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

Question

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?

Answer 1

Write the equation as :

$\frac{dx}{dv}=\frac{v}{F(x)}$

Then integrating, you get

$v^2(x)/2=\int_0^x F(p)dp$

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