Relativistic Ermakov–Milne–Pinney Systems and First Integrals

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.

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Relativistic Ermakov-Milne-Pinney Systems and First Integrals

10.20944/preprints202101.0143.v1 ◽

2021 ◽

Author(s):

Fernando Haas

Keyword(s):

Physical Interpretation ◽

Relativistic Effects ◽

First Integral ◽

First Integrals ◽

Nonlinear Superposition ◽

Relativistic Limit ◽

Guiding Principle ◽

Relativistic Analog ◽

Relativistic Extension ◽

Pseudo Potential

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 as 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.

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Optical solitons for the resonant nonlinear Schrödinger's equation with time-dependent coefficients by the first integral method

Optik ◽

10.1016/j.ijleo.2014.01.013 ◽

2014 ◽

Vol 125 (13) ◽

pp. 3107-3116 ◽

Cited By ~ 86

Author(s):

M. Eslami ◽

M. Mirzazadeh ◽

B. Fathi Vajargah ◽

Anjan Biswas

Keyword(s):

Integral Method ◽

First Integral ◽

Optical Solitons ◽

Time Dependent ◽

First Integral Method ◽

Schrödinger’S Equation ◽

Schrödinger's Equation ◽

Nonlinear Schrodinger’S Equation ◽

Nonlinear Schrödinger’S Equation

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A New Generalized Nonlinear Superposition Theory and its Applications in Modeling Corrosion-Fatigue

Volume 6A: Materials and Fabrication ◽

10.1115/pvp2013-97633 ◽

2013 ◽

Cited By ~ 3

Author(s):

Zhigang Wei ◽

Limin Luo ◽

Marek Rybarz ◽

Kamran Nikbin

Keyword(s):

Synergistic Effect ◽

Stress Corrosion ◽

Corrosion Fatigue ◽

Stress Corrosion Cracking ◽

Failure Mechanisms ◽

Corrosion Cracking ◽

Time Dependent ◽

Linear Superposition ◽

Nonlinear Superposition ◽

Cycle Dependent

Corrosion-fatigue and stress corrosion cracking have long been recognized as the principal degradation and failure mechanisms of materials under combined corrosive environment and sustained/cyclic loading conditions. These phenomena are strongly material and environment dependent, and cycle-dependent fatigue and time-dependent matter diffusion/chemical reaction at the crack tip can be operational simultaneously. How to include these cycle-dependent and time-dependent phenomena in a single model and how to study the failure mechanisms interaction are big challenges posed to material scientists and engineers. In this paper the current linear superposition theories for modeling cycle-dependent and time-dependent corrosion-fatigue and stress corrosion cracking phenomena are reviewed first. Subsequently, a generalized nonlinear superposition theory is proposed to incorporate possible nonlinear interaction or synergistic effect among the underlying mechanisms. The unified model derived from the new theory, depending on the specific materials and loading condition and environment, can be reduced to pure corrosion, pure fatigue, stress corrosion cracking and corrosion-fatigue. Finally, for the first time, a new breakthrough parameter is defined in this paper to quantitatively describe the interaction or synergistic effect between two different operating mechanisms, such as time- and cycle-dependent mechanisms.

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Integrals of the motion and green functions for time-dependent mass harmonic oscillators

Revista Mexicana de Física ◽

10.31349/revmexfis.64.30 ◽

2018 ◽

Vol 64 (1) ◽

pp. 30

Author(s):

Surarit Pepore

Keyword(s):

Harmonic Oscillator ◽

Harmonic Oscillators ◽

Time Dependent ◽

Green Functions ◽

Feynman Path Integral ◽

Feynman Path ◽

Damped Harmonic Oscillator ◽

System Trajectory ◽

Dependent Mass ◽

Motion Method

The application of the integrals of the motion of a quantum system in deriving Green function or propagator is established. The Greenfunction is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phasespace. The explicit expressions for the Green functions of the damped harmonic oscillator, the harmonic oscillator with strongly pulsatingmass, and the harmonic oscillator with mass growing with time are obtained in co-ordinate representations. The connection between theintegrals of the motion method and other method such as Feynman path integral and Schwinger method are also discussed.

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On combined optical solitons of the one-dimensional Schrödinger’s equation with time dependent coefficients

Open Physics ◽

10.1515/phys-2016-0003 ◽

2016 ◽

Vol 14 (1) ◽

pp. 65-68 ◽

Cited By ~ 13

Author(s):

Bulent Kilic ◽

Mustafa Inc ◽

Dumitru Baleanu

Keyword(s):

Soliton Solutions ◽

First Integral ◽

Optical Solitons ◽

Time Dependent ◽

Ansatz Method ◽

One Dimensional ◽

Optical Metamaterials ◽

Schrödinger’S Equation ◽

The One ◽

Schrödinger's Equation

AbstractThis paper integrates dispersive optical solitons in special optical metamaterials with a time dependent coefficient. We obtained some optical solitons of the aforementioned equation. It is shown that the examined dependent coefficients are affected by the velocity of the wave. The first integral method (FIM) and ansatz method are applied to reach the optical soliton solutions of the one-dimensional nonlinear Schrödinger’s equation (NLSE) with time dependent coefficients.

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Time-dependent coupled harmonic oscillators: Classical and quantum solutions

International Journal of Modern Physics E ◽

10.1142/s0218301314500487 ◽

2014 ◽

Vol 23 (09) ◽

pp. 1450048 ◽

Cited By ~ 2

Author(s):

D. X. Macedo ◽

I. Guedes

Keyword(s):

Canonical Transformation ◽

Unitary Transformation ◽

Coupling Parameter ◽

Wave Functions ◽

Harmonic Oscillators ◽

Time Dependent ◽

Classical Solutions ◽

Arbitrary System ◽

Coupled Harmonic Oscillators ◽

The Masses

In this work we present the classical and quantum solutions for an arbitrary system of time-dependent coupled harmonic oscillators, where the masses (m), frequencies (ω) and coupling parameter (k) are functions of time. To obtain the classical solutions, we use a coordinate and momentum transformations along with a canonical transformation to write the original Hamiltonian as the sum of two Hamiltonians of uncoupled harmonic oscillators with modified time-dependent frequencies and unitary masses. To obtain the exact quantum solutions we use a unitary transformation and the Lewis and Riesenfeld (LR) invariant method. The exact wave functions are obtained by solving the respective Milne–Pinney (MP) equation for each system. We obtain the solutions for the system with m1 = m2 = m0eγt, ω1 = ω01e-γt/2, ω2 = ω02e-γt/2 and k = k0.

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Relativistic extension of a charge-conservative finite element solver for time-dependent Maxwell-Vlasov equations

Physics of Plasmas ◽

10.1063/1.5004557 ◽

2018 ◽

Vol 25 (1) ◽

pp. 013109 ◽

Cited By ~ 14

Author(s):

D.-Y. Na ◽

H. Moon ◽

Y. A. Omelchenko ◽

F. L. Teixeira

Keyword(s):

Finite Element ◽

Time Dependent ◽

Vlasov Equations ◽

Relativistic Extension ◽

A Charge

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A general time-dependent invariant for and integrability of the quadratic system

Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences ◽

10.1098/rspa.1994.0044 ◽

1994 ◽

Vol 444 (1922) ◽

pp. 643-650 ◽

Cited By ~ 4

Keyword(s):

Direct Method ◽

First Integral ◽

Quadratic System ◽

Time Dependent ◽

Polynomial Invariant ◽

Polynomial Invariants ◽

Linear Polynomial ◽

A Direct Method ◽

Special Case ◽

General Time

We derive a general time-dependent invariant (first integral) for the quadratic system (QS) that requires only one condition on the coefficients of the QS. The general invariant could yield asymptotic behaviour of phase-space trajectories. With more conditions imposed on the coefficients, the general invariant reduces to polynomial form and is equivalent to polynomial invariants found using a direct method. For the special case of a linear polynomial invariant where one of the variables is analytically invertible, the solution of the QS is reduced to a quadrature.

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