# Problem in Arnold's Mathematical Methods of Classical Mechanics regarding Lissajous figures

In Arnolds book Mathematical Methods of Classical Mechanics he defines the system

$$\ddot{x}_1=-x_1,\,\,\,\ddot{x}_2=-\omega^2x_2^2.$$

The potential energy is $$U(x_1,x_2)=\frac{1}{2}(x_1^2+\omega^2x_2^2)$$. Because both energies $$E_1=\frac{1}{2}\dot{x}_1^2+\frac{1}{2}x_1^2$$, $$E_2=\frac{1}{2}\dot{x}_2^2+\frac{1}{2}\omega^2x_2^2$$ preserved the variable $$x_i$$ is bounded by $$|x_i|<\sqrt{2E_i}$$ for $$i=1,2$$. This defines a rectangle in $$\mathbb{R}^2$$.

Now the problem given in the book is to show that this rectangle lies completely in the ellipse given by $$U(x_1,x_2)=E$$.

But if I check at the corners of the rectangle $$(\sqrt{2E_1},\sqrt{2E_2})$$ I get that $$U(\sqrt{2E_1},\sqrt{2E_2})=E_1+\omega^2 E_2 > E$$, so they dont lie in the ellipse.

Can anyone help finding where Ive done something wrong?

I think instead of $$x_i \le E_i$$, we have $$x_1 \le E_1$$ and $$\omega x_2 \le E_2$$, from which the conclusion readily follows.
• Which also makes more sense in the context of the problem than $x_i\leq E_i$. Thank you very much! – TwoStones Oct 27 '19 at 8:18