## PHY354 Advanced Classical Mechanics. Problem set 1 (ungraded).

Posted by peeterjoot on February 3, 2012

# Disclaimer.

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

Taking derivatives

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 have

So our Lagrangian is

and our action is

To minimize this action with respect to we take the derivative

Integrating we have

or

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.

Let’s write

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

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