Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am stuck with the following ODE problem:

Let $V \in C^2(\mathbb{R}^n,\mathbb{R})$ and $x_0 \in \mathbb{R}^n$, where $n \ge 2$. Let $\phi_{x_0}: (t^-,t^+) \to \mathbb{R}^n$ be the maximal solution to the initial value problem $$ \dot{x}=-\nabla V(x),\quad x(0)=x_0. $$

  1. Prove that the function $V\circ\phi_{x_0}$ is non-increasing on $(t^-,t^+)$.
  2. Suppose $\phi_{x_0}(p)=\phi_{x_0}(0)$ for some $p>0$. Prove that $t^+=-t^-=\infty$, and that $\phi_{x_0}(t)=x_0$ for every $t$.

My problem is with the second question. Any help will be appreciated.

share|cite|improve this question
up vote 1 down vote accepted

We know by theorems of uniqueness for autonomous equations that if the solution self intersecting($\phi_{x_0}(0)=\phi_{x_0}(p)$ then the solution must be periodic. Therefore your solution $\phi_{x_0}$ is periodic of period $p$(defined in all $\mathbb R$). By a) we know that

$$ V(\phi_{x_0}(t))=V(\phi_{x_0}(0))$$ for all $t\in[0,p]$.

Differentiating both sides we have

$$<\text{grad}V(\phi_{x_0}(t),\phi_{x_0}'(t)>=0\implies <-\phi_{x_0}'(t),\phi'_{x_0}(t)>=0$$


$ \phi'_{x_0}(t)=0$ for all $t$

share|cite|improve this answer
This helped you? – user52188 Jan 17 '13 at 23:13
Yeah, thanks for your help! – albmiz-mth Jan 17 '13 at 23:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.