Asymptotics of the Solution of a Variational Problem on a Large Interval

The variational problem of minimizing the energy functional that results in a second-order nonlinear differential equation of pendulum type on a finite interval with natural boundary conditions is analyzed. It is shown that the number of solutions of the boundary-value problem depends on the length $L$ of the interval and unboundedly increases as $L\to\infty$. The solutions on which the energy minimum is realized converge as $L\to\infty$ to the solution of a variational problem in the class of periodic functions.