scholarly journals Lagrange geometry on tangent manifolds

2003 ◽  
Vol 2003 (51) ◽  
pp. 3241-3266 ◽  
Author(s):  
Izu Vaisman
Keyword(s):  
Tangent Bundle ◽  
Tensor Field ◽  
Finsler Geometry ◽  
Interesting Case ◽  
Lagrangian Function ◽  
Total Space ◽  
Lagrangian Functions ◽  
Tangent Manifold

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a nondegenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper, we study a generalization which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples and find the cohomological obstructions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, while giving an index-free presentation of these connections.

scholarly journals (Pseudo) generalized Kaluza–Klein G-spaces and Einstein equations

2014 ◽  
Vol 25 (12) ◽  
pp. 1450116 ◽  
Author(s):  
Constantin M. Arcuş ◽  
Esmaeil Peyghan
Keyword(s):  
Vector Bundle ◽  
Tangent Bundle ◽  
Ricci Tensor ◽  
Tensor Field ◽  
General Framework ◽  
Bianchi Type ◽  
Einstein Equations ◽  
Lie Algebroid ◽  
Linear Connections ◽  
Kaluza Klein

Introducing the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza–Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza–Klein G-spaces and we develop the Einstein equations in this general framework.

scholarly journals The second-order tangent bundle with deformed 2nd lift metric

2019 ◽  
Vol 16 (04) ◽  
pp. 1950062
Author(s):  
Abdullah Magden ◽  
Kubra Karaca ◽  
Aydin Gezer
Keyword(s):  
Tangent Bundle ◽  
Tensor Field ◽  
Complex Structure ◽  
Sufficient Conditions ◽  
Second Order ◽  
Riemannian Curvature Tensor ◽  
Almost Complex ◽  
Horizontal Part ◽  
Order Tangent ◽  
Second Order Tangent Bundle

Let [Formula: see text] be a pseudo-Riemannian manifold and [Formula: see text] be its second-order tangent bundle equipped with the deformed [Formula: see text]nd lift metric [Formula: see text] which is obtained from the [Formula: see text]nd lift metric by deforming the horizontal part with a symmetric [Formula: see text]-tensor field [Formula: see text]. In the present paper, we first compute the Levi-Civita connection and its Riemannian curvature tensor field of [Formula: see text]. We give necessary and sufficient conditions for [Formula: see text] to be semi-symmetric. Secondly, we show that [Formula: see text] is a plural-holomorphic [Formula: see text]-manifold with the natural integrable nilpotent structure. Finally, we get the conditions under which [Formula: see text] with the [Formula: see text]nd lift of an almost complex structure is an anti-Kähler manifold.

Lagrangian methods in plasma dynamics. Part 2. Construction of Lagrangians for plasmas

1974 ◽  
Vol 11 (2) ◽  
pp. 331-346 ◽  
Author(s):  
J. P. Dougherty
Keyword(s):  
Stress Tensor ◽  
Cold Plasma ◽  
Lagrangian Function ◽  
Covariant Formalism ◽  
Lagrangian Functions ◽  
Plasma Dynamics ◽  
Lagrangian Methods ◽  
Hot Plasma ◽  
The World

Most of this paper is concerned with the construction of suitable Lagrangian functions for the dynamics of a cold plasma, in such a way as to retain the relativistically covariant formalism. In one method, this is achieved by the introduction of a set of three variables which label the world lines of the particles. A second method results in a Clebsch type of representation. Sturrock's relativistic Lagrangian and Low's hot plasma Lagrangian are also briefly discussed in the context of the present work. The behaviour of the canonical stress tensor is considered. The applicability of many of the general results of part 1 is ensured by establishing the existence of the Lagrangian function.

Lagrangian Formulation of Lorenz and Chen Systems

2021 ◽  
Vol 31 (04) ◽  
pp. 2150055
Author(s):  
Palanisamy Vijayalakshmi ◽  
Zhiheng Jiang ◽  
Xiong Wang
Keyword(s):  
Fractional Derivatives ◽  
Lagrangian Function ◽  
Conserved Quantity ◽  
Lagrangian Formulation ◽  
Lagrangian Functions ◽  
Differentiable Functions ◽  
Derivatives Of

This paper presents the formulation of Lagrangian function for Lorenz, Modified Lorenz and Chen systems using Lagrangian functions depending on fractional derivatives of differentiable functions, and the estimation of the conserved quantity associated with the respective systems.

GRAVITY IN CURVED PHASE-SPACES, FINSLER GEOMETRY AND TWO-TIMES PHYSICS

2012 ◽  
Vol 27 (12) ◽  
pp. 1250069 ◽  
Author(s):  
CARLOS CASTRO
Keyword(s):  
Phase Space ◽  
Tangent Bundle ◽  
Finsler Geometry ◽  
Space Time ◽  
Vacuum Field ◽  
Field Equations ◽  
Problem Of Time ◽  
Gravitational Field Equations ◽  
Cotangent Space ◽  
Kaluza Klein

The generalized (vacuum) field equations corresponding to gravity on curved 2d-dimensional (dim) tangent bundle/phase spaces and associated with the geometry of the (co)tangent bundle TMd-1, 1(T*Md-1, 1) of a d-dim space–time Md-1, 1 are investigated following the strict distinguished d-connection formalism of Lagrange–Finsler and Hamilton–Cartan geometry. It is found that there is no mathematical equivalence with Einstein's vacuum field equations in space–times of 2d dimensions, with two times, after a d+d Kaluza–Klein-like decomposition of the 2d-dim scalar curvature R is performed and involving the introduction of a nonlinear connection [Formula: see text]. The physical applications of the 4-dim phase space metric solutions found in this work, corresponding to the cotangent space of a 2-dim space–time, deserve further investigation. The physics of two times may be relevant in the solution to the problem of time in quantum gravity and in the explanation of dark matter. Finding nontrivial solutions of the generalized gravitational field equations corresponding to the 8-dim cotangent bundle (phase space) of the 4-dim space–time remains a challenging task.

scholarly journals Modified Hamiltonian Formalism for Regge-Teitelboim Cosmology

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Pinaki Patra ◽  
Md. Raju ◽  
Gargi Manna ◽  
Jyoti Prasad Saha
Keyword(s):  
Cosmological Model ◽  
Tangent Bundle ◽  
Dynamical Equation ◽  
Hamiltonian Formalism ◽  
Equation Of Motion ◽  
Scale Factor ◽  
Correct Equation ◽  
Higher Derivative ◽  
Alternative Approach ◽  
Tangent Manifold

The Ostrogradski approach for the Hamiltonian formalism of higher derivative theory is not satisfactory because the Lagrangian cannot be viewed as a function on the tangent bundle to coordinate manifold. In this paper, we have used an alternative approach which leads directly to the Lagrangian which, being a function on the tangent manifold, gives correct equation of motion; no new coordinate variables need to be added. This approach can be used directly to the singular (in Ostrogradski sense) Lagrangian. We have used this method for the Regge-Teitelboim (RT) minisuperspace cosmological model. We have obtained the Hamiltonian of the dynamical equation of the scale factor of RT model.

scholarly journals A Smooth Pseudo-Gradient for the Lagrangian Action Functional

2009 ◽  
Vol 9 (4) ◽  
Author(s):  
Alberto Abbondandolo ◽  
Matthias Schwarzy
Keyword(s):  
Tangent Bundle ◽  
Lagrangian Function ◽  
Quadratic Growth ◽  
Action Functional ◽  
Hilbert Manifold ◽  
Lagrangian Action

AbstractWe study the action functional associated to a smooth Lagrangian function on the tangent bundle of a manifold, having quadratic growth in the velocities. We show that, although the action functional is in general not twice differentiable on the Hilbert manifold consisting of H

scholarly journals The projective curvature of the tangent bundle with natural diagonal metric

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 401-410 ◽  
Author(s):  
Cornelia-Livia Bejan ◽  
Simona-Luiza Druţă-Romaniuc
Keyword(s):  
Riemannian Manifold ◽  
Tangent Bundle ◽  
Space Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Total Space ◽  
Projectively Flat ◽  
Projective Curvature ◽  
Necessary And Sufficient

Our study is mainly devoted to a natural diagonal metric G on the total space TMof the tangent bundle of a Riemannian manifold (M, 1). We provide the necessary and sufficient conditions under which (TM,G) is a space form, or equivalently (TM,G) is projectively Euclidean. Moreover, we classify the natural diagonal metrics G for which (TM,G) is horizontally projectively flat (resp. vertically projectively flat).

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