Semi-basic 1-forms and  Helmholtz conditions for the inverse
problem of the calculus of variations

Bucataru, Ioan; Dahl, Matias F.

doi:10.3934/jgm.2009.1.159

Cited by 37 publications

(71 citation statements)

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“…The tangent bundle carries some canonical structures, such as the vertical distribution, the Liouville vector field, and the vertical endomorphism. The differential calculus associated to these structures, using the Frölicher-Nijenhuis theory developed in the previous subsection, plays an important role in the geometry of a system of second-order ordinary differential equations, [4,5,14,15,17,20,24].…”

Section: Vertical Calculus On T M and Semi-basic Formsmentioning

confidence: 99%

“…The problem was formulated rigorously in 1960's by Rapcsák [27] and Klein and Voutier [17]. Yet, the projective metrizability problem is far from being solved, and in the last decade it has been intensively studied [1,5,10,11,29,30,31,32].…”

Section: Vertical Calculus On T M and Semi-basic Formsmentioning

confidence: 99%

“…One of this approaches seeks for the existence of a multiplier matrix that satisfies four Helmholtz conditions [19,28]. In [5], these four Helmholtz conditions where reformulated in terms of a semi-basic 1-form. For the particular case of the projective metrizability problem, it has been shown in [5] that only two of the four Helmholtz conditions are independent.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], these four Helmholtz conditions where reformulated in terms of a semi-basic 1-form. For the particular case of the projective metrizability problem, it has been shown in [5] that only two of the four Helmholtz conditions are independent. In this work we discuss the formal integrability of these two Helmholtz conditions using two sufficient conditions provided by Cartan-Kähler theorem.…”

Section: Introductionmentioning

confidence: 99%

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Projective Metrizability and Formal Integrability

Bucataru

Muzsnay

2011

SIGMA

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The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential operator P_1 and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P_1 using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P_1 is involutive and hence one of the two conditions is always satisfied. While discussing the second condition, in Theorem 4.3 we prove that there is only one obstruction to the formal integrability of P_1, and this obstruction is due to the curvature tensor of the induced nonlinear connection. When the curvature obstruction is satisfied, the projective metrizability problem reduces to the discussion of the algebraic conditions, which as we show are always satisfied in the analytic case. Based on these results, we recover all classes of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are projectively metrizable without being Finsler metrizable

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Section: Vertical Calculus On T M and Semi-basic Formsmentioning

confidence: 99%

Section: Vertical Calculus On T M and Semi-basic Formsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Projective Metrizability and Formal Integrability

Bucataru

Muzsnay

2011

SIGMA

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“…Secondly, there are similarities between the formulation of the Helmholtz conditions for sprays in the autonomous setting and respectively for semisprays in the nonautonomous one [5,6]. This is natural because J 1 π can be embedded in T M (the tangent bundle with the zero section removed).…”

Section: Introductionmentioning

confidence: 99%

Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

Constantinescu¹

2012

SIGMA

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Abstract. We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds.

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