- #1
Tangent87
- 148
- 0
Say we have a Hamiltonian H(q,p,t) and we then transform from p and q to P=P(q,p,t) and Q=Q(q,p,t), with:
[tex]P\dot{Q}-K=p\dot{q}-H+\frac{d}{dt}F(q,p,Q,P,t)[/tex]
where K is the new Hamiltonian. How do we show that P and Q obey Hamilton's equations with Hamiltonian K? I have tried partial differentiating both sides of the above w.r.t Q and P but I'm not sure what to differentiate (i.e. do we consider p and q independent from P and Q?). I also expanded the big dF/dt but it didn't seem to help.
[tex]P\dot{Q}-K=p\dot{q}-H+\frac{d}{dt}F(q,p,Q,P,t)[/tex]
where K is the new Hamiltonian. How do we show that P and Q obey Hamilton's equations with Hamiltonian K? I have tried partial differentiating both sides of the above w.r.t Q and P but I'm not sure what to differentiate (i.e. do we consider p and q independent from P and Q?). I also expanded the big dF/dt but it didn't seem to help.