An extension of the explicit calculation of the principle of least action to a power series potential

We present the general formula of x(t) of a particle moving under any potential expressed by a power series of x without solving the equations of motion. If x(t) is assumed to be expanded by a power series of t with unknown coefficients, one can explicitly perform the time integration of the action. By imposing the boundary conditions and the extremum conditions to the action, all the expansion coefficients and the function of x(t) are determined. This approach will be an alternative way to analyze the motion in classical mechanics based not on the variational principle and the Euler-Lagrange equation. Key words: analytical mechanics, inverted pendulum, Morse potential, power series solution, principle of least action. (author)