  It is shown that any singular Lagrangian theory: 1) can be formulated without
the use of constraints by introducing a Clairaut-type version of the
Hamiltonian formalism; 2) leads to a special kind of nonabelian gauge theory
which is similar to the Poisson gauge theory; 3) can be treated as the
many-time classical dynamics. A generalization of the Legendre transform to the
zero Hessian case is done by using the mixed (envelope/general) solution of the
multidimensional Clairaut equation. The corresponding system of motion
equations is equivalent to the Lagrange equations and has a linear algebraic
subsystem for "unresolved" velocities. Then the equations of motion are written
in the Hamilton-like form by introducing new antisymmetric brackets. This is a
"shortened" formalism, since it does not contain the "nondynamical"
(degenerate) momenta at all, and therefore there is no notion of constraint. It
is outlined that any classical degenerate Lagrangian theory (in its
Clairaut-type Hamiltonian form) is equivalent to the many-time classical
dynamics. Finally, the relation between the presented formalism and the Dirac
approach to constrained systems is given.
