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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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  • $\begingroup$ 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. $\endgroup$ Feb 22, 2013 at 18:03
  • $\begingroup$ I've tried that to no result. Also, I don't have F; it must be assumes as a continuous function. $\endgroup$
    – Marra
    Feb 22, 2013 at 18:18

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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

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  • $\begingroup$ Well done :) Thanks! $\endgroup$
    – Marra
    Feb 22, 2013 at 19:21

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