6. The general case.

We define the n canonical momentums 


for the general action integral


If we assume that


we can via the implicit function theorem solve the system of momentums for the variables


and get

We now define the Hamiltonian by


By using similar arguments as in the previous part we get Hamilton’s equations:


This is a system of 2n first order ordinary differential equations in the unknown funktions y1,y2,…,yn, and the unknown momentums p1,p2,…,pn. The system contains the same information as the corresponding n second order Euler-Lagrange’s equations