Hamilton equations for gauge-invariant problems

The Hamiltonian and Lagrangian formalisms in field theories are equivalent when the Lagrangian is regular. Nevertheless, there are many interesting instances where regularity is not guaranteed. This is the case of those variational problems on connections invariant under the action of the gauge group. The Yang-Mills Lagrangian is the best known example of this situation. The goal of the talk is to show in this situation the fibered nature of the set of solutions of the Hamilton equations over the set of solutions of the Euler-Lagrange equations. Moreover, this structure is studied for the Jacobi fields and the moduli spaces under the gauge groups. Some physical considerations will be also analyzed.