Lagrangians Galore

The last multiplier of Jacobi [1,2,3,4,5] is probably nowadays the generally most forgotten of Jacobi's contributions to Mathematics. Even after Lie [6] showed that his newly introduced symmetries provided a very direct route to the calculation of the multiplier, its use in practice was slight despite excellent descriptions of its properties and usage in such classics as the text of Bianchi [7]. For a listing of the applications of the Jacobi Last Multiplier see the bibliography of [8]. Although Lie groups became of central importance in some areas of Theoretical Physics, the primary idea of using infinitesimal transformations to elucidate the properties of differential equations fell into disuse apart from the bowdlerised variation known as Buckingham's Theorem which is widely appreciated by engineers. With the decline in interest in the Lie algebraic properties of differential equations the last multiplier became a mathematical oddity known only to a select few.

The revival of interest in the Lie algebraic analysis of differential equations began some fifty years ago, but its widespread use as a standard tool is only from about half that period ago. One of the obstacles to the use of the techniques of symmetry analysis has always been the high effort in the computation of the symmetries. Their calculation could be characterised as being tedious without the reward of substantial intellectual stimulation. The advent of symbolic manipulation codes [9,10] has allowed the tedium to be transferred to the computer and freed the brain of the operator for thinking.

A similar problem of computation besets the calculation of the Jacobi Last Multiplier. Unfortunately there seems not to have been the same incentive to use the multiplier as there was in the application of symmetry methods to the equations of gas dynamics and it is only in the last decade that Jacobi’s Last Multiplier has seen application in its rightful place with the beneficial assistance of the computer [11,12,13,8,14,15].

Jacobi’s Last Multiplier is a solution of the linear partial differential equation [3,4,5,16],

where ∂ t + N i=1 a i ∂ x i is the vector field of the set of first-order ordinary differential equations for the N dependent variables x i . The ratio of any two multipliers is a first integral of the system of first-order differential equations and in the case that this system is derived from the Lagrangian of a one-degree-of-freedom system one has that [5,16]

Consequently a knowledge of the multipliers of a system enables one to construct a number of Lagrangians of that system. We recall that Lie’s method [6,17] for the calculation of the Jacobi Last Multiplier is firstly to find the value of

in which the matrix is square with the elements e ij being the vector field of the set of first-order differential equations by which the system is described and the elements, s ij , being the coefficient functions of the number of symmetries of the given system necessary to make the matrix square. If ∆ is not zero, the corresponding multiplier is M = ∆ -1 . Here we consider that the vector fields of the system of equations and symmetries are known and that we seek the multiplier. From another direction one could know the multiplier and all but one of the symmetries. From (3) the remaining symmetry may be determined [11]. Moreover one can use equation (1) to raise the order of the system and find nonlocal symmetries of the original equation [8].

When one has a Lagrangian, it is natural to consider the Noether symmetries of its Action Integral. These constitute a subset, not necessarily proper, of the Lie symmetries of the corresponding Lagrangian equation of motion. The Lagrangians obtained may be equivalent or inequivalent. The distinction can be variously expressed. We use the Noether point symmetries. If two Lagrangians have the same number and algebra of Noether point symmetries, they are equivalent. Otherwise they are inequivalent.

If we have different Lagrangians, be they equivalent or inequivalent, it is natural to consider the relationships among the Lagrangians, the symmetries which produce them through the Jacobi Last Multiplier, the Noether symmetries of the Lagrangians and the associated Noether integrals. This last group returns us to the Jacobi Last Multipliers since the integrals are ratios of multipliers.

In this paper we explore the connections mentioned above. For the purposes of our investigation we choose some simple one-degree-of-freedom problems with much symmetry. These are the simple harmonic oscillator with equation of motion q + k 2 q = 0,

the simple harmonic oscillator in disguise [18] [ex 18, p 433] with equation of motion

and the damped linear oscillator with equation of motion

where c and k are constants and in the case of the damped oscillator the coefficient of q has been written as such to simplify later expressions. We have made our selection for several reasons. Firstly the simple harmonic oscillator

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