Euler-Poincaré Equation for Lie Groups with Non Null Symplectic Cohomology. Application to the Mechanics

Abstract

Let G be a symplectic group on a symplectic manifold $\mathcal {N}$. To any momentum map $\psi : \mathcal {N} \rightarrow \mathfrak {g}^*$, one can associate a class of symplectic cohomology cocs. It does not depend on the choice of the momentum map but only on the structure of the Lie group G. It forwards an affine left action $\mu = a \cdot \mu ' = Ad^* (a)\, \mu ' + cocs (a)$ of G on $\mathfrak {g}^*$.

If G is a Lie subgroup of the affine group, one can define an associated affine connection as a field of $\mathfrak {g}$-valued 1-forms $\tilde{\varGamma }$ on a G-principal bundle of affine frames $\pi : \mathcal {F} \rightarrow \mathcal {M}$. Let $\omega$ be a smooth field of 2-form on $\mathfrak {g}^* \times \mathcal {F}$ defined by: $(\mu , f) \mapsto \frac{1}{2}\, d \mu \wedge \tilde{\varGamma }$.

On each orbit $\varvec{\mu }$, $\omega$ is the pull-back of Kirillov-Kostant-Souriau symplectic form by the projection $\psi _{\varvec{\mu }}: (\mu , f) \mapsto \mu$. The G-principal bundle $\mathfrak {g}^* \times \mathcal {F}$ is a presymplectic bundle of symplectic form $\omega$ and $\psi _{\varvec{\mu }}$ is a momentum map.

The equation of motion $d(\mu , f)\in Ker (\omega )$ expresses the fact that the momentum is parallel-transported. It generalizes Euler-Poincaré equation when the class of symplectic cohomology of the group is not null, especially for the important case of Galileo’s group.