The Geometry of Higher-Order Lagrange Spaces

Miron, Radu

doi:10.1007/978-94-017-3338-0

Cited by 73 publications

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“…In (25) and (26), L (α)i (β)jµ and C (α)i (β)j (γ )k can be reduced to L (α)i (α)jµ and C (α)i (α)j (γ )k and g (α)i(β)j can be reduced to g (α)i(α)j , under some convenient conditions (Miron, 1997). Thus it is understood from the above that we can consider many interesting modified connection structures by generalizing the independent variables (or the adapted frame) in various ways.…”

Section: Modified Connection Structures-iiimentioning

confidence: 99%

“…Further, the Finslerian metrical structure is given by (Miron and Anastasiei, 1997) G ≡ G AB dx A dx B = g λκ dx λ ⊗ dx κ + g ij dx i ⊗ dx j . The metrical conditions g λκ|µ = 0, g λκ | l = 0, g ij |µ = 0 and g ij | l = 0 can be imposed, if necessary.…”

Section: Finslerian Connection Structurementioning

confidence: 99%

“…From the vector bundle-like standpoint (Miron and Anastasiei, 1997), the Finslerian gravitational field is considered the unified field over the total space of the vector bundle whose base manifold is the (x)-field and fiber at each point x is the (y)-field. That is to say, the Finslerian gravitational field can be treated by means of differential geometry of the total space of the vector bundle mentioned above.…”

Section: Introductionmentioning

confidence: 99%

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Connection Considerations of Gravitational Field in Finsler Spaces

Stavrinos

Ikeda

2006

Int J Theor Phys

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Section: Modified Connection Structures-iiimentioning

confidence: 99%

Section: Finslerian Connection Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Connection Considerations of Gravitational Field in Finsler Spaces

Stavrinos

Ikeda

2006

Int J Theor Phys

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“…By configuration manifold we mean a sub-manifold M of a 2n-manifold N such that TM = N. With this definition we adopt the formalism of Lagrange spaces ( [7]) (indeed dual of Lagrange spaces), instead of considering the formalism of higher order Lagrange spaces( [8]). The relation between Finsler structures and deterministic systems is based on the following points:…”

Section: Finslerian Deterministic Quantum Models At the Planck Scalementioning

confidence: 99%

A Finslerian version of ‘t Hooft deterministic quantum models

Torromé¹

2006

Journal of Mathematical Physics

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Using the Finsler structure living in the phase space associated to the tangent bundle of the configuration manifold, deterministic models at the Planck scale are obtained. The Hamiltonian function are constructed directly from the geometric data and some assumptions concerning time inversion symmetry. The existence of a maximal acceleration and speed is proved for Finslerian deterministic models. We investigate the spontaneous symmetry breaking of the orthogonal symmetry SO(6N ) of the Hamiltonian of a deterministic system. This symmetry break implies the non-validity of the argument used to obtain Bell's inequalities for spin states. It is introduced and motivated in the context of Randers spaces an example of simple 't Hooft model with interactions.

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“…This monograph is the sixth one resulting from 50 years of research activity of the prominent Romanian school on Finsler geometry, Lagrange-Hamilton spaces, and their higher-order generalizations [1][2][3][4][5]. The new book presents an overview of the higher-order Hamilton spaces with applications to higher-order mechanics following the canonical non-linear connection (N-connection) and (semi-)spray formalism and the geometry of induced almost complex or contact Riemann-Cartan structures.…”

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confidence: 99%

Radu Miron: The geometry of higher-order Hamilton spaces

Vacaru

2005

Gen Relativ Gravit

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Radu Miron: The geometry of higher-order Hamilton spacesKluwer Academic Publishers, Dordrecht, 2003, 264 p., This monograph is the sixth one resulting from 50 years of research activity of the prominent Romanian school on Finsler geometry, Lagrange-Hamilton spaces, and their higher-order generalizations [1][2][3][4][5]. The new book presents an overview of the higher-order Hamilton spaces with applications to higher-order mechanics following the canonical non-linear connection (N-connection) and (semi-)spray formalism and the geometry of induced almost complex or contact Riemann-Cartan structures. The variational principles for higher-order Hamiltonians, HamiltonJacobi equations and related conservation laws and symmetries are described and investigated.In this review, I would like to discuss some relations of this original approach to geometry and physics and reflect on applications to modern gravity and string theory.There are two general directions towards geometrization of mechanics on the tangent or cotangent bundle and their higher-order extensions: The first one is based on the idea of describing classical mechanics and classical field theory in terms of symplectic geometry following certain procedures of geometrization of the Euler-Lagrange and Hamilton equations by using the (multi-)symplectic formalism, differential forms, jets, etc. (see, for instance, [6,7]). In a quite different way, in the second direction Lagrange and Hamilton mechanics was geometrized using Riemann-Finsler spaces and, in a more general context, certain types of non-holonomic manifolds enabled with non-integrable distributions defined by N-connections. The general ideas and methods originate from E. Cartan [8] and A. Kawaguchi [9, 10] who were the first to consider metric-compatible models of Finsler spaces and N-connection structures. In this way, the geometric constructions are not related to certain particular properties of the Euler-Lagrange, or

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The Geometry of Higher-Order Lagrange Spaces

Cited by 73 publications

References 0 publications

Connection Considerations of Gravitational Field in Finsler Spaces

Connection Considerations of Gravitational Field in Finsler Spaces

A Finslerian version of ‘t Hooft deterministic quantum models

Radu Miron: The geometry of higher-order Hamilton spaces

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