A NOTE ON INTEGRAL PROPERTIES OF PERIODIC-ORBITS

For the class of periodic functions u(x) that can be associated with closed orbits in the (u, u') phase-plane described by a differential equation of the form u'2 = p - F(u), around some local minimum of F(u), a useful integral expression is obtained for any derivative with respect to the parameter p of any suitably differentiable function of u and p integrated over one period in x. That is, an expression is found for (d/dp)k integral-lambda(p)/0 f(u, p) dx, where lambda(p) is the period. Some illustrative examples are presented concerning relationships between moments of u(x) and some forms of the function F(u).