Lagrangians galore

Abstract

Searching for a Lagrangian may seem either a trivial endeavour or an impossible task. In this paper we show that the Jacobi last multiplier associated with the Lie symmetries admitted by simple models of classical mechanics produces (too?) many Lagrangians in a simple way. We exemplify the method by such a classic as the simple harmonic oscillator, the harmonic oscillator in disguise [H Goldstein, Classical Mechanics, 2nd edition (Addison-Wesley, Reading, 1980)] and the damped harmonic oscillator. This is the first paper in a series dedicated to this subject.

Lie groups and quantum mechanics

“…The number of Noether point symmetries associated with these Lagrangians varies [9]. There are five for L 12 , three for L 13 and L 23 and two for L 34 , i.e.…”

“…Not one of the fourteen Lagrangians has four, an additional three have three, seven more have two and there are none with one or zero Noether point symmetries [9]. For each of these Lagrangians one may construct an Hamiltonian.…”

The nonlinear pendulum always oscillates

“…where 1 and 2 are functions of t and x, which have to satisfy a single partial differential equation related to (3.8) [25]. As was shown in [25], 1 , 2 are related to the gauge function F = F(t, x).…”

“…This may require searching for the Lagrangian yielding the maximum possible number of Noether point symmetries [25][26][27][28]. (3) Construct the Schrödinger equation f admitting these Noether point symmetries as Lie point symmetries, namely…”

On the inverse problem of calculus of variations for fourth-order equations

“…Some of these acknowledge the seminal work of Jacobi (see references in Nucci (2005) and Nucci & Leach (2007)), but most failed to do so. Fels (1996) derived the necessary and sufficient conditions under which a fourth-order equation…”