'FALLING CAT' CONNECTIONS CONSTRUCTED FROM THE MOMENTUM MAP

Mari\'an FECKO

1991 Math. Subject Classification: 70D10, 58F05, 53B15.

Key words: Lagrangian system with symmetry on $TM$, connection on principal
        bundle, momentum map.

We consider a standard symplectic dynamics on $TM$ generated by a natural
Lagrangian $L$. The Lagrangian is assumed to be invariant with respect
to the action $TR_g$ of a Lie group $G$ lifted from the free and proper
action $R_g$ of $G$ on $M$. It is shown that under these conditions
a connection on principal bundle $\pi : M \rightarrow M/G$ can be
constructed based on the momentum map corresponding to the action $TR_g$.
A simple explicit formula for the connection form is given. For the
special case of the standard action of $G$ = SO(3) on
$M$ = ${\Bbb R}^3 \times \dots \times {\Bbb R}^3$ corresponding to a
rigid rotation of a N-particle system the formula obtained earlier
by Guichardet and Shapere and Wilczek is reproduced.