We know that for system with one degree of freedom movement in potential field satisfies the equation $\frac{\overset{.}{x}^2}{2} + U(x) = E$, where $U(x)$ is potential energy and $E$ is total energy of the system (constant).

Let also $x_1$ be a local maximum of $U(x)$. Setting $E = U(x_1)$ we get a phase trajectory called separatrix. I want to prove that it takes infinity time to move system from some position $x_1$ to $x_0$.

So, we can rewrite deiiferential equation: $$\overset{.}{x} = \pm \sqrt{2(E - U(x))}$$

By separating variables we can get something like following: $$\int_{x_0}^{x_1}\frac{dx}{\sqrt{2(E - U(x))}} = \int_0^{t_1}dt = t_1$$

But know I need some mathemtical argument to show that $t_1$ can't be a number. It's clear that $\frac{1}{\sqrt{2(E - U(x))}} \rightarrow \infty$ when $x \rightarrow x_1$, and integral is improper, but we know very little about $U(x)$.

Any hints? Thanks!


You better hope to know something about $U(x)$ as this is false in general.

For example, let $U(x) = - |x|$. Then the maximum is $E = 0$ which occurs at $x_1 = 0$. For $x_0 > 0$, the integral is proportional to $$\int_{x_0}^0 \frac{dx}{\sqrt x}$$which is easily seen to be finite.


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