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Given a function $A: \mathbb{R}^2 \to \mathbb{R}$, and that $A$ is $C^2$ and the autonomous system of ODE's:

$$\dot{x} = -\frac{\partial A}{\partial x}(x, v)$$ $$\dot{v} = -\frac{\partial A}{\partial v}(x, v)$$

Show that this system has no nonconstant periodic solutions.

My attempt

I was looking for materials in this stack exchange that could help me solve this, and I've found this solution. It seems that my problem is the 2D generalization of the solution posted on that link. But I have no idea to work with what was posted in that link. Does anyone have any hint on the correct way to approach this problem?

Tian Vlašić
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Occhima
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  • In the contrary case, $a(t)=A(x(t),v(t))$ would be at the same time periodic and non-increasing. Which are (almost completely) contradicting properties. – Lutz Lehmann Jun 23 '23 at 09:27

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For convenience I'll define $\mathbf{x} = (x, v)$ so that the equation is $\dot{\mathbf{x}} = -\nabla A$. Now, since $\dot{\mathbf{x}}\cdot\nabla A = dA/dt$, we have that $dA/dt = -|\nabla A|^2$. In particular, this means $A(\mathbf{x}(t))$ is nonincreasing for any solution $\mathbf{x}(t)$. Since the only periodic functions that are nonincreasing are constant, a periodic solution must have $d A/dt = 0$ throughout. This means $|\nabla A|^2 = dA/dt = 0$ at all points of the solution, and thus $\dot{\mathbf{x}} = -\nabla A = 0$. So $\mathbf{x}$ itself is constant.

eyeballfrog
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