Nonstandard conserved Hamiltonian structures in dissipative/damped systems: Nonlinear generalizations of damped harmonic oscillator

Abstract

Articles you may be interested inAn observation of quadratic algebra, dual family of nonlinear coherent states and their non-classical properties, in the generalized isotonic oscillator A simple and unified approach to identify integrable nonlinear oscillators and systems Bifurcation in the entrained state of a nonlinear oscillator under sinusoidal perturbation with the 2nd harmonics AIP Conf.In this paper we point out the existence of a remarkable nonlocal transformation between the damped harmonic oscillator and a modified Emden-type nonlinear oscillator equation with linear forcing, ẍ + ␣xẋ + ␤x 3 + ␥x = 0, which preserves the form of the time independent integral, conservative Hamiltonian, and the equation of motion. Generalizing this transformation we prove the existence of nonstandard conservative Hamiltonian structure for a general class of damped nonlinear oscillators including Liénard-type systems. Further, using the above Hamiltonian structure for a specific example, namely, the generalized modified Emden equation ẍ + ␣x q ẋ + ␤x 2q+1 = 0, where ␣, ␤, and q are arbitrary parameters, the general solution is obtained through appropriate canonical transformations. We also present the conservative Hamiltonian structure of the damped Mathews-Lakshmanan oscillator equation. The associated Lagrangian description for all the above systems is also briefly discussed.

A class of exact solutions of the Liénard-type ordinary nonlinear differential equation

“…Generally, a linear oscillation can be described by the equationẍ + f (t)ẋ + g(t)x = 0. The mathematical properties of Liénard-type equations have been intensively investigated from both mathematical and physical points of view, and their study remains an active field of research in mathematical physics [20][21][22][23][24][25][26][27][28][29][30]. Several methods of integrability, such as the Lie symmetries method [31,32] and the Weierstrass integrability, introduced in [33], were used to study the Liénard equation and the relations between the Riccati and Liénard equations, respectively.…”

Lie point symmetries classification of the mixed Liénard-type equation

“…Equation (51) admits a time-independent Hamiltonian for all values of f 1 and g 1 [31] and hence, one can establish the integrability of Eq. (49) also.…”

“…Equation (43) admits a time-independent Hamiltonian for all values of α and β [30,31]. Consequently, the integrability of Eq.…”

Position-dependent mass Lagrangians: nonlocal transformations, Euler–Lagrange invariance and exact solvability

“…For more details on this issue the reader may refer to [17,22]. Obviously, moreover, equation (2) is a quadratic Liénard-type differential equation (quadratic in terms ofẋ 2 in (2)) which serves as a very interesting model in both physics and mathematics (cf., e.g., the sample of references [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37] and related references cited therein).…”