Variational Principles for Second-Order Differential Equations - Application of the Spencer Theory to Characterize Variational Sprays

Grifone, Joseph; Muzsnay, Zoltán

doi:10.1142/9789812813596

Cited by 21 publications

(33 citation statements)

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“…The formal integrability of the field equations for first-order Lagrangians in classical field theory has been extensively studied (e.g., see [1,3,6,8,10,13]). Finally, Section 6 is devoted to study the formal integrability of the field equations of second-order Lagrangians with projectable P-C form to first order in their Hamiltonian form.…”

Section: Introductionmentioning

confidence: 99%

Integrability of second-order Lagrangians admitting a first-order Hamiltonian formalism

María¹,

Masqué²

2014

Differential Geometry and its Applications

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Second-order Lagrangian densities admitting a first-order Hamiltonian formalism are studied; namely, i) necessary and sufficient conditions for the Poincaré-Cartan form of a second-order Lagrangian on an arbitrary fibred manifold p: E → N to be projectable onto J 1 E are explicitly determined; ii) for each of such Lagrangians, a first-order Hamiltonian formalism is developed and a new notion of regularity is introduced; iii) the variational problems of this class defined by regular Lagrangians are proved to be involutive.

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Section: Introductionmentioning

confidence: 99%

Integrability of second-order Lagrangians admitting a first-order Hamiltonian formalism

María¹,

Masqué²

2014

Differential Geometry and its Applications

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“…Since d J L = ∂L ∂y i dx i , the above relation is equivalent to Let X = S and K = J in the formula i X d K = −d K i X + L KX + i [K,X] , ( [14], Appendix A, page 215). Then…”

Section: Deformations Of Lagrangians: Main Resultsmentioning

confidence: 99%

“…The study of the solutions of such a SODE, from geometric point of view, consists in identifying the system (1.1) with a second order vector field or a semi-spray. This means that S is a vector field on T M , (M an n-dimensional differentiable manifold), with JS = C, where J is the vertical endomorphism and C the Liouville vector field [14]. The solutions of the above system are called the geodesics of the semispray.…”

Section: Introductionmentioning

confidence: 99%

Alternative Lagrangians obtained by scalar deformations

Constantinescu

Taha

2020

Int. J. Geom. Methods Mod. Phys.

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We study mechanical systems that can be recast into the form of a system of genuine Euler-Lagrange equations. The equations of motions of such systems are initially equivalent to the system of Lagrange equations of some Lagrangian L, including a covariant force field. We find necessary and sufficient conditions for the existence of a differentiable function Φ : R → R such that the initial system is equivalent to the system of Euler-Lagrange equations of the deformed Lagrangian Φ(L). We construct various examples of such deformations.

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“…Since F is a holonomy invariant function, the It is clear that the above spray S is projectively flat. Moreover, one can show that (4.4) is also R-flat and by [9] it is locally Finsler metrizable. It should be noted that the (global) Finsler metrizability of (4.4) is questioned in [14,Chapter 10.3].…”

Section: Metrizability Of Holonomic Projective Deformationsmentioning

confidence: 94%

Metrizability of Holonomy Invariant Projective Deformation of Sprays

Elgendi¹,

Muzsnay²

2020

Can. Math. Bull.

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In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray S and a holonomy invariant function P, we investigate the metrizability property of the projective deformation S = S −2λPC. We prove that for any holonomy invariant nontrivial function P and for almost every value λ ∈ R, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the holonomy invariant function P is necessarily one of the principal curvatures of the geodesic structure.2010 Mathematics Subject Classification. 53C60, 53B40, 58B20, 49N45, 58E30.

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Variational Principles for Second-Order Differential Equations - Application of the Spencer Theory to Characterize Variational Sprays

Cited by 21 publications

References 0 publications

Integrability of second-order Lagrangians admitting a first-order Hamiltonian formalism

Integrability of second-order Lagrangians admitting a first-order Hamiltonian formalism

Alternative Lagrangians obtained by scalar deformations

Metrizability of Holonomy Invariant Projective Deformation of Sprays

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