The Euler-Lagrange equations give a necessary condition for the action be extremal given some lagrangian which depends on some function to be varied over. The basic form assumes fixed endpoints for the function to be varied over, but we can extend to cases in which one or both endpoints are free to vary by introducing additional transversality conditions. I would like to know what the transversality condition(s) would be in the case that the values at the two boundaries are equal, but otherwise free to vary. The reason for this is that I want to consider functions which map the circle onto the real numbers, and use the calculus of variations approach to maximise an action based on a lagrangian of this function. Thanks!