Hamilton's Canonical Equations of motion:-
- Consider the transformations
- For Qj and Pj to be canonical they should be able to be expressed in Hamiltonian form of equations of motion i.e.,
- Qj and Pj to be canonical must also satisfy modified Hamilton's principle i.e.,
- Using same principle for old set qj and pj
- Term ∂F/∂t in 1 contributes to the variation of the action integral only at end points and will therefore vanish if F is a function of (q,p,t) or (Q,P,t) or any mixture of phase space co-ordinates since they have zero variation at end points.
- F is useful for specifying the exact form of anonical transformations only when half of the variables (except time) are from the old set and half from the new set.
- F acts as bridge between two sets of canonical variables and is known as generating function of transformations.