Dynamical Noether invariants for time-dependent nonlinear systems

Abstract

Dynamical invariants are derived for time-dependent systems with nonlinear equations of motion including nonharmonic damped systems. The concept of a dynamical algebra is discussed and its utility for the construction of dynamical invariants for nonharmonic systems is demonstrated. Finally we show the existence of dynamical invariants for some nonlinear quantum systems.

Comment on A note on the construction of the Ermakov Lewis invariant

“…These works, however, do not deal with truly twodimensional, coupled Ermakov systems, as in the case of references [1,2]. Rather, these papers [5]- [8] deal with Ermakov systems in which one of the equations plays the principal role, while the other, decoupled from the first, is treated as an auxiliary equation. In these cases, the Lagrangian description is effectively one-dimensional and the Ermakov invariant cannot be obtained as a result from an associated dynamical Noether symmetry.…”

“…As a final remark, the work by Moyo and Leach also ignores the papers [5]- [8], dedicated to the analysis of uncoupled Ermakov systems in the light of Noether's theorem. These works, however, do not deal with truly twodimensional, coupled Ermakov systems, as in the case of references [1,2].…”

“…This can be seen by comparing the potential functions from equations (11) in the work by Haas and Goedert with the potential given by equation (3.18) in the work by Moyo and Leach. This later work also ignores the Hamiltonian descriptions for Ermakov systems developed earlier in [3], for the case of frequency functions depending on time only, and in [4], for the case of frequency functions depending also on dynamical variables.As a final remark, the work by Moyo and Leach also ignores the papers [5]- [8], dedicated to the analysis of uncoupled Ermakov systems in the light of Noether's theorem. These works, however, do not deal with truly twodimensional, coupled Ermakov systems, as in the case of references [1,2].…”

Reflections on Friction in Quantum Mechanics

“…"Dynamical invariants" are quantities that remain constant, but are explicit functions of time-dependent parameters and of (implicitly time-dependent) expectation values; for the harmonic oscillator, the first such invariant was noted by Lewis [20]. Korsch and Koshual used dynamical algebras to derive the dynamical invariants which lie within the algebra [21,22] (i.e., invariants which are linear in the expectation values of the algebra's operators), and Sarris and Proto demonstrated that for our state (Equation 2) it is possible to generalize further and derive dynamical invariants that are outside the algebra [23], consisting of higher powers of the the algebra's expectation values (they actually consider maximum entropy states,ρ = exp n λ n (t)L n , but a product form and exponential sum are interchangeable (see [24], corollary to theorem 2)). One such invariant in our case was found to be [25] …”

Exact solutions for the general nonstationary oscillator with a singular perturbation

“…Previously this singular oscillator (SO) has been treated in a number of paper [6,7,8,9,10,11,12,13,14], exact invariants and wave functions being obtained for the case of stationary SO (constant m, ω and g) in [12,14] and of SO with varying frequency ω(t) (but constant m, g) in [11,13]. The more general cases treated in [7,9] corresponds (as in [2]) to m(t)g(t) = const.…”