A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients

Cieśliński, Jan L.; Nikiciuk, Tomasz

doi:10.1088/1751-8113/43/17/175205

Cited by 95 publications

(162 citation statements)

References 43 publications

(100 reference statements)

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“…so the discrete curve x obtained by iteration of the contact map satisfies the discrete Herglotz variational principle for L. Note that x need not be a scalar: the Lagrangian L = 1 2 |ẋ| 2 − V (x) − αz yields the analogous multi-component equation. This contrasts many other variational descriptions of the damped harmonic oscillator, which only apply to the scalar case [44,15]. The same comment applies to the following discretization, which we write down for scalar x but can easily be adapted to higher dimensions.…”

Section: Discrete Herglotz Variational Principlementioning

confidence: 93%

Contact variational integrators

Vermeeren¹,

Bravetti²,

Seri³

2019

J. Phys. A: Math. Theor.

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We present geometric numerical integrators for contact flows that stem from a discretization of Herglotz' variational principle. First we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle. Then we discuss the backward error analysis of our variational integrators, including the construction of a modified Lagrangian. Throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues.

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Section: Discrete Herglotz Variational Principlementioning

confidence: 93%

Contact variational integrators

Vermeeren¹,

Bravetti²,

Seri³

2019

J. Phys. A: Math. Theor.

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“…After the oscillators considered in Sections 3 and 4, we change gears slightly and consider two classes of variable-coefficients dissipative oscillators that have been treated recently in [6].…”

Section: Dissipative Oscillatorsmentioning

confidence: 99%

“…Section 4 investigates a special case of the generalized anharmonic oscillator given in [7]. Section 5 considers two classes of dissipative dynamical systems with variable coefficients [6]. The Lane-Emden equation [10] is treated in Section 6.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier

Tanriver

Choudhury

Gambino

2015

International Journal of Non-Linear Mechanics

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In this paper, we employ the technique of Jacobi Last Multiplier (JLM) to derive Lagrangians for several important and topical classes of non-linear second-order oscillators, including systems with variable and parametric dissipation, a generalized anharmonic oscillator, and a generalized Lane–Emden equation. For several of these systems, it is very difficult to obtain the Lagrangians directly, i.e., by solving the inverse problem of matching the Euler–Lagrange equations to the actual oscillator equation. In order to facilitate the derivation of exact solutions, and also investigate possible isochronous behavior in the analyzed systems, we next invoke some recent theoretical results and attempt to map the potential term to either the simple harmonic oscillator or the isotonic potential for specific values of the coefficient parameters of each non-linear oscillator. We find non-trivial parameter sets corresponding to isochronous dynamics in some of the considered systems, but none in others. Finally, the Lagrangians obtained here are coupled to Noether׳s theorem, leading to non-trivial conservation laws for several of the oscillators

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“…They were introduced by Arnold since 1978 [1], R. A. El-Nabulsi (B) College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641112, Sichuan, China e-mail:nabulsiahmadrami@yahoo.fr but they were ignored for a good period of time due to the lack of their Hamiltonian formalisms. Their roles in the theory of differential equations [2][3][4], dissipative systems [5][6][7][8][9][10][11][12] and theoretical physics [13][14][15][16][17] are well appreciated. Despite the fact that a solid and comprehensive Hamiltonian formulation is missed, NSL are considered to be good candidates to explain dissipative dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Non-standard Lagrangians in rotational dynamics and the modified Navier–Stokes equation

El-Nabulsi

2014

Nonlinear Dyn

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We report some of the implications of nonstandard Lagrangians in rotational dynamics. After deriving a new form of the Euler-Lagrange equation from the variational principle for the case of a particle moving in a non-inertial frame and subject to a velocity constraint, we deduce the modified Coriolis force, the modified centrifugal force and the modified transverse force. We then discuss the modified equation of motion relative to Earth, the free-fall problem and the Foucault pendulum issue. We show that the modified dynamics results on a modified Navier-Stokes equation where some features were raised and discussed accordingly.

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A direct approach to the construction of standard and non-standard Lagrangians for dissipative-like dynamical systems with variable coefficients

Cited by 95 publications

References 43 publications

Contact variational integrators

Contact variational integrators

Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier

Non-standard Lagrangians in rotational dynamics and the modified Navier–Stokes equation

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