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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.

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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

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

Therefore

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

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