We introduce a version of the Hamiltonian formalism based on the Clairaut equation theory, which allows us a self-consistent description of systems with degenerate (or singular) 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 initial 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.
