SimpleSystems.m

Andrew D. Lewis

16 August 1998

1 SimpleSystems.m for simple mechanical systems with symmetry

If you do not know about simple mechanical systems with symmetry, look at, for example, [1]. We are concerned here with things like connection forms, locked inertia tensors, etc.

2 What do you get?

Well, something like this cannot be perfect. It does require some setting up before hand. But hopefully it will be useful. The first thing you need to do is set up your coordinates in a local trivialisation properly. We assume the group acts freely and properly on Q so we have a nice principal bundle Q → Q/G. The first dim(Q) − dim(G) coordinates should locally parameterise the base space. The last dim(G) coordinates should parameterise the group fibres. Let us denote these coordinates by (x^{1}, …, x^{n−r}, y^{1}, …, y^{r}); thus dim(Q) = n and dim(G) = r. We also need to choose a basis {ξ_{1}, …, ξ_{r}} for the Lie algebra of G. This must be done in conjunction with the choice of coordinates (y^{1}, …, y^{r}). Specifically, choose the coordinates (y^{1}, …, y^{r}) so that (0, …, 0) corresponds to the identity in the group in the local trivialisation. Further, you need to have

ξ_{a,Q}(x^{1}, …, x^{n−r}, 0, …, 0) = \frac{∂}{∂y^{a}} .

In simple examples, this is not so bad… In any case, it has to be done for the commands in SimpleSystems.m to make sense.

Okay, so here are the functions in alphabetical order. Certain notation holds throughout for variable names.

1. Adg is the matrix for the adjoint representation in the chosen basis {ξ_{1}, …, ξ_{r}} for the Lie algebra of G. This will be a function of the group coordinates (y^{1}, …, y^{r}) and so is problematic.

2. g are the components of the Riemannian metric.

3. InfGen is an r × n matrix whose ath column are the components for the ath infinitesimal generator.

4. LASC are the Lie algebra structure constants for the basis {ξ_{1}, …, ξ_{r}} for the Lie algebra of G.

5. x are the base space coordinates.

I’m ready now.

ConnectionForm[HBun,InfGen] HBun is the basis for the horizontal distribution. The result is the connection form in the form of a r × n matrix, i.e., contract it with a vector field on the right, and you get the Lie algebra element corresponding to action on this vector field with the connection one-form.

EPEquations[LIT,LASC,xi,xidot] xi are the names of the components

in the Lie algebra relative to the basis {ξ_{1}, …, ξ_{r}}, and xidot are the names for their time-derivatives. The result is an r vector containing the Euler-Poincaré equations for the Lagrangian on the Lie algebra formed using the inner product LIT, the locked inertia tensor

HorizontalBundle[g,InfGen] This returns an (n − r) × n matrix whose columns are basis vectors for the horizontal subbundle for the mechanical connection.

LocalConnectionForm[Conn,Adg] This is the local connection form, and so is a r × (n − r) matrix.

CurvatureForm[Conn] Conn is the connection form, and you get back the curvature form. This the result has three indices, the first running from 1 to r, and the last two from 1 to n.

LocalCurvatureForm[Conn,LASC,x] The local form of the curvature. This

guy returns a triply indexed array, the first index running through {1, …, r}, and the last two running over {1, …, n − r}.

LocalLockedInertia[LIT,Adg] Here LIT is the locked inertia tensor, and the result is the local form of the same. Thus it is an r × r matrix function of x.

LockedInertia[g,InfGen] This is the locked inertia tensor. Thus it is an r × r matrix function of (x, y).

ReducedEquations[gred,LIT,Conn,LASC,x,xdot,xddot,xi,xidot]

gred is the Riemannian metric on the base space, LIT is the locked inertia tensor, Conn is the connection form, xdot are the names for the base space velocities, xddot are the names for the base space accelerations, xi are the names for the Lie algebra components relative to the basis {ξ_{1}, …, ξ_{r}}, ad xidot are the names of the time-derivatives of these components. The result is a vector whose first n − r components are the base space equations, and the last r are the Lie algebra equations.

ReducedMetric[g,Connloc,LITloc] Connloc and LITloc are the local

forms of the connection form and the locked inertia tensor, respectively. You get back the induced Riemannian metric on the base space.

References

[1] Jerrold E. Marsden and Tudor S. Ratiu. Introduction to Mechanics and Symmetry. Number 17 in Texts in Applied Mathematics. Springer-Verlag, New York-Heidelberg-Berlin, 1994.