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


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?


1 Answer 1


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.

  • $\begingroup$ Which also makes more sense in the context of the problem than $x_i\leq E_i$. Thank you very much! $\endgroup$
    – TwoStones
    Oct 27, 2019 at 8:18

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