On the Stability of a Nonlinear Oscillator with Higher Derivatives

Kaparulin, D. S.; Lyakhovich, S. L.

doi:10.1007/s11182-015-0419-7

Cited by 18 publications

(38 citation statements)

References 24 publications

(35 reference statements)

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“…This fact has been established in [10,11] (see also Ref. [12]) where the corresponding families of Hamiltonian structures have been constructed. The main advantage of the alternative Hamiltonian formulation obtained in such a way is that this may correspond to a positive-definite Hamiltonian.…”

Section: Introductionmentioning

confidence: 75%

The odd-order Pais–Uhlenbeck oscillator

Masterov

2016

Nuclear Physics B

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Section: Introductionmentioning

confidence: 75%

The odd-order Pais–Uhlenbeck oscillator

Masterov

2016

Nuclear Physics B

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“…We suppose that we have the coordinates x µ , µ = 0, 1, 2 such that x i , i = 1, 2 are the space coordinates, while x 0 is the time. The ADM variables (17), (18), (19) are very convenient to describe the metric once the spacetime is decomposed in space and time.…”

Section: The First Order Formulationmentioning

confidence: 99%

“…It has been also found that the interactions can be included into the PU equations leaving the dynamics stable [1]. Furthermore, the stable higher derivative PU equation with interaction still admits Hamiltonian formulation in the corresponding first order formalism, though the stable vertices are non-Lagrangian in the higher derivative equations of motion [19].One of the frequently discussed higher derivative field theories is the third order extension of Chern-Simons [20]. It has * g kl ( F 0 F 0 + * g sr F s F r ) .…”

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confidence: 99%

Third order extensions of 3d Chern–Simons interacting to gravity: Hamiltonian formalism and stability

Kaparulin¹,

Karataeva²,

Lyakhovich³

2018

Nuclear Physics B

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We consider inclusion of interactions between 3d Einstein gravity and the third order extensions of Chern-Simons. Once the gravity is minimally included into the third order vector field equations, the theory is shown to admit a two-parameter series of symmetric tensors with on-shell vanishing covariant divergence. The canonical energy-momentum is included into the series. For a certain range of the model parameters, the series include the tensors that meet the weak energy condition, while the canonical energy is unbounded in all the instances. Because of the on-shell vanishing covariant divergence, any of these tensors can be considered as an appropriate candidate for the right hand side of Einstein's equations. If the source differs from the canonical energy momentum, the coupling is non-Lagrangian while the interaction remains consistent with any of the tensors. We reformulate these not necessarily Lagrangian third order equations in the first order formalism which is covariant in the sense of 1 + 2 decomposition. After that, we find the Poisson bracket such that the first order equations are Hamiltonian in all the instances, be the original third order equations Lagrangian or not. The brackets differ from canonical ones in the matter sector, while the gravity admits the usual PB's in terms of ADM variables. The Hamiltonian constraints generate lapse, shift and gauge transformations of the vector field with respect to these Poisson brackets. The Hamiltonian constraint, being the lapse generator, is interpreted as strongly conserved energy. The matter contribution to the Hamiltonian constraint corresponds to 00-component of the tensor included as a source in the right hand side of Einstein equations. Once the 00-component of the tensor is bounded, the theory meets the usual sufficient condition of classical stability, while the original field equations are of the third order. IntroductionVarious higher derivative field theories are studied once and again over many decades for several reasons. Among the most frequently mentioned advantages of the higher derivative systems are the better convergence properties at classical and quantum level comparing to the analogues without higher derivatives. For discussion of various types of higher derivative models we refer to the paper [1] and references therein. The higher derivative theories are also notorious for the instability problem. The simplest stability test -boundedness of energy -is usually failed by the models with higher order equations of * dsc@phys.tsu.ru † karin@phys.tsu.ru ‡ sll@phys.tsu.ru motion. The best known exception -f (R)-gravity [2] -is stable due to very strong second class constraints. Because of that, on the constrained surface, the Hamiltonian is bounded, so the theory meets the sufficient condition for stability.Even if the energy is unbounded for general higher-derivative dynamics, the theory is not necessarily unstable. If another bounded conserved quantity exists, it stabilizes dynamics at least at classical level. For example, the free fo...

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“…It is also worth to notice that the PU oscillator equation of motion admits the interaction vertices such that do not spoil the classical stability [19,39]. These vertices are nonLagrangian, while the interacting higher-derivative equations, being brought to the first-order formalism, still remain Hamiltonian with positive Hamilton function [23,24]. In this way, the PU oscillator equation admits inclusion of interactions such that leave the dynamics stable beyond the free level and admit Hamiltonian formulation.…”

Section: Introductionmentioning

confidence: 98%

“…Also notice that all the considered examples [19,22] of stable higher-derivative models admit the interactions such that do not spoil classical stability. Further examples of stable interactions can be found in [23][24][25] for various higher-derivative models. In all these models, the canonical energy is unbounded at free level, while the stability is due to another bounded conserved quantity.…”

Section: Introductionmentioning

confidence: 99%

Multi-Hamiltonian formulations and stability of higher-derivative extensions of 3d Chern–Simons

Abakumova¹,

Kaparulin²,

Lyakhovich³

2018

Eur. Phys. J. C

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Most general third-order 3d linear gauge vector field theory is considered. The field equations involve, besides the mass, two dimensionless constant parameters. The theory admits two-parameter series of conserved tensors with the canonical energy-momentum being a particular representative of the series. For a certain range of the model parameters, the series of conserved tensors include bounded quantities. This makes the dynamics classically stable, though the canonical energy is unbounded in all the instances. The free third-order equations are shown to admit constrained multi-Hamiltonian form with the 00-components of conserved tensors playing the roles of corresponding Hamiltonians. The series of Hamiltonians includes the canonical Ostrogradski's one, which is unbounded. The Hamiltonian formulations with different Hamiltonians are not connected by canonical transformations. This means, the theory admits inequivalent quantizations at the free level. Covariant interactions are included with spinor fields such that the higher-derivative dynamics remains stable at interacting level if the bounded conserved quantity exists in the free theory. In the first-order formalism, the interacting theory remains Hamiltonian and therefore it admits quantization, though the vertices are not necessarily Lagrangian in the third-order field equations.

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On the Stability of a Nonlinear Oscillator with Higher Derivatives

Abstract: The stability of a Pais-Uhlenbeck nonlinear oscillator with higher derivatives is examined. The stability of the linear theory is demonstrated and a nonlinearity preserving the stability of the system is found.

Cited by 18 publications

References 24 publications

The odd-order Pais–Uhlenbeck oscillator

The odd-order Pais–Uhlenbeck oscillator

Third order extensions of 3d Chern–Simons interacting to gravity: Hamiltonian formalism and stability

Multi-Hamiltonian formulations and stability of higher-derivative extensions of 3d Chern–Simons

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