Geometric Hamilton–jacobi Theory

Cariñena, José F.; Gràcia, Xavier; Marmo, Giuseppe; Martínez, Eduardo; Muñoz-Lecanda, Miguel C.; Román‐Roy, Narciso

doi:10.1142/s0219887806001764

Cited by 96 publications

(178 citation statements)

References 33 publications

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“…Suppose that γ = dS where S is a function S : Q −→ R. In this case, the condition dγ ∈ I(D 0 ) is trivially satisfied. Moreover, we note that in previous approximations to Hamilton-Jacobi theory [7,19,15,17,18] the considered sections are of the form 6) and the coefficientsλ i are determined through the nonholonomic constraint equations…”

Section: Hamilton-jacobi Theory For Nonholonomic Mechanical Systemsmentioning

confidence: 99%

Towards a Hamilton–Jacobi theory for nonholonomic mechanical systems

Ponte¹,

León²,

Diego³

2007

J. Phys. A: Math. Theor.

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Section: Hamilton-jacobi Theory For Nonholonomic Mechanical Systemsmentioning

confidence: 99%

Towards a Hamilton–Jacobi theory for nonholonomic mechanical systems

Ponte¹,

León²,

Diego³

2007

J. Phys. A: Math. Theor.

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“…As a (Lagrangian) counterpart of the Remark that was made in Sect.1, we have now the following [14] Remark 4. If: X ∈ X (Q) , X : Q → T Q is a solution of Eq.…”

Section: 2mentioning

confidence: 89%

“…Besides many original papers on this vast subject, we shall rely on some work by A.Vinogradov [53], some more recent work by Grabowski and Poncin [25], a previous paper of ours [38] and recent paper on the Hamilton-Jacobi theory in a Lagrangian setting [14]. Besides many original papers on this vast subject, we shall rely on some work by A.Vinogradov [53], some more recent work by Grabowski and Poncin [25], a previous paper of ours [38] and recent paper on the Hamilton-Jacobi theory in a Lagrangian setting [14].…”

Section: Comments and Plan Of The Papermentioning

confidence: 99%

The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics

Marmo¹,

Morandi²,

Mukunda³

2009

Journal of Geometric Mechanics

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We review here some conventional as well as less conventional aspects of the time-independent and time-dependent Hamilton-Jacobi (HJ) theory and of its connections with Quantum Mechanics. Less conventional aspects involve the HJ theory on the tangent bundle of a configuration manifold, the quantum HJ theory, HJ problems for general differential operators and the HJ problem for Lie groups.2000 Mathematics Subject Classification. 70H20, 70S05, 70FXX, 70HXX. Contents 1 This method was first introduced in the discussion of problems in wave propagation by lord Rayleigh back in 1912 and then applied to wave mechanics by H.Jeffreys in 1923 and, later on and simultaneously, by L.Brillouin, H.A.Kramers and G.Wentzel in 1926. 2 Many authors [42] prefer to incorporate the prefactor A into the definition of S, writing then: ψ (x.t) = exp(iS (x, t) / , where the S of Eq.(16) is replaced by S − i ln A and is no more real. 6 G. MARMO G. MORANDI AND N. MUKUNDA (with both A and S real). Substituting into the Schrödinger equation, one ends up with the (exact) coupled equations [22]:Received xxxx 20xx; revised xxxx 20xx.

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“…The geometric version of the Hamilton-Jacobi theorem recalled above (which is actually a simple remark in geometric terms) has been generalized by the Geometric Mechanics community to many different contexts: Lagrangian mechanics [7], non-holonomic systems [18,12,35,8,22], almost Poisson…”

Section: 32mentioning

confidence: 99%

Characteristics, bicharacteristics and geometric singularities of solutions of PDEs

Vitagliano

2014

Int. J. Geom. Methods Mod. Phys.

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Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics and in Physics. I will review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e. those aspects which are invariant under general changes of coordinates. After a basically analytic introduction, I will pass to a modern, geometric point of view, presenting characteristics within the jet space approach to PDEs. In particular, I will discuss the relationship between characteristics and singularities of solutions and observe that: "wave-fronts are characteristic surfaces and propagate along bicharacteristics". This remark may be understood as a mathematical formulation of the wave/particle duality in optics and/or quantum mechanics. The content of the paper reflects the three-hour mini-course that I gave at the XXII International Fall Workshop on Geometry and Physics, September 2–5, 2013, Évora, Portugal.

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Geometric Hamilton–jacobi Theory

Cited by 96 publications

References 33 publications

Towards a Hamilton–Jacobi theory for nonholonomic mechanical systems

Towards a Hamilton–Jacobi theory for nonholonomic mechanical systems

The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics

Characteristics, bicharacteristics and geometric singularities of solutions of PDEs

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