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Subsections
In Hamiltonian mechanics we use generalized momentum in place of
velocity as a coordinate. The generalized momentum is defined in terms
of the Lagrangian and the coordinates (q,) :

p = .
 (21)

The Hamiltonian is defined in terms of the Lagrangian
as follows:

H(p,q) = p  L(q,)
 (22)

where in the above equation is replaced with a function of
(p,q) by solving the definining equation 21
with respect to . We note that this task may be rather hard if
the dependence of L on is complicated. Fortunately,
in most interesting siuations, L is quadratic in
(i.e. T . the kinetic energy is a quadratic function of the
velocity). Thus, the equation for in terms of (p,q) is
linear.
The Lagrangian equation of motion 5 becomes a
pair of equations known as the Hamiltonian system of equations:
The second equation of this system is easy to explain. It is simply
the Lagrangian equation 5.
The second equation can be explained by using duality. For the sake
of this argument we need to assume that L(q,) is strictly
convex as a function of . It is sufficient to assume that the
Hessian

 (24)

is positive definite. We note that for Newtonian equations this matrix
is I . The function H in this case is the Legendre transform
of L , i.e.

H(p,q) = [p  L(p,)]
 (25)

where the infimum is taken over all
p .
One can show that L is a Legendre transform of H as well, i.e

L(q,) = [p  H(p,q)].
 (26)

In particular, the minimum is attained for

= .
 (27)

This is exactly the second equation of the system
23.
Next: 5 Motion of a
Up: Lagrangian and Hamiltonian mechanics
Previous: 3 The variational principle
Marek Rychlik
9/2/1997