FINAL ADVANCED DYNAMICS – PHY 4241/5227 (April 29, 2009)

PROBLEM 1 (30 points)

Assume a Lagrangian L = L({q_{i}}, {q̇_{i}}, t) where q_{i}, i = 1, …, n are generalized coordinates, q̇_{i}, i = 1, …, n are generalized velocities and t is the time.

1. Write down the principle of least action.

2. Derive the Euler-Lagrange equations from the principle of least action.

3. Assume that the Lagrangian is invariant under translations q_{i} → q′_{i} = q_{i} + ϵ_{i}. Find the corresponding conserved quantities.

PROBLEM 2 (30 points)

An electromagnetic field tensor (in arbitrary units) is given by

(F^{αβ}) = ( \begin{matrix}0 & 1 & 2 & −1 \\ −1 & 0 & −2 & 1 \\ −2 & 2 & 0 & 3 \\ 1 & −1 & −3 & 0 \end{matrix} ) .

Find (the Einstein summation convention is assumed for identical indices):

1. F _{α}^{α}.

2. The matrix (F_{αβ}).

3. F_{αβ}F^{αβ}.

PROBLEM 3 (40 points)

1. Write down the three components, E^{1}, E^{2} and E^{3} of the electric field of a static point charge q, which is located at the origin of a coordinate system described by x^{0}, x^{1}, x^{2} and x^{3}.

2. What is the magnetic field B⃗ of this point charge?

3. Write the down the electromagnetic field tensor (F^{αβ}) of this point charge.

4. Use the transformation law F′(x) = A F (x) A^{†} and the Lorentz matrix

A = ( \begin{matrix}γ & −γβ & 0 & 0 \\ −γβ & γ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} )

to calculate the electromagnetic field tensor (F^{′αβ}) for an inertial frame K′, which moves with velocity v along the x-axis of the rest frame of the point charge (the orientations of the axis of the two frames are parallel and their origins coincide at time t = t′ = 0).

5. The previous expression F′(x) is still written in x-coordinates. Use now x′- coordinates to write down the electric E^{′i}(x′) and magnetic B^{′i}(x′), i = 1, 2, 3 field components in the frame K′.