On the Lie symmetries of a class of generalized Ermakov systems

Abstract

The symmetry analysis of Ermakov systems is extended to the generalized case where the frequency depends on the dynamical variables besides time. In this extended framework, a whole class of nonlinearly coupled oscillators are viewed as Hamiltonian Ermakov system and exactly solved in closed form

Lie point symmetries for reduced Ermakov systems

“…for S defined in (19). Because (32) has to be satisfied for arbitrary r, it can be split in two parts, one corresponding to r, the other to r −3 ,…”

“…The point symmetry group of Ermakov systems has been identified as the SL(2, R) group [15]- [19]. More recently, using the converse to Noether's theorem, it has been shown that the Ermakov invariant can be associated to a dynamical symmetry, in the cases where the Ermakov system admits a variational formulation [20].…”

On the linearization of the generalized Ermakov systems

“…As already pointed out before [7,9], the concept of a generalized Ermakov system has originated from the observation that ω may depend arbitrarily on the dynamic variables and that, as a consequence, only two and not three arbitrary functions are necessary to specify the Ermakov systems. Indeed, redefining…”

“…These subclasses are frequently more flexible and may be tailored to suite some particular application or special purpose. Among others, we quote applications such as the identification of the Hamiltonian character in special circumstances [6,7]; the determination of a second constant of motion in certain other particular cases [8]; the identification of the structure of associated Lie group of point symmetries [9,10] and the extension of the Ermakov systems concept itself to higher dimensions [10]- [13].…”

Perturbation to Symmetry and Adiabatic Invariants of General Discrete Holonomic Dynamical Systems

“…Since Noether unveiled the profound relations between symmetries and conservation laws, many researches on them were done [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Recently, symmetry theories have been extended to discrete mechanics and equations [16][17][18][19][20][21][22][23][24][25].…”