A variational principle motivated by the optimal rod theory

In this work we show that a number of well known nonlinear second order ODE appearing in theoretical physics provide the necessary condition far the m inimum of the Functional I = integral(a)(b) L(x, (x) over double dot, t)dt with rhs Lagrangian L = (-lambda F(t) x/(x) over dot)(alpha). Also we prove that those second-order differential equations may be viewied as conservat ion laws for the corresponding Euler-Lagrange equations that are the fourth -order ODE. Several special cases that have importance in physics, mechanic s and optimal rod theory are studied in detail.