set7

PHYS6001, Classical Mechanics, Fall semester 2015 Homework set 7, due Oct 29 2015

1. [10pts] Show that the Lorentz force ! can be obtained as a generalized force from the “potential”

! ,

where ! and ! denote the scalar potential and vector potential, ! the charge, and ! the velocity of the particle. The electromagnetic field, i.e. the electric and the magnetic fields are related to the potentials by

!

2. [10pts] The Lagrangian of a system is given in terms of generalized coordinates ! and ! by

!

Obtain the Hamiltonian and Hamilton's equations of motion and compare with Lagrange’s

equations of motion.

3. [10pts] A spherical pendulum of length ! with a massless rod is supported at the origin of a coordinate system. Use spherical coordinates as indicated.

!

F = qE + qv × B

φ A q v

E = −∇φ(r )− ∂A∂t

B = ∇× A

q1 q2

L = !q12 +

!q22

a + bq12 + kq2

2 + k2 !q1 !q2

l

a) Write the kinetic and potential energy and the Lagrangian in terms of the polar and azimuthal angles ! and !

b) Determine the generalized momenta and the Hamiltonian

c) Write down Hamilton’s equations of motion and compare with previous homework

4) Proof Jacobi's identity for Poisson brackets !

ϕ θ

[u,[v,w]]+ [v,[w,u]]+ [w,[u,v]]= 0