Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

Abstract

We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given.

Some Applications of Affine in Velocities Lagrangians in Two-Dimensional Systems

“…In this section we analyse this problem and conclude that, in general, it has always a positive answer; moreover, there are infinitely many affine Lagrangians for a given first-order system. The main result is expressed in terms of the so called Jacobi multipliers [33,50,61,62,63], a notion which we present in geometrical terms [10,21].…”

“…Section 4 is devoted to analyse the twodimensional inverse problem for systems of first-order differential equations. As the main result is to be expressed in terms of a Jacobi multiplier of the given system, we first review in Subsection 4.1 the theory of Jacobi multipliers in geometrical terms [2,3,10,18,19,21,28,33,40,41,42,43,44,45,46,50,61,62,63]. The main result asserts that in order to have a Lagrangian description for a given system of two firstorder differential equations it is necessary and sufficient to establish the existence of a Jacobi multiplier for the system.…”

“…The equations determining the multipliers have always a local solution, so this inverse problem has a positive answer; in fact, there are infinitely many (not necessarily gauge-equivalent) Lagrangians for the given system, opening in this way the possibility of finding constants of motion (see e.g. [10,61,62]) and alternative Lagrangians [13]. For more information on the rôle of Jacobi multipliers in integrability and with other approaches to integrability see e.g [55,56,57,58].…”

Infinitesimal time reparametrisation and its applications

“…For examples of applications of Jacobi multipliers in integrability and the inverse problem of mechanics see e.g. the recent review paper [23] and references therein.…”

Reduction and integrability: a geometric perspective

“…The nonvanishing functions R such that L X (R Ω) = L R X (Ω) = 0, i.e. div(R X) = 0, are called Jacobi multipliers, and (6.2) shows that the nonvanishing function R is a Jacobi multiplier if and only if (see [32,33] and references therein)…”