A unified framework for mechanics: Hamilton–Jacobi equation and applications

Balseiro, Paula; Marrero, Juan Carlos; Diego, David Martı́n de; Padrón, Edith

doi:10.1088/0951-7715/23/8/006

Cited by 24 publications

(41 citation statements)

References 43 publications

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“…[24] for this system (seen as an almost-Poisson system), and for the natural fibration Π := π Q | D : D → Q, coincides with the so-called nonholonomic Hamilton-Jacobi equation of Refs. [11,23,5]. Then, based on the last corollary, we can conclude that the nonholonomic Hamilton-Jacobi equation is precisely our π Q | D -HJE (restricted to weakly Lagrangian and Lagrangian sections).…”

Section: Remark 217mentioning

confidence: 73%

A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds

Grillo¹,

Padrón²

2016

Journal of Geometry and Physics

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In this paper we develope, in a geometric framework, a Hamilton-Jacobi Theory for general dynamical systems. Such a theory contains the classical theory for Hamiltonian systems on a cotangent bundle and recent developments in the framework of general symplectic, Poisson and almost-Poisson manifolds (including some approaches to a Hamilton-Jacobi theory for nonholonomic systems). Given a dynamical system, we show that every complete solution of its related Hamilton-Jacobi Equation (HJE) gives rise to a set of first integrals, and vice versa. From that, and in the context of symplectic and Poisson manifolds, a deep connection between the HJE and the (non)commutative integrability notion, and consequently the integrability by quadratures, is stablished. Moreover, in the same context, we find conditions on the complete solutions of the HJE that also ensures integrability by quadratures, but they are weaker than those related to the (non)commutative integrability. Examples are developed along all the paper in order to illustrate the theoretical results.

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Section: Remark 217mentioning

confidence: 73%

A Hamilton–Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds

Grillo¹,

Padrón²

2016

Journal of Geometry and Physics

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“…Therefore, a HJ theory finds solutions on the lower dimensional manifold Q and retrieves them on the higher dimensional manifold T * Q by the existence of a section γ of the cotangent bundle which is the solution γ of the Hamilton-Jacobi equation (1). This picture (3) can be devised in different situations, as it is the case of nonholonomic systems [14,23,32,37,39,50,51], geometric mechanics on Lie algebroids [5] and almost-Poisson manifolds, singular systems [41], Nambu-Poisson framework [44], control theory [7], classical field theories [38,40,45], partial differential equations in general [64], the geometric discretization of the Hamilton-Jacobi equation [43,52], and others [6,13].…”

Section: Introductionmentioning

confidence: 99%

A Hamilton–Jacobi formalism for higher order implicit Lagrangians

Esen¹,

León²,

Sardón³

2020

J. Phys. A: Math. Theor.

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In this paper, we present a generalization of a Hamilton-Jacobi theory to higher order implicit differential equations. We propose two different backgrounds to deal with higher order implicit Lagrangian theories: the Ostrogradsky approach and the Schmidt transform, which convert a higher order Lagrangian into a first order one. The Ostrogradsky approach involves the addition of new independent variables to account for higher order derivatives, whilst the Schmidt transform adds gauge invariant terms to the Lagrangian function. In these two settings, the implicit character of the resulting equations will be treated in two different ways in order to provide a Hamilton-Jacobi equation. On one hand, the implicit differential equation will be a Lagrangian submanifold of a higher order tangent bundle and it is generated by a Morse family. On the other hand, we will rely on the existence of an auxiliary section of a certain bundle that allows the construction of local vector fields, even if the differential equations are implicit. We will illustrate some examples of our proposed schemes, and discuss the applicability of the proposal.

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“…For instance, the statement and applications of the Hamilton-Jacobi method for non-holonomic and holonomic mechanical systems is done in [8,16,40,32,49,50], the geometric treatment of the theory for dynamical systems described by singular Lagrangians is analyzed in [37,42,43], the application to control theory is given in [6,61,62], and the generalization for higher-order dynamical systems is established in [18,19]. Moreover, the Hamilton-Jacobi theory has been extended for mechanical systems which are described using more general geometrical frameworks, such as Lie algebroids [4,36], almost-Poisson manifolds [44], and fiber bundles in general [17], and the relationship between the Hamilton-Jacobi equation and some other geometric structures in mechanics are analyzed in [7,15]. Finally, the geometric discretization of the Hamilton-Jacobi equation is also considered in [5,51].…”

Section: Introductionmentioning

confidence: 99%