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9 pages, 250 KiB

Open AccessArticle

On the Damped Pinney Equation from Noether Symmetry Principles

by Fernando Haas

Symmetry 2024, 16(10), 1310; https://doi.org/10.3390/sym16101310 - 4 Oct 2024

Viewed by 874

There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work, these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether’s theorem for [...] Read more.

There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work, these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether’s theorem for the usual time-dependent harmonic oscillator as a guiding principle for derivation of the Pinney equation also works in the damped case, selecting a Noether symmetry-based damped Pinney equation. The results are extended to general nonlinear damped Ermakov systems. A certain time-rescaling always allows to remove the damping from the final equations. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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28 pages, 435 KiB

Open AccessArticle

Quadratic First Integrals of Constrained Autonomous Conservative Dynamical Systems with Fixed Energy

by Antonios Mitsopoulos and Michael Tsamparlis

Symmetry 2022, 14(9), 1870; https://doi.org/10.3390/sym14091870 - 7 Sep 2022

Cited by 3 | Viewed by 1489

Abstract

We consider autonomous conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent and autonomous, of these systems in terms of [...] Read more.

We consider autonomous conservative dynamical systems which are constrained with the condition that the total energy of the system has a specified value. We prove a theorem which provides the quadratic first integrals (QFIs), time-dependent and autonomous, of these systems in terms of the symmetries (conformal Killing vectors and conformal Killing tensors) of the kinetic metric. It is proved that there are three types of QFIs and for each type we give explicit formulae for their computation. It is also shown that when the autonomous QFIs are considered, then we recover the known results of previous works. For a zero potential function, we have the case of constrained geodesics and obtain formulae to compute their QFIs. The theorem is applied in two cases. In the first case, we determine potentials which admit the second of the three types of QFIs. We recover a superintegrable potential of the Ermakov type and a new integrable potential whose trajectories for zero energy and zero QFI are circles. In the second case, we integrate the constrained geodesic equations for a family of two-dimensional conformally flat metrics. Full article

(This article belongs to the Special Issue Noether and Space-Time Symmetries in Physics)

16 pages, 327 KiB

Open AccessArticle

Dynamical Invariants for Generalized Coherent States via Complex Quantum Hydrodynamics

by Moise Bonilla-Licea and Dieter Schuch

Dynamics 2021, 1(2), 155-170; https://doi.org/10.3390/dynamics1020009 - 8 Oct 2021

Cited by 4 | Viewed by 2899

Abstract

For time dependent Hamiltonians like the parametric oscillator with time-dependent frequency, the energy is no longer a constant of motion. Nevertheless, in 1880, Ermakov found a dynamical invariant for this system using the corresponding Newtonian equation of motion and an auxiliary equation. In [...] Read more.

For time dependent Hamiltonians like the parametric oscillator with time-dependent frequency, the energy is no longer a constant of motion. Nevertheless, in 1880, Ermakov found a dynamical invariant for this system using the corresponding Newtonian equation of motion and an auxiliary equation. In this paper it is shown that the same invariant can be obtained from Bohmian mechanics using complex Hamiltonian equations of motion in position and momentum space and corresponding complex Riccati equations. It is pointed out that this invariant is equivalent to the conservation of angular momentum for the motion in the complex plane. Furthermore, the effect of a linear potential on the Ermakov invariant is analysed. Full article

19 pages, 800 KiB

Open AccessArticle

Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

by Qiliang Zhao, Pengming Zhang and Peter A. Horvathy

Symmetry 2021, 13(10), 1866; https://doi.org/10.3390/sym13101866 - 3 Oct 2021

Cited by 8 | Viewed by 2230

Abstract

The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” [...] Read more.

The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile. Full article

(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)

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16 pages, 338 KiB

Open AccessReview

A Review in Ermakov Systems and Their Symmetries

by Jose M. Cerveró and Pilar G. Estévez

Symmetry 2021, 13(3), 493; https://doi.org/10.3390/sym13030493 - 17 Mar 2021

Cited by 4 | Viewed by 2139

Abstract

A review of the mathematical and physical aspects of the Ermakov systems is presented. The main properties of Lie algebra invariants are extensively used. The two and tridimensional Ermakov systems are fully analyzed and the correspondent invariants found. Then, we go over quantization [...] Read more.

A review of the mathematical and physical aspects of the Ermakov systems is presented. The main properties of Lie algebra invariants are extensively used. The two and tridimensional Ermakov systems are fully analyzed and the correspondent invariants found. Then, we go over quantization with special emphasis in the two dimensional case. An application to Nonlinear Optics is hereby developed. We also treat the so-called “one dimensional” case, which is easily solved in the classical case but offers special interest in the quantum realm, where one can find exactly the Feynman propagator. We finish with the stationary phase approximation which contains also some interesting features when compared with the exact solution. Some prospects for future research are also discussed. Full article

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12 pages, 346 KiB

Open AccessArticle

Relativistic Ermakov–Milne–Pinney Systems and First Integrals

by Fernando Haas

Physics 2021, 3(1), 59-70; https://doi.org/10.3390/physics3010006 - 12 Feb 2021

Cited by 4 | Viewed by 2901

Abstract

The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding [...] Read more.

The Ermakov–Milne–Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time-dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov–Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov–Milne–Pinney equation and associated first integral. The special relativistic extension of the Ray–Reid system and invariant is obtained. General properties of the relativistic Ermakov–Milne–Pinney are analyzed. The conservative case of the relativistic Ermakov–Milne–Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered to be well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov–Milne–Pinney equation has additional nonlinearities, due to the relativistic effects. Full article

(This article belongs to the Section Classical Physics)

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9 pages, 270 KiB

Open AccessArticle

Solution to the Time-Dependent Coupled Harmonic Oscillators Hamiltonian with Arbitrary Interactions

by Alejandro R. Urzúa, Irán Ramos-Prieto, Manuel Fernández-Guasti and Héctor M. Moya-Cessa

Quantum Rep. 2019, 1(1), 82-90; https://doi.org/10.3390/quantum1010009 - 22 Jul 2019

Cited by 26 | Viewed by 5726

Abstract

We show that by using the quantum orthogonal functions invariant, we found a solution to coupled time-dependent harmonic oscillators where all the time-dependent frequencies are arbitrary. This system may be found in many applications such as nonlinear and quantum physics, biophysics, molecular chemistry, [...] Read more.

We show that by using the quantum orthogonal functions invariant, we found a solution to coupled time-dependent harmonic oscillators where all the time-dependent frequencies are arbitrary. This system may be found in many applications such as nonlinear and quantum physics, biophysics, molecular chemistry, and cosmology. We solve the time-dependent coupled harmonic oscillators by transforming the Hamiltonian of the interaction using a set of unitary operators. In passing, we show that N time-dependent and coupled oscillators have a generalized orthogonal functions invariant from which we can write a Ermakov–Lewis invariant. Full article

15 pages, 1647 KiB

Open AccessArticle

Exactly Solvable One-Qubit Driving Fields Generated via Nonlinear Equations

by Marco Enríquez and Sara Cruz y Cruz

Symmetry 2018, 10(11), 567; https://doi.org/10.3390/sym10110567 - 1 Nov 2018

Cited by 10 | Viewed by 2662

Abstract

Using the Hubbard representation for

S U (2),

we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of nonlinear coupled equations. In order to [...] Read more.

Using the Hubbard representation for

S U (2),

we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of nonlinear coupled equations. In order to find exact solutions, we use an inverse approach and find families of time-dependent Hamiltonians whose off-diagonal elements are connected with the Ermakov equation. A physical model with the so-obtained Hamiltonians is discussed in the context of the nuclear magnetic resonance phenomenon. Full article

(This article belongs to the Special Issue Symmetry Breaking in Quantum Phenomena)

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14 pages, 254 KiB

Open AccessArticle

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

by Rutwig Campoamor-Stursberg

Symmetry 2016, 8(3), 15; https://doi.org/10.3390/sym8030015 - 17 Mar 2016

Cited by 6 | Viewed by 4255

Abstract

A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints [...] Read more.

A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems. Full article

(This article belongs to the Special Issue Symmetry and Integrability)

23 pages, 289 KiB

Open AccessArticle

The Riccati System and a Diffusion-Type Equation

by Erwin Suazo, Sergei K. Suslov and José M. Vega-Guzmán

Mathematics 2014, 2(2), 96-118; https://doi.org/10.3390/math2020096 - 15 May 2014

Cited by 15 | Viewed by 6600

Abstract

We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in [...] Read more.

We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance. Full article

(This article belongs to the Special Issue Mathematics on Partial Differential Equations)

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