What path will the system take if there are more than one path for which the action integral takes a stationary value? Hamilton's principle states that "The true evolution $q(t)$ of a system described by $N$ generalized coordinates $q = (q_1, q_2, ..., q_N)$ between two specified states $q_1 = q(t_1)$ and $q_2 = q(t_2)$ at two specified times $t_1$ and $t_2$ is a stationary point of the action integral.
But is there any way to prove that there is only one path for which the action integral takes a stationary value? (Or) Does the principle say that, "For the real path, the action integral takes a stationary value and the converse of the principle, is not true?"
 A: A classical path is a path for which the variation of the action vanishes. That means that the action becomes extreme at that point. Mostly a minimum. Thats why it is also called principle of least action. For this to happen the solution must satisfy the Euler-Lagrange equations. All solutions to this equation for that particular system are minima of the action. And how many there are depends on the system. Mainly on how many minimas the potential term has. But a nonlinear system for example can also have topological solutions but they dont need to be minimas of the action rather than stationary points of the euclidean path integral
A: Multiple different evolutions corresponding to different solutions of the equations of motions (+ the initial conditions, since these are required to define the Cauchy problem for time evolution of a dynamical system) may exist, and usually this condition occurs when the system has some symmetries.
Hamilton's principle gives you the equations of motion, but it doesn't give you the initial conditions.
There must be several paths, but it's likely that the initial conditions discriminate which path the system takes in the state of space.
Moreover, in real world where disturbances and imperfections exist, even small imperfections of the system or small external disturbances may discriminate the path taken by the system.
Think at an inverted pendulum in the vertical position corresponding to the unstable equilibrium. If the initial condition reads zero angular velocity, the system is perfect and no external disturbance exists, the pendulum doesn't move. If the initial angular velocity make the pendulum rotating in the clockwise direction, it rotates in the clockwise direction. Otherwise, if the initial angular velocity make the pendulum rotating in the counter-clockwise direction, it rotates in the counter-clockwise direction. Same governing equations, different paths in the space of states.
