Which Principle Is It?

According to the classical principle of least action, if a particle goes from $(x_1,y_1,z_1,t_1)$ to $(x_2,y_2,z_2,t_2)$ , it will do so along a path which minimizes the action $S$ (where $E$ is the kinetic energy and $U$ is the potential energy).

$\large{S = \int_{t_1}^{t_2} (E - U) \, dt}$

In a potential-free universe (which also means that there are no forces), the action reduces to:

$\large{S = \int_{t_1}^{t_2} E \, dt}$

In the specific case under consideration, the particle is at the same point in space at different points in time. The minimum action between the starting and ending points obviously corresponds to $E = 0$ (zero velocity) at all instants in time. This solution is consistent with the particle remaining in place.

So given the starting and ending points, and the absence of potentials (forces), the least action principle dictates that the particle's velocity remains zero. By extension, the absence of potentials (forces) implies that the velocity must be constant (zero). This is essentially a re-statement of Newton's 1st law of motion.

See this note for a more general treatment.

According to the given data, the position of the particle remains invariant with time. That is, the particle is at rest relative to the assumed frame during the time interval [ $t_1$ , $t_2$ ]. Hence in accordance with Newton's law of Inertia, the particle will be at rest for ever, that is, it's kinetic energy will remain zero at all times, and hence within that interval, relative to that frame.