set7
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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
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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