Physics 321

1. Physics 321 Hour 20 Hamiltonians

2. whereas • For simple systems, H = T + U • There are two first order equations of motion for each variable: • The Lagrangian method gives one second order equation for each variable. The Hamiltonian

3. Take Therefore The Hamiltonian - Origins

4. Let If has no explicit time dependence, then H is a conserved quantity. The Hamiltonian - Origins

5. The Hamiltonian is a function of p and q. But p is not ‘the momentum,’ it is the generalized momentum conjugate to q. • The general expression is: • That means you generally have to find the Lagrangian before you can find he Hamiltonian! The Hamiltonian - Notes

6. The Hamiltonian is H=T+U unless • The Lagrangian has explicit time dependence • The transormation between the coordinatiates and Cartesian coordinates have explicit time dependence • But … you have to write the Hamiltonian in terms of the correct generalized momenta – so you usually still need the Lagrangianfirst! The Hamiltonian - Notes

7. hamilton.nb Example