(i) 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 to $S$, 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 which arise in the general solution.

(ii) 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 $J$.