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

Abstract

The Lagrange formalism is developed for Bateman oscillators, which includes both damped and amplified systems, and a novel method to derive the Caldirola-Kanai and null Lagrangians is presented. For the null Lagrangians, the corresponding gauge functions are obtained. It is shown that the gauge functions can be used to convert the undriven Bateman oscillators into the driven ones. Applications of the obtained results to quantizatation of the Bateman oscillators are briefly discussed.

Non-standard Null Lagrangians and Forces in Classical Mechanics

“…The derived equation of motion demonstrates how the NSNL can be used to turn an undriven system to a driven one. The main difference between the previous results [25][26][27] and those presented in this Letter is that we show here how classical forces can be defined using the non-standard Lagrangians given by Eqs. ( 2) and ( 4).…”

“…Specifically, recent physical applications involved restoring Galilean invariance of Lagrangians in Newtonian dynamics [23,24], and introducing forces to CM [25][26][27], which was done independently from the original Newton approach and others [28,29]. Most of the previous work was done by using standard null Lagrangians [24,25].…”

Nonstandard Null Lagrangians and gauge Functions for Newtonian Law of Inertia

“…The presented methods of finding general standard and nonstandard ENLs and their EGFs can be extended to all second-order ODEs of the form Dx(t) = 0, which includes the equations of motion of undamped and damped oscillators, and other dynamical systems. In previous work [43], the general standard ENLs and EGFs were derived for the Bateman oscillators; however, the nonstandard ENLs and EGFs are yet to be obtained. The presented methods may also be generalized to partial differential equations of quantum mechanics, such as the Schrödinger equation.…”

General Null Lagrangians and Their Novel Role in Classical Dynamics

“…The third family of null Lagrangians (NLs) has the following two main equivalent characteristics: (i) they must satisfy identically the Euler-Lagrange (E-L) equation: the null condition, and (ii) they must be expressed as the total derivative of any scalar function known as gauge function: the gauge condition. The properties and applications of these null Lagrangians have been extensively explored in different fields of mathematics (e.g., [13][14][15][16][17][18][19][20]) as well as in some physical applications (e.g., [21,22]) that include restoring Galilean invariance of Lagrangians in Newtonian dynamics [23,24], and introducing forces to CM [25][26][27].…”

“…Several methods of constructing the NLs have been proposed and most of them rely on specifying a gauge function and using it to obtain the resulting null Lagrangian [13,14,26,27]. However, in this paper, we develop a new method that is based on a generating function that differs from the gauge function.…”