  An extension of the Legendre transform to non-convex functions with vanishing
Hessian as a mix of envelope and general solutions of the Clairaut equation is
proposed. Applying this to systems with constraints, the procedure of finding a
Hamiltonian for a degenerate Lagrangian is just that of solving a corresponding
Clairaut equation with a subsequent application of the proposed
Legendre-Clairaut transformation. In this way the unconstrained version of
Hamiltonian equations is obtained. The Legendre-Clairaut transformation
presented is involutive. We demonstrate the origin of the Dirac primary
constraints, along with their explicit form, and this is done without using the
Lagrange multiplier method.
