On the extended Euler system and the Jacobi and Weierstrass elliptic functions

Crespo, Francisco; Ferrer, S.

doi:10.3934/jgm.2015.7.151

Cited by 6 publications

(14 citation statements)

References 15 publications

(29 reference statements)

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“…Then, the solution of (3) are given by means of those functions, using the method of undetermined coefficients. For some readers could be useful to consult our paper [2] where we have studied the extended Euler system…”

Section: On Euler System Jacobi Functions and 3-eesmentioning

confidence: 99%

“…For some readers could be useful to consult our paper [2] where we have studied the extended Euler system…”

Section: On Euler System Jacobi Functions and 3-eesmentioning

confidence: 99%

“…the (1) for N = 3, considering generic values for coefficients α i and initial conditions defining the system. Relying on the work of Tricomi [13], Hille [7] and Meyer [11] dedicated to system (4), we have shown in a straightforward manner how Jacobi and Weierstrass elliptic functions in the real domain are connected with this system [2], although the tradition is to treat them separately because of the their intrinsic differences in the complex domain (see for instance Whittaker and Watson [14] and Lawden [9]). Here we will apply the same approach to the system in N -dimensions.…”

Section: On Euler System Jacobi Functions and 3-eesmentioning

confidence: 99%

“…Because the functions u i , i = 1, 2, 3 satisfy (25), they belong to the set of functions defined by the 'Jacobi elliptic functions' sn, cn, dn and their ratios. Then, following Crespo and Ferrer [2], we know our system corresponds to one of the four possible cases (Glashier systems), depending on the sign of the integrals.…”

Section: Proposition 32 For Suitable Ic the 4-ees (18) Has As Solutio...mentioning

confidence: 99%

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On the $$\varvec{N}$$ N -extended Euler system: generalized Jacobi elliptic functions

2016

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We study the integrable system of first order differential equations ω i (v) = α i j =i ω j (v), (1 ≤ i, j ≤ N ) as an initial value problem, with real coefficients α i and initial conditions ω i (0). The analysis is based on its quadratic first integrals. For each dimension N , the system defines a family of functions, generically hyperelliptic functions. When N = 3, this system generalizes the classic Euler system for the reduced flow of the free rigid body problem, thus we call it N -extended Euler system (N -EES). In this Part I the cases N = 4 and N = 5 are studied, generalizing Jacobi elliptic functions which are defined as a 3-EES. Taking into account the nested structure of the N -EES, we propose reparametrizations of the type dv * = g(ω i ) dv that separate geometry from dynamic. Some of those parametrizations turn out to be generalization of the Jacobi amplitude. In Part II we consider geometric properties of the N -system and the numeric computation of the functions involved. It will be published elsewhere.keywords: Integrable systems Generalized Euler system Jacobi and Weierstrass elliptic functions third Legendre elliptic integral arXiv:1505.06142v1 [math.DS]

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Section: On Euler System Jacobi Functions and 3-eesmentioning

confidence: 99%

“…For some readers could be useful to consult our paper [2] where we have studied the extended Euler system…”

Section: On Euler System Jacobi Functions and 3-eesmentioning

confidence: 99%

Section: On Euler System Jacobi Functions and 3-eesmentioning

confidence: 99%

Section: Proposition 32 For Suitable Ic the 4-ees (18) Has As Solutio...mentioning

confidence: 99%

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On the $$\varvec{N}$$ N -extended Euler system: generalized Jacobi elliptic functions

2016

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“…This system of differential equations is named as the N -extended Euler system (N -EES) and has been studied before in [6,8]. We have shown above that the N -EES is precisely a set of Nambu-Hamilton equations of motion and therefore it is also a Poisson-Hamiltonian system.…”

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confidence: 99%

Poisson and integrable systems through the Nambu bracket and its Jacobi multiplier

Crespo¹,

Molero²,

Ferrer³

2016

JGM

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Poisson and integrable systems are orbitally equivalent through the Nambu bracket. Namely, we show that every completely integrable system of differential equations may be expressed into the Poisson-Hamiltonian formalism by means of the Nambu-Hamilton equations of motion and a reparametrisation related by the Jacobian multiplier. The equations of motion provide a natural way for finding the Jacobian multiplier. As a consequence, we partially give an alternative proof of a recent theorem in [13]. We complete this work presenting some features associated to Hamiltonian maximally superintegrable systems.

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