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

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

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

1-D Case General 1-D motion solvable Equations of motion

Simple Oscillator Lagrangian Generalized momentum Also called the canonical momentum Hamiltonian

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

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

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

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

Example: Mass on a Cone

Lagrangian and Hamiltonian

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

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