in both Finslerian and Riemannian geometries (and in C++), but I'm B-level in Maple. I'm hoping for suggestions as to the best way to approach Finslerian computations in Maple.
My context here is Finsler geometry with Cartan connection. I use ##x^\mu## for the usual spacetime position coordinates, and ##u^\mu \equiv \dot x^\mu## for velocity coordinates (the overdot denotes differentiation by an arbitrary parameter, not necessarily proper time). To explain the problem I need to introduce some further auxiliary definitions:

## \eta_{\mu\nu}## is an ordinary Minkowski metric , of signature ##(-1,+1,+1,+1)~,##
## A := \sqrt{-\eta_{\mu\nu} u^\mu u^\nu} ~,##
## \hat u^\alpha := u^\alpha/A ~,## is a "unit" velocity vector,
##\epsilon := \hat u^0 \equiv u^0/A ~## is called the local time direction of an observer at ##x~,##
##\Upsilon## is a constant with dimensions of inverse length ,
##\rho := (1 - \epsilon\Upsilon x^0)## is just a convenient shorthand.

(On the ordinary mass hyperboloid, ##\epsilon## is just sign##(u^0)##, but I need to perform off-shell computations, e.g., differentiations, hence I need the more complicated expression above.)

In my particular Finsler geometry, the fundamental function is given via
$$L(x,u) \equiv F^2(x,u) ~:=~ \frac{\eta_{00} + \eta_{jk}\epsilon^2 \Upsilon^2 x^j x^k}{\rho^4}\, (u^0)^2 ~+~ \frac{2\epsilon\Upsilon \eta_{jk} x^j\, u^k u^0}{\rho^3} ~+~ \frac{\eta_{jk} u^j u^k}{\rho^2}.$$
The series of computations I need to perform are as follows:
$$g_{\mu\nu} ~=~ \frac12 \, \frac{\partial^2 L}{\partial u^\mu \partial u^\nu} ~,$$ and ##g^{\mu\nu}## as the usual inverse matrix thereof. Then I need:
$$G_\alpha ~:=~ \frac12\left( \frac{\partial^2 L}{\partial x^\nu \partial u^\alpha}\, u^\nu ~-~ \frac{\partial L}{\partial x^\alpha} \right) ~,~~~~~~ G^\lambda ~:=~ g^{\lambda\alpha} G_\alpha ~,$$ $$N^\lambda_{~\nu} ~:=~ \frac{\partial G^\lambda}{\partial u^\nu} ~,~~~~~~ B^\lambda_{~\nu\sigma} ~:=~ \frac{\partial N^\lambda_{~\nu}}{\partial u^\sigma} ~,$$ $$R^\lambda_{~\nu} ~=~u^\mu \frac{\partial N^\lambda_{~\nu}}{\partial x^\mu} - 2 \frac{\partial G^\lambda}{\partial x^\nu} + N^\sigma_{~\nu} N^\lambda_{~\sigma} - 2 G^\sigma B^\lambda_{~\nu\sigma} ~.$$
In an older Finsler textbook, I've seen Maple programs for simpler cases that just use explicit coordinates ##t,x,y,z## and velocities ##T,X,Y,Z##. All tensor components and contractions are written out in full. But my case is much more complicated.

I'd appreciate suggestions on what programming paradigm I should try here. Explicit components? Matrices? Other?

Or other symbolic math packages likely to be better than Maple for my problem?

I'm aware a Maple package "New Finsler", as described in this paper by Youssef and Elgendi, but they seem not to say where I can download it. I've attempted to contact them, but so far no response.

I'm also aware of a Mathematica package for Finsler geometry, but it seems to be limited to the case of a homogeneous Finsler metric.Any suggestions would be much appreciated. Thank you all in advance!

1. What is Finsler Geometry?

Finsler Geometry is a branch of mathematics that studies geometric structures on differentiable manifolds. It is an extension of Riemannian Geometry, which studies smooth curved spaces, by allowing the metric tensor to vary from point to point on the manifold.

2. How can Maple be used to explore Finsler Geometry calculations?

Maple is a powerful mathematical software that can perform complex calculations and graphing. It has built-in functions and packages specifically designed for Finsler Geometry, making it an ideal tool for exploring and visualizing this mathematical concept.

3. What are the advantages of using Maple for Finsler Geometry calculations?

One of the main advantages of using Maple for Finsler Geometry calculations is its ability to handle symbolic calculations, allowing for more flexibility and accuracy. Additionally, Maple's user-friendly interface and extensive documentation make it easy to learn and use for mathematical research.

4. Can Maple be used to visualize Finsler Geometry?

Yes, Maple has built-in functions for graphing Finsler manifolds and geodesics. These visualizations can help researchers gain a deeper understanding of the geometric structures and properties of Finsler manifolds.

5. Are there any resources available for learning how to use Maple for Finsler Geometry calculations?

Yes, there are various online tutorials, forums, and textbooks available that provide step-by-step instructions on how to use Maple for Finsler Geometry calculations. Additionally, Maple's official website offers documentation and examples for using its Finsler Geometry functions and packages.