Variational symmetries, conserved quantities and identities for several equations of mathematical physics

Donchev, Veliko

doi:10.1063/1.4867626

Cited by 13 publications

(13 citation statements)

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“…18 This approach was developed by Gustav Herglotz, 19 and it has recently been explored by several authors (see, for example, Refs. 18,[20][21][22][23][24][25][26][27]. A Lagrangian that depends explicity on S can be used to derive equations of motion that describe certain types of non-conservative and dissipative systems, and it contains the standard conservative Lagrangian as a special case.…”

Section: Sec II B)mentioning

confidence: 99%

“…A Lagrangian that depends explicity on S can be used to derive equations of motion that describe certain types of non-conservative and dissipative systems, and it contains the standard conservative Lagrangian as a special case. Some investigators have produced generalized versions of Noether's theorems stemming from the symmetries of the Herglotz Lagrangian, 20,23,24 and have used these to discover conservation laws for non-conservative systems 27 (see also Ref. 25 for an analysis of adiabatic invariants for non-conservative systems, and Ref.…”

Section: Sec II B)mentioning

confidence: 99%

“…We therefore need to be careful in handling the terms proportional to f in eq. (27). At some arbitrary time t, eq.…”

Section: B Second Variationmentioning

confidence: 99%

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When action is not least for systems with action-dependent Lagrangians

Ryan¹

2022

Preprint

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The dynamics of some non-conservative and dissipative systems can be derived by calculating the first variation of an action-dependent action, according to the variational principle of Herglotz. This is directly analogous to the variational principle of Hamilton commonly used to derive the dynamics of conservative systems. In a similar fashion, just as the second variation of a conservative system's action can be used to infer whether that system's possible trajectories are dynamically stable, so too can the second variation of the actiondependent action be used to infer whether the possible trajectories of non-conservative and dissipative systems are dynamically stable. In this paper I show, generalizing earlier analyses of the second variation of the action for conservative systems, how to calculate the second variation of the action-dependent action and how to apply it to two physically important systems: a time-independent harmonic oscillator and a time-dependent harmonic oscillator.

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Section: Sec II B)mentioning

confidence: 99%

Section: Sec II B)mentioning

confidence: 99%

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When action is not least for systems with action-dependent Lagrangians

Ryan¹

2022

Preprint

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“…Donchev applied the generalized variational principle of Herglotz type and its Noether's theorem to Bö cher equation and nonlinear Schrö dinger equation, etc. that these equations do not have a variational description with the classical variational principle [4]. Santos et al studied the variational problem of Herglotz type with time delay and obtained the corresponding Noether's theorem [5].…”

Section: Introductionmentioning

confidence: 99%

Noether's symmetry and conserved quantity for a time-delayed Hamiltonian system of Herglotz type

Zhang¹

2018

R. Soc. open sci.

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The variational problem of Herglotz type and Noether's theorem for a time-delayed Hamiltonian system are studied. Firstly, the variational problem of Herglotz type with time delay in phase space is proposed, and the Hamilton canonical equations with time delay based on the Herglotz variational problem are derived. Secondly, by using the relationship between the non-isochronal variation and the isochronal variation, two basic formulae of variation of the Hamilton–Herglotz action with time delay in phase space are derived. Thirdly, the definition and criterion of the Noether symmetry for the time-delayed Hamiltonian system are established and the corresponding Noether's theorem is presented and proved. The theorem we obtained contains Noether's theorem of a time-delayed Hamiltonian system based on the classical variational problem and Noether's theorem of a Hamiltonian system based on the variational problem of Herglotz type as its special cases. At the end of the paper, an example is given to illustrate the application of the results.

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“…In recent years, the Herglotz variational principle and its symmetries have been applied in finite and infinite dimensional non-conservative dynamic systems, quantum systems, thermodynamics, optimal control theory, and other fields. [24][25][26][27][28][29][30][31][32][33] In Ref. [34], the simple and physically meaningful Lagrangians of Herglotz type were constructed, which describe a wide range of non-conservative classical and quantum systems, for example, vibrating string under viscous forces, non-conservative electromagnetic theory, non-conservative Schrödinger equation, non-conservative Klein-Gordon equation, etc.…”

Section: Introductionmentioning

confidence: 99%

Conservation laws for Birkhoffian systems of Herglotz type

Zhang¹,

Tian²

2018

Chinese Phys. B

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Conservation laws for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are studied. We propose a new differential variational principle, called the Pfaff-Birkhoff-d'Alembert principle of Herglotz type. Birkhoff's equations for both the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are obtained. According to the relationship between the isochronal variation and the nonisochronal variation, the conditions of the invariance for the Pfaff-Birkhoff-d'Alembert principle of Herglotz type are given. Then, the conserved quantities for the Birkhoffian system and the constrained Birkhoffian system of Herglotz type are deduced. Furthermore, the inverse theorems of the conservation theorems are also established.

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Variational symmetries, conserved quantities and identities for several equations of mathematical physics

Cited by 13 publications

References 6 publications

When action is not least for systems with action-dependent Lagrangians

When action is not least for systems with action-dependent Lagrangians

Noether's symmetry and conserved quantity for a time-delayed Hamiltonian system of Herglotz type

Conservation laws for Birkhoffian systems of Herglotz type

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