Gauge Functions in Classical Mechanics: From Undriven to Driven Dynamical Systems

Abstract

Novel gauge functions are introduced to non-relativistic classical mechanics and used to define forces. The obtained results show that the gauge functions directly affect the energy function and allow for converting an undriven physical system into a driven one. This is a novel phenomenon in dynamics that resembles the role of gauges in quantum field theories.

Bateman Oscillators: Caldirola-Kanai and Null Lagrangians and Gauge Functions

“…The results given by Equation ( 19) through ( 22) are new general gauge functions for the Bateman oscillators and they generalize those previously obtained for Newton's law of inertia [20] and for linear undamped oscillators [21] for which b = 0.…”

“…The general null Lagrangian L gn [ẋ(t), x(t), t] and its partial null Lagrangians reduce to those previously found [20,21] when b = 0. These Lagrangians depend on four functions, which can be constrainted by the initial conditions if they are specified.…”

“…and its substitution into the EL equation gives Equation (6). Let us now follow [20,21] and consider the following Lagrangian…”

“…where C 1 , C 2 , C 4 and C 6 are constants. It is easy to verify that L n [ẋ 1 (t), x 1 (t), t] is the null Lagrangian with the constants being arbitrary, and that this NL is constructed to the lowest order of its dynamical variables [20,21]. The NL can be added to L s [ẋ 1 (t), x 1 (t)] without changing the form of the equation of motion (see Equation ( 6)) resulting from it.…”

“…Despite the fact that the partial null Lagrangians were used in deriving the total energy functions, these null Lagrangians were derived from the partial gauge functions given by Equation (19) through (21). Based on definitions given by Equation ( 34) through (36), it can be seen that the partial gauge function φ gn1 [x(t), t] contributes only toω 2 o (t), and that F(t) is determined by φ gn2 [x(t), t] and also by φ gn3 [x(t), t], which contributes partially to both F(t) and G(t).…”