Geometric Mechanics

The Cartan development takes a Lie algebra valued 1-form satisfying the Maurer–Cartan equation on a simply connected manifold $M$ to a smooth mapping from $M$ into the Lie group. In this paper, this is generalized to infinite dimensional $M$ for infinite dimensional regular Lie groups. The Cartan development is viewed as a generalization of the evolution map of a regular Lie group. The tangent mapping of a Cartan development is identified as another Cartan development.

The main result of this paper establishes a new polynomial relation among several projections of the angular momentum vector on the spatial and moving frames involved in the rotational dynamics of rigid bodies. This relation is surprisingly simple to derive and provides a geometric insight into the attitude dynamics of bodies with axial symmetry, explaining the bounds for the nutation angle $𝜃$.

In this paper, we study coisotropic reduction in multisymplectic geometry. On the one hand, we give an interpretation of Hamiltonian multivector fields as Lagrangian submani- folds and prove that $k$-coisotropic submanifolds induce a Lie subalgebra in the algebra of Hamiltonian $(k-1)$-forms, similar to how coisotropic submanifolds in symplectic geometry induce a Lie subalgebra under the Poisson bracket. On the other hand, we extend the classical result of symplectic geometry of projection of Lagrangian submanifolds in coisotropic reduction to bundles of forms, which naturally carry a multisymplectic structure.