Invariants for forced time-dependent oscillators and generalizations

Ray, John R.; Reid, James L.

doi:10.1103/physreva.26.1042

Cited by 38 publications

(27 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The above procedure is a simple derivation of the quantum Ermakov Lewis invariant, which has otherwise been obtained using rather more complex mathematical methods [9]. The non Hermitian linear invariant Î c introduced by Malkin et al written in terms of the orthogonal functions invariants iŝ…”

Section: Classical and Quantum Invariantsmentioning

confidence: 99%

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Fernández‐Guasti

Moya‐Cessa

2003

Phys. Rev. A

View full text Add to dashboard Cite

The correspondence between classical and quantum invariants is established. The Ermakov Lewis quantum invariant of the time dependent harmonic oscillator is translated from the coordinate and momentum operators into amplitude and phase operators. In doing so, Turski's phase operator as well as Susskind-Glogower operators are generalized to the time dependent harmonic oscillator case. A quantum derivation of the Manley-Rowe relations is shown as an example.

show abstract

Section: Classical and Quantum Invariantsmentioning

confidence: 99%

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Fernández‐Guasti

Moya‐Cessa

2003

Phys. Rev. A

View full text Add to dashboard Cite

show abstract

“…The multidimensional analog of Eq. was introduced by Ray and Reid who subsequently analyzed their properties . In this subsection, we find necessary and sufficient conditions under which an Ermakov system can be transform to a scalar equation by hypercomplexification.…”

Section: Integration Of Systems Of Ordinary Differential Equations Bymentioning

confidence: 94%

Hypercomplex analysis and integration of systems of ordinary differential equations

Soh

Mahomed

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

We review the theory of hypercomplex numbers and hypercomplex analysis with the ultimate goal of applying them to issues related to the integration of systems of ordinary differential equations (ODEs). We introduce the notion of hypercomplexification, which allows the lifting of some results known for scalar ODEs to systems of ODEs. In particular, we provide another approach to the construction of superposition laws for some Riccati‐type systems, we obtain invariants of Abel‐type systems, we derive integrable Ermakov systems through hypercomplexification, we address the problem of linearization by hypercomplexification, and we provide a solution to the inverse problem of the calculus of variations for some systems of ODEs. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

“…where ω0 is a constant. The pair of Equations (20a) and (20b) corresponds to an Emarkov (1880) system, wherein the information about the time-evolution of the potential is carried by the auxiliary Equation (20b) (see Ray & Reid 1982). From Equation (8) the energy can be written as IHO = 1/2(dr /dτ ) 2 + 1/2ω 2 0 r 2 .…”

Section: Time-dependent Harmonic Oscillatormentioning

confidence: 99%

Dynamical invariants and diffusion of merger substructures in time-dependent gravitational potentials

Peñarrubia¹

2013

Monthly Notices of the Royal Astronomical Society

View full text Add to dashboard Cite

This paper explores a mathematical technique for deriving dynamical invariants (i.e. constants of motion) in time-dependent gravitational potentials. The method relies on the construction of a canonical transformation that removes the explicit time-dependence from the Hamiltonian of the system. By referring the phase-space locations of particles to a coordinate frame in which the potential remains 'static' the dynamical effects introduced by the time evolution vanish. It follows that dynamical invariants correspond to the integrals of motion for the static potential expressed in the transformed coordinates. The main difficulty of the method reduces to solving the differential equations that define the canonical transformation, which are typically coupled with the equations of motion. We discuss a few examples where both sets of equations can be exactly de-coupled, and cases that require approximations. The construction of dynamical invariants has far-reaching applications. These quantities allow us, for example, to describe the evolution of (statistical) microcanonical ensembles in time-dependent gravitational potentials without relying on ergodicity or probability assumptions. As an illustration, we follow the evolution of dynamical fossils in galaxies that build up mass hierarchically. It is shown that the growth of the host potential tends to efface tidal substructures in the integral-of-motion space through an orbital diffusion process. The inexorable cycle of deposition, and progressive dissolution, of tidal clumps naturally leads to the formation of a 'smooth' stellar halo.

show abstract

Invariants for forced time-dependent oscillators and generalizations

Cited by 38 publications

References 22 publications

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Hypercomplex analysis and integration of systems of ordinary differential equations

Dynamical invariants and diffusion of merger substructures in time-dependent gravitational potentials

Contact Info

Product

Resources

About