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Cruz, Manuel Jesús de la Torre; Gaspar, Néstor; Jiménez-Lara, Lidia; Linares, Román

doi:10.1016/j.aop.2017.02.016

Cited by 6 publications

(19 citation statements)

References 30 publications

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“…In this section we give a short summary of the characteristics of the system that are relevant for our analysis of the relation between the so called extended rigid body and the simple pendulum. We base our discussion in [19,20,33,37].…”

Section: The Euler Equations and Its Sl(2 R) Symmetrymentioning

confidence: 99%

“…In this latter case the Casimir surface can also have the geometry of an elliptic cone or a hyperbolic cylinder in the proper limit situations. A complete classification of the geometrical shapes of the Casimir surface C 1 can be found in [37]. Regarding the solutions of the Euler equations (12), given the ordering (10) of the dimensionless inertia parameters, the solutions depend of the relative value between e 0 and e 2 .…”

Section: A the Euler Equations And The Casimir Functionsmentioning

confidence: 99%

“…The whole classification can be found, for instance, in [37]. Because is relevant for our discussion, it is important to emphasize that if instead we take the Casimir surface N as the Hamiltonian surface, then the generator X G is given by…”

Section: A the Euler Equations And The Casimir Functionsmentioning

confidence: 99%

“…Even more, according to [33] the pendulum can be obtained from the extended rigid body if, from the geometrical point of view, the surfaces of the two new Casimir functions have the shape of elliptic cylinders, which leads to the fact that the corresponding Lie algebra is ISO (2). Very recently using a representation of the rigid body in terms of two free parameters e 0 and κ [37] instead of the usual five parameters: energy E, magnitude of the angular momentum L, and the three principal moments of inertia I 1 , I 2 and I 3 , the classification of the inequivalent SL(2, R) combinations of the Casimir functions was discussed both algebraically and geometrically. It turns out that whereas the geometry of the Casimir function that represents the square of the angular momentum continues being an S 2 sphere, the geometry of the Casimir function associated to the kinetic energy which is an ellipsoid in the five parameters representation of the rigid body, is replaced by an elliptic hyperboloid that can be either of one or two sheets depending on the numerical values of both e 0 and κ, making the classification of the SL(2, R) combinations of the Casimir functions richer.…”

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confidence: 99%

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The Extended Rigid Body and the Pendulum Revisited

Cruz¹,

Gaspar²,

Linares³

2020

Nelin. Dinam.

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In this paper we revisit the construction by which the SL(2, R) symmetry of the Euler equations allows to obtain the simple pendulum from the rigid body. We begin reviewing the original relation found by Holm and Marsden in which, starting from the two Casimir functions of the extended rigid body with Lie algebra ISO(2) and introducing a proper momentum map, it is possible to obtain both the Hamiltonian and equations of motion of the pendulum. Important in this construction is the fact that both Casimirs have the geometry of an elliptic cylinder. By considering the whole SL(2, R) symmetry group, in this contribution we give all possible combinations of the Casimir functions and the corresponding momentum maps that produce the simple pendulum, showing that this system can also appear when the geometry of one of the Casimirs is given by a hyperbolic cylinder and the another one by an elliptic cylinder. As a result we show that from the extended rigid body with Lie algebra ISO(1, 1), it is possible to obtain the pendulum but only in circulating movement. Finally, as a by product of our analysis we provide the momentum maps that give origin to the pendulum with an imaginary time. Our discussion covers both the algebraic and the geometric point of view. I. INTRODUCTIONThe simple pendulum and the torque free rigid body are two well understood physical systems in both classical and quantum mechanics. The first systematic study of the pendulum is attributed to Galileo Galilei around 1602 and its dynamical description culminated with the development of the elliptic functions by Abel [1] and Jacobi [2,3], which turn out to be the analytical solutions to the equation of motion of the pendulum (for a review of elliptic functions see for instance [4][5][6][7][8][9] and [10][11][12] for the solutions of the pendulum). The quantization of the pendulum is based on the equivalence between the Schrödinger equation and the Mathieu differential equation, result developed originally by Condon in 1928 [13] and source of subsequent analysis of different aspects of the quantum system [14,15]. On the other hand in 1758 Euler showed that the equations of motion that describe the rotation of a rigid body form a vectorial quasilinear first-order ordinary differential equations set [16]. A geometric construction of the solution was given later on by Poinsot [17] and analytically these solutions are given, as for the simple pendulum, by elliptic functions (see for instance [18][19][20][21] and references therein). The quantization of the problem was attacked first by Kramers and Ittmann [22] and since then many authors have contributed to understand deeper many aspects of the problem [23][24][25][26][27][28][29][30].Despite the old age of these problems, from time to time there are some new physical aspects uncovered about these systems that contribute to our knowledge and understanding of physics in general. The list is long and here we point out just four examples: i) In 1973 Y. Nambu, taking the Liouville theorem as a guiding principle, ...

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Section: The Euler Equations and Its Sl(2 R) Symmetrymentioning

confidence: 99%

Section: A the Euler Equations And The Casimir Functionsmentioning

confidence: 99%

Section: A the Euler Equations And The Casimir Functionsmentioning

confidence: 99%

mentioning

confidence: 99%

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The Extended Rigid Body and the Pendulum Revisited

Cruz¹,

Gaspar²,

Linares³

2020

Nelin. Dinam.

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“…associated to the 3-dimensional Lie algebra sl(2), used in the classification of rigid motions [14], and in other mechanical systems [43,7]. To evaluate the matrix of Π sl(2) at points of Q 3 in (3) we compute:…”

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confidence: 99%

On computational Poisson geometry II: Numerical methods

Evangelista-Alvarado¹,

Ruíz-Pantaleón²,

Suárez–Serrato³

2021

JCD

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We present twelve numerical methods for evaluation of objects and concepts from Poisson geometry. We describe how each method works with examples, and explain how it is executed in code. These include methods that evaluate Hamiltonian and modular vector fields, compute the image under the coboundary and trace operators, the Lie bracket of differential 1-forms, gauge transformations, and normal forms of Lie-Poisson structures on R 3 . The complexity of each of our methods is calculated, and we include experimental verifications on examples in dimensions two and three.

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Symmetries and conservation laws for the generalized n‐dimensional Ermakov system

Paliathanasis

León

2022

Math Methods in App Sciences

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We revise recent results on the classification of the generalized three-dimensional Hamiltonian Ermakov system. We show that a statement published recently is incorrect, while the solution for the classification problem was incomplete. We present the correct classification for the three-dimensional system by using results which related the background space with the dynamics.Finally, we extend our results for the generalized n -dimensional Hamiltonian Ermakov system.

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Classification of the classicalSL(2,R)gauge transformations in the rigid body

Cited by 6 publications

References 30 publications

The Extended Rigid Body and the Pendulum Revisited

The Extended Rigid Body and the Pendulum Revisited

On computational Poisson geometry II: Numerical methods

Symmetries and conservation laws for the generalized n‐dimensional Ermakov system

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