# Why are the periodic solutions of conservative unidimensional systems symmetrical regarding the $x$ axis?

Consider the equation $$x''=F(x)$$ which is equivalent to $$\begin{array}{l} x'=v\\ v'=F(x) \end{array}$$ I have already shown that all the equilibrium points of the system are on the $x$ axis, and that all the periodic orbits of it intersect the $x$ axis.

How do I show that the periodic orbits are symmetrical with respect to the $x$ axis? Can I solve this using the fact that the Total Energy of the system is a first integral for it?

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Do you have a formula for F(x) ? For symmetry, you would want to show that substituting -y for y doesn't change the equation. – nonlinearism Feb 22 '13 at 18:03
I've tried that to no result. Also, I don't have F; it must be assumes as a continuous function. – Marra Feb 22 '13 at 18:18

$\frac{dx}{dv}=\frac{v}{F(x)}$
$v^2(x)/2=\int_0^x F(p)dp$
Hence, for every value of $x$, you have a +v and -v value of $v(x)$. Hence there is symmetry w.r.t to x-axis