On a class of integrable time-dependent dynamical systems

Bartuccelli, Michele V.; Gentile, Guido

doi:10.1016/s0375-9601(02)01731-0

Cited by 8 publications

(10 citation statements)

References 6 publications

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“…This is one of the main results of this paper, since it strongly suggests that our statistical mechanical theory of a closed universe might complement the standard cosmological model of an oscillating universe based on purely general relativity grounds [1]. The implications of this theory on the concept of time can be extracted by first noticing that the dynamics (22) for the entropy of the universe corresponds the Hamiltonian [18] H S,Ṡ, t = 1 2…”

Section: Entropic Oscillatormentioning

confidence: 71%

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Statistical Mechanical Theory of a Closed Oscillating Universe

Pérez‐Madrid

Santamaría‐Holek

2009

Found Phys

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Based on Newton's laws reformulated in the Hamiltonian dynamics combined with statistical mechanics, we formulate a statistical mechanical theory supporting the hypothesis of a closed oscillating universe. We find that the behaviour of the universe as a whole can be represented by a free entropic oscillator whose lifespan is nonhomogeneous, thus implying that time is shorter or longer according to the state of the universe itself given through its entropy. We conclude that time reduces to the entropy production of the universe and that a nonzero entropy production means that local fluctuations could exist giving rise to the appearance of masses and to the curvature of the space.

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Section: Entropic Oscillatormentioning

confidence: 71%

“…(15) that the distribution vector f (t) corresponding to the closed universe possesses an oscillatory behavior and in Section 3, we have established the relation between f (t) and the entropy S through Eq. (18). Therefore, as a consequence of these S(t) will also have an oscillatory behavior which can be described by d dt…”

Section: Entropic Oscillatormentioning

confidence: 99%

Statistical Mechanical Theory of a Closed Oscillating Universe

Pérez‐Madrid

Santamaría‐Holek

2009

Found Phys

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“…We shall focus on the theory on an infinite spatial interval (a, b) = (−∞, ∞). 15 Picking the 'negative' branch 16 of the square root, then the spectral function z(w; τ, σ) in (2.13) has the finite limit z = w+σ+τ w+σ−τ → z ∞ = −1 at σ → ±∞. Hence, assuming J ± is bounded for large |σ|, the Lax connection L ± = 1 2 (1 + z ±1 )J ± vanishes at spatial infinity.…”

Section: Jhep11(2020)020mentioning

confidence: 99%

“…Then for any (polynomial) decaying boundary conditions on the fields at spatial infinity, the periodicity condition (4.4) on L τ will be broken at sufficiently higher order in this expansion. 15 It seems much harder to satisfy the periodicity condition (4.4) on a spatial circle due to the explicit non-periodic σ dependence in the Lax connections for the time-dependent models. 16 We note that it is consistent to 'pick a branch' here: e.g., by assuming that w has an imaginary part, it then follows that the sign of the square root does not change from σ = −∞ to σ = ∞.…”

Section: Jhep11(2020)020mentioning

confidence: 99%

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Sigma models with local couplings: a new integrability-RG flow connection

Hoare

Levine

Tseytlin

2020

J. High Energ. Phys.

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We consider several classes of σ-models (on groups and symmetric spaces, η-models, ⋋-models) with local couplings that may depend on the 2d coordinates, e.g. on time τ . We observe that (i) starting with a classically integrable 2d σ-model, (ii) formally promoting its couplings hα to functions hα(τ ) of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that hα(τ ) must solve the 1-loop RG equations of the original theory with τ interpreted as RG time. This provides a novel example of an ‘integrability-RG flow’ connection. The existence of a Lax connection suggests that these time-dependent σ-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such σ-models with D-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a (D + 2)-dimensional conformal σ-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.

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Integrable modulation, curl forces and parametric Kapitza equation with trapping and escaping

Guha

Garai

2021

Nonlinear Dyn

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In this present communication, the integrable modulation problem has been applied to study parametric extension of the Kapitza rotating shaft problem, which is a prototypical example of curl force as formulated by Berry and Shukla in (JPA 45:305201, 2012) associated with simple saddle potential. The integrable modulation problems yield parametric timedependent integrable systems. The Hamiltonian and first integrals of the linear and nonlinear parametric Kapitza equation (PKE) associated with simple and monkey saddle potentials have been given. The construction has been illustrated by choosing ω(t) = a + b cos t and that maps to Mathieu-type equations, which yield Mathieu extension of PKE. We study the dynamics of these equations. The most interesting finding is the mixed mode of particle trapping and escaping via the heteroclinic orbits depicted with the parametric Mathieu-Kapitza equation, which are absent in the case of nonparametric cases.Dedicated to Sir Michael Berry on his 80-th birthday with great respect and admiration.

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On a class of integrable time-dependent dynamical systems

Cited by 8 publications

References 6 publications

Statistical Mechanical Theory of a Closed Oscillating Universe

Statistical Mechanical Theory of a Closed Oscillating Universe

Sigma models with local couplings: a new integrability-RG flow connection

Integrable modulation, curl forces and parametric Kapitza equation with trapping and escaping

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