Boltzmann-Hamel equations, quick way of generating Hamel coefficients?

[TheFerruccio]

Most of this post is an overview. The actual question is at the end. Basically: If I am given crazy summations with lots of indices cycling through multiple matrices, is there a quick way I can implement this in Mathematica?


So, I am doing a little bit of research into nonholonomic constraints in Lagrange equations, and I ran across this particular topic. It seems a lot more rigorous and standardized than trying to seek out Lagrange multipliers. It involves finding a matrix $\Psi$ which converts generalized velocities into instantaneous (quasi-) velocities, for the coordinates which there exist nonholonomic constraints.

If you have a system whose generalized coordinates are mostly independent of each other, this matrix will be mostly diagonal populated with 1's. However, for the coordinates that have nonholonomic constraints attached to them, nondiagonal terms show up. This makes perfect sense.

However, finding the Hamel Coefficients is an absolute dog of a task. Once I find the Hamel coefficients, then everything works itself out pretty nicely.

These are the equations in question. The first equation is the Boltzmann-Hamel equation for the r'th generalized quasicoordinate. Solving this equation shows how the r'th coordinate evolves over time. $\overline{Q}_r$ is the force acting in the generalized quasicoordinate direction, and $\tilde{n}$ is the number of generalized quasicoordinates. The number of equations needed to fully describe the evolution of the entire system is $\tilde{n}-\tilde{c}$ where $\tilde{c}$ is the number of nonholonomic constraints, so the index r for the equation only goes up to $\tilde{n}-\tilde{c}$.

I need some good method for generating the $\gamma$ Hamel coefficients. They act on an $\tilde{n}\times\tilde{n}$ matrix $\Phi$ that is the inverse of $\Psi$

If I have the matrix $\Phi$ written up in Mathematica, is there a good way that I can apply these summations in there, to quickly generate the coefficients $\gamma$ I need?

 

[TheFerruccio]

Here is some Mathematica code I used in attempt to solve for the Hamel coefficients $\gamma$

ClearAll["Global`*"];
Clear[Derivative];
S = {
   {1, 0, 0, 0, 0},
   {0, 1, 0, 0, 0},
   {0, 0, 1, 0, 0},
   {0, 0, -a, Cos[\[Phi]], Sin[\[Phi]]},
   {0, 0, 0, -Sin[\[Phi]], Cos[\[Phi]]}
  };
P = Inverse[S];
q = {\[Phi], \[Theta], \[Psi], u, v};
gu = IdentityMatrix[5];
gv = IdentityMatrix[5];
For[r = 1, r < 5, r++, 
 For[l = 1, l < 5, l++, 
  gu[[{r}, {l}]] = 
   Sum[Sum[(D[S[[{4}, {j}]], q[[{k}]]] - 
        D[S[[{4}, {k}]], q[[{j}]]]) P[[{k}, {l}]] P[[{j}, {r}]], {k, 
      1, 5}], {j, 1, 5}]]]
For[r = 1, r < 5, r++, 
 For[l = 1, l < 5, l++, 
  gv[[{r}, {l}]] = 
   Sum[Sum[(D[S[[{5}, {j}]], q[[{k}]]] - 
        D[S[[{5}, {k}]], q[[{j}]]]) P[[{k}, {l}]] P[[{j}, {r}]], {k, 
      1, 5}], {j, 1, 5}]]]
Simplify[gu] // TableForm
Simplify[gv] // TableForm

The two For loops are where most of the information is stuffed. I would have preferred to have a single triple embedded For loop, but I do not know how Mathematica deals with higher dimensional arrays. I did not thoroughly skim through the help section enough.

These successfully generated two layers of the 3-dimensional $\gamma$ tensor, focusing on where I wished to solve the problem at hand: index 4 and 5. S and P are the matrices $\Psi$ and $\Phi$, respectively.