ssume that the function
solves the Euler-Lagrange equation. We introduce the
function
Assume further that the
equation

(generalized impulse equation)

has a unique smooth
solution

(generalized impulse solution)

It follows
that
consequently
We define a Hamiltonian associated with the Lagrangian
as
follows:

Claim

Under assumption that the
solves the EL equation (
Euler Lagrange
equation
) and the
and
are defined as above, the
and
solve the Hamilton
equations:
and

Proof

We have the
relationships
satisfied at
and the
definition
We calculate the derivatives
,
and
at
accordingly: