A note on generalization of the Ermakov–Lewis invariant and its demystification

2019 ◽  
Vol 34 (03) ◽  
pp. 1950021 ◽  
Author(s):  
Partha Guha ◽  
Anindya Ghose-Choudhury
Keyword(s):  
Physical Basis ◽  
Time Dependent ◽  
Nonlocal Transformation

Recently, a physical basis for the Ermakov–Lewis invariant has been pointed out by Padmanabhan [Mod. Phys. Lett. A 33, 1830005 (2018)] and the most general form of the equations admitting such an invariant has been identified. Further generalizations to other equations beyond the Ermakov–Lewis system are derived by considering a more general form of a nonlocal transformation of the Sundman type and the corresponding invariants are derived. Reductions to the time-dependent isotonic oscillator are also obtained.

scholarly journals Nonlocal transformations of the generalized Liénard type equations and dissipative Ermakov-Milne-Pinney systems

2019 ◽  
Vol 16 (07) ◽  
pp. 1950107 ◽  
Author(s):  
Partha Guha ◽  
A. Ghose-Choudhury
Keyword(s):  
Harmonic Oscillator ◽  
Transformation Method ◽  
Natural Generalization ◽  
First Integrals ◽  
Time Dependent ◽  
Liénard Equation ◽  
Linear Harmonic Oscillator ◽  
Linearization Problem ◽  
Lienard Equation ◽  
Nonlocal Transformation

We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Liénard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov–Lewis invariant for a time-dependent oscillator to (coupled) Liénard and Liénard type equations. We also study the linearization problem for the coupled Liénard equation using nonlocal transformations and derive coupled dissipative Ermakov-Milne-Pinney equation. As an offshoot of this nonlocal transformation method when the standard Liénard equation, [Formula: see text], is mapped to that of the linear harmonic oscillator equation, we obtain a relation between the functions [Formula: see text] and [Formula: see text] which is exactly similar to the condition derived in the context of isochronicity of the Liénard equation.

The physical basis of mind.

1877 ◽  
Author(s):  
George Henry Lewes
Keyword(s):  
Physical Basis

The Physical Basis of Heredity

1922 ◽  
Vol 126 (3) ◽  
pp. 177-178 ◽  
Author(s):  
James B. Kelly
Keyword(s):  
Physical Basis
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