A nonlinear Hamiltonian formalism for singular Lagrangian theories

We introduce a “nonlinear” version of the Hamiltonian formalism which allows a self-consistent description of theories with degenerate Lagrangian. A generalization of the Legendre transform to the case when the Hessian is zero is done using the mixed (envelope/general) solutions of the multidimensional Clairaut equation. The corresponding system of equations of motion is equivalent to the Lagrange equations, but contains “nondynamical” momenta and unresolved velocities. This system is reduced to the physical phase space and presented in the Hamiltonian form by introducing a new (non-Lie) bracket.