PHY354 Advanced Classical Mechanics. Problem set 1 (ungraded).
Posted by peeterjoot on February 3, 2012
Ungraded solutions to posted problem set 1 (I’m auditing half the lectures for this course and won’t be submitting any solutions for grading).
Problem 1. Lorentz force Lagrangian.
Evaluate the Euler-Lagrange equations.
This problem has two parts. The first is to derive the Lorentz force equation
using the Euler-Lagrange equations using the Lagrangian
In coordinates, employing summation convention, this Lagrangian is
This must equal
So we have
The first term is just . If we expand out we see that matches
A substition, and comparision of this with the Euler-Lagrange result above completes the exersize.
Show that the Lagrangian is gauge invariant.
With a gauge transformation of the form
show that the Lagrangian is invariant.
We really only have to show that
is invariant. Making the transformation we have
We see then that the Lagrangian transforms as
and differs only by a total derivative. With the lemma from the lecture, we see that this gauge transformation does not have any effect on the end result of applying the Euler-Lagrange equations.
Problem 2. Action minimization problem for surface gravity.
Here we are told to guess at a solution
for the height of a particle thrown up into the air. With initial condition we have
and with a final condition of we also have
So our Lagrangian is
and our action is
To minimize this action with respect to we take the derivative
Integrating we have
which is the result we are required to show.
Problem 3. Change of variables in a Lagrangian.
Here we want to show that after a change of variables, provided such a transformation is non-singular, the Euler-Lagrange equations are still valid.
Our “velocity” variables in terms of the original parameterization are
so we have
Computing the LHS of the Euler Lagrange equation we find
For our RHS we start with
but , so this is just
The Euler-Lagrange equations become
Since we have an assumption that the transformation is non-singular, we have for all
so we have the Euler-Lagrange equations for the new abstract coordinates as well