Explicit Calculation of Principle of Least Action

Yuji Kajiyama Gifu Shotoku Gakuen University, 1-1, Takakuwa-Nishi, Yanaizu, Gifu, 501-6194, Japan

Full Text PDF

In the standard way of analytical mechanics, the Euler-Lagrange equation as an equation of motion is derived from the principle of least action based on the variational principle. In this paper, we present an alternative way to describe the motion by explicitly solving an extreme value problem of the action without the variational principle. We assume that the position x(t) is expressed by power series of time t with an infinite number of unknown coefficients, and show that these coefficients can be correctly determined by the boundary conditions and the extremum conditions of the action by explicit calculations, which describe the actual motion. We will present the motion of a free particle, the motion under constant gravity, a harmonic oscillation, the time-dependent Lagrangian, and a non-linear force as examples.

DOI:10.12693/APhysPolA.139.704 topics: principle of least action, harmonic oscillator, non-linear force, analytical mechanics