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

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

12. Equations of Motion z direction θ direction
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 pθ is evaluated using the initial conditions