HW# 2 (Submitted: 2023.03.20; Due date: 2023.03.27)
[Prb 1.] Consider a particle moving in the +x-direction with a speed of gt+1 in a certain inertial frame, where g is a constant having the dimension of acceleration and t > 0 is the time in this frame.
a) Put the factors of c to get a dimensionally correct expression for the speed.
b) Find u as a function of t.
c) Show that gT = ln(gt+1).
d) Express u as a function of T.
e) Express u as a function of both t and T.
f) Find expressions for x and t.
[Prb 2.] The action for a relativistic particle is S = -mc^2∫(1 - u^2/c^2)^0.5 dt, where T is the proper time.
a) Using this action, find the Lagrangian of a relativistic particle for a particular Lorentz observer moving with uniform velocity u relative to the particle.
b) Find the low-speed limit of this Lagrangian.
c) Find the canonical momentum and Hamiltonian of the relativistic particle.
[Prb 3.] Recall that the electromagnetic force on a point charge Q is F = QE + B in SI. Show that this equation can be obtained from the spatial components of the following equation dp/dt = QF + nv x B, where T is the proper time. What is the temporal component of this equation?