| course |
course_year |
question_number |
tags |
title |
year |
Variational Principles |
IB |
80 |
IB |
2018 |
Variational Principles |
|
Paper 4, Section II, B |
2018 |
(a) A two-dimensional oscillator has action
$$S=\int_{t_{0}}^{t_{1}}\left{\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \dot{y}^{2}-\frac{1}{2} \omega^{2} x^{2}-\frac{1}{2} \omega^{2} y^{2}\right} d t$$
Find the equations of motion as the Euler-Lagrange equations associated with $S$, and use them to show that
$$J=\dot{x} y-\dot{y} x$$
is conserved. Write down the general solution of the equations of motion in terms of $\sin \omega t$ and $\cos \omega t$, and evaluate $J$ in terms of the coefficients that arise in the general solution.
(b) Another kind of oscillator has action
$$\widetilde{S}=\int_{t_{0}}^{t_{1}}\left{\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \dot{y}^{2}-\frac{1}{4} \alpha x^{4}-\frac{1}{2} \beta x^{2} y^{2}-\frac{1}{4} \alpha y^{4}\right} d t$$
where $\alpha$ and $\beta$ are real constants. Find the equations of motion and use these to show that in general $J=\dot{x} y-\dot{y} x$ is not conserved. Find the special value of the ratio $\beta / \alpha$ for which $J$ is conserved. Explain what is special about the action $\widetilde{S}$ in this case, and state the interpretation of $J$.