Jacobi Last Multiplier and Lie Symmetries: A Novel Application of an Old Relationship

Abstract

After giving a brief account of the Jacobi last multiplier for ordinary differential equations and its known relationship with Lie symmetries, we present a novel application which exploits the Jacobi last multiplier to the purpose of finding Lie symmetries of first-order systems. Several illustrative examples are given.

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

“…A property of the Jacobi last multiplier is that if one knows a Jacobi last multiplier and a first integral, then their product gives another multiplier (Nucci 2005 If we apply Lie group analysis to this equation, we find that it admits a threedimensional Lie symmetry algebra generated by the following three operators: (3.23) which means that we can reduce equation (3.22) to a first-order equation, i.e. (3.26) This means that equation (3.22), i.e.…”

“…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…”

“…What seems less known is that the key is the Jacobi last multiplier (Jacobi 1844(Jacobi , 1845(Jacobi , 1886Whittaker 1999), which has many interesting properties, a list of which can be found in Nucci (2005).…”

Application of Jacobi’s last multiplier for construction of Hamiltonians of certain biological systems

“…Besides, the JLM allows us to determine the Lagrangian of a second-order ODE in many cases [4,[6][7][8][9]20]. In recent years a number of articles have dealt with this particular aspect [3,11,12]. However, when a planar system of ODEs cannot be reduced to a second-order differential equation the question of interest arises whether the JLM can provide a mechanism for finding the Lagrangian of the system.…”

The Method of Jacobi Last Multiplier for Integrating Nonholonomic Systems

“…This paper presents a new integration method of nonholonomic systems, i.e., the method of Jacobi last multiplier. The method of Jacobi last multiplier is very important in the problems of differential equations, and some important results have been obtained [14][15][16][17][18]. However, there are few works involving in the application of the Jacobi last multiplier to the nonholonomic systems.…”