Variational principle in Hamiltonian mechanics

Special features of the variational principle in Hamiltonian mechanics, that is the problem of covariant formulation and boundary conditions, are analyzed. The geometric setting of Hamiltonian mechanics presumes the existence of an invariant action with given symplectic matrix ωμv. Action is obtained from Lagrangian mechanics, which differs fundamentally from Lagrangian mechanics. The initial point of Lagrangian mechanics is specified by coordinates and velocities. The variational principle with action and corresponding boundary conditions gives the equation of motion with standard symplectic invariant. Invariants can be used, which requires twice as many independent boundary conditions on coordinates and momenta as when action is required. Thus, in Hamiltonian mechanics, the variational principle must be treated with taking into account the specifics of this mechanics.