.
If we make a canonical transformation with a generating function F2(q,P,t)
such that Assume we have a system with a Hamiltonian .
If we make a canonical transformation with a generating function F2(q,P,t)
such that
and 
we obtain the Hamilton-Jacobi equation. The function F2 is called Hamilton’s principal function S, and the Hamilton-Jacobi equation is
.
Since K=0,
,
i.e. all new variables are constant in time. Therefore 
, where the a’s are the
transformed momenta. We have
.
To solve a problem using the Hamilton-Jacobi method
 | solve the Hamilton-Jacobi equation  . |
| The a’s are the transformed momenta.
Find the
transformed coordinates from  |
| These equations give the coordinates q as a function of 2n constants a and b.
Solve for q. Solve for p
by differentiating  |
If H=H(q,p), then
Here a1 is a constant, often it is the energy. W(q,a) is called Hamilton’s characteristic function. W(q,a) is a generating function for its own canonical transformation.
To solve a problem using the Hamilton’s characteristic function
| solve  . |
| The a’s are the transformed momenta.
Find the
transformed coordinates from  |
| These equations give the coordinates q as a function of 2n constants a and b.
Solve for q. Solve for p
by differentiating 
|
Assume a conservative system with one degree of freedom which has periodic motion.
Let H(q,p)=a, p=p(q,a).
Define
The integration is over one complete period of the motion.
J
is called the action variable, and it is taken
to be the new constant momentum in Hamilton’s principal function 
Now we look at the canonical transformation generated by W(q,J).
J is the new constant momentum and w the new
coordinate. We have
w is called the angle variable.
n is the
frequency associated with the periodic motion. We can find n
without finding a complete solution of the problem.
