Physics 319 Classical Mechanics

1. Physics 319Classical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 23

2. Hamiltonian Mechanics • Built on Lagrangian Mechanics • In Hamiltonian Mechanics • Generalized coordinates and generalized momenta are the fundamental variables • Equations of motion are first order with time as the independent variable • In EE, very much like “state space” formalism • You will see this trick similarly in relativistic quantum mechanics when go from Klein-Gordon type of equation to Dirac Equation • Very general choices and transformations of the coordinates and momenta are allowed

3. General Procedure • Write down Lagrangian for the problem • Determine the generalized momenta • Determine the Hamiltonian function (energy for simple problems) and write it in terms of the coordinates and momenta • Solve equations of motion in Hamiltonian form

4. 1-D Case • General 1-D motion solvable • Equations of motion

5. Simple Oscillator • Lagrangian • Generalized momentum Also called the canonical momentum • Hamiltonian

6. Hamilton’s Equations of Motion • Equations of motion in Hamiltonian form are • General proof • Argument works even when the Lagrangian/Hamiltonian depends explicitly on time

7. Hamiltonian Conserved • The time dependence of the Hamiltonian function is given by • When Langrangian/Hamiltonian does not explicitly depend on time, the Hamiltonian (energy) is conserved

8. Phase Space • The set of variables describing the system in Hamiltonian form is called phase space Phase space variables for particle i • Think of the motion occurring through phase space

9. Central Force Again • Hamiltonian equations of motion • Ignorable coordinate gives conservation law. Reduction!

10. Example: Mass on a Cone

11. Lagrangian and Hamiltonian • Lagrangian • Hamiltonian

12. Equations of Motion • z direction • θdirection Conservation of angular momentum again. Centrifugal barrier at z =0 • Balanced condition

13. Ignorable Coordinates • As in Lagrangian theory independence of the Lagrangian/Hamiltonian on a coordinate guarantees a conserved quantity • In Hamiltonian theory, reduce the number of degrees of freedom in the problem • Simply evaluate conserved quantity using initial conditions and substitute into Hamiltonian • In most recent example is a perfectly good 1-D potential for the z motion, once pθis evaluated using the initial conditions