  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.
