Canonical formalism for SO(2) is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide with the Lie equations. It is shown that the constraints satisfy CCR. Consistency of the constrain...

Let (M,ω) be a connected, compact symplectic manifold equipped with a Hamiltonian G action, where G is a connected compact Lie group. Let φ be the moment map. In [12], we proved the following result for G = S action: as fundamental groups of topological spaces, π1(M) = π1(Mred), where Mred is the symplectic quotient at any value of the moment map φ, and = denotes “isomorphic to”. In this paper,...

3 Invariants of Hamiltonian group actions 17 3.1 An action functional . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Symplectic reduction . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Hamiltonian perturbations . . . . . . . . . . . . . . . . . . . . 24 3.4 Moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 Fredholm theory . . . . . . . . . . . . . . . . . . ...

This paper develops the Liapunov-Schmidt procedure for systems with additional constraints such as having a first integral, being Hamiltonian, or being a gradient system. Similar developments for systems with symmetry, including reversibility, are well known, and the method of this paper augments and is consistent with that approach. One of the results states that the bifurcation equation for H...