scholarly journals JACOBI VECTOR FIELDS FOR LAGRANGIAN SYSTEMS ON ALGEBROIDS

2013 ◽  
Vol 10 (05) ◽  
pp. 1350011 ◽  
Author(s):  
MICHAŁ JÓŹWIKOWSKI
Keyword(s):  
Tangent Bundle ◽  
Jacobi Equation ◽  
Vector Fields ◽  
Conjugate Points ◽  
Lagrangian Systems ◽  
Lie Algebroid ◽  
Second Variation ◽  
Jacobi Vector ◽  
The Second Variation ◽  
Geometric Nature

We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler–Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence we obtain a simple non-computational proof of the relation between the null subspace of the second variation of the action and the presence of JVFs (and conjugate points) along an extremal. We work in the framework of skew-symmetric algebroids.

Natural Lagrangian systems without conjugate points

1994 ◽  
Vol 14 (1) ◽  
pp. 169-180
Author(s):  
Nobuhiro Innami
Keyword(s):  
Jacobi Equation ◽  
Vector Fields ◽  
Variational Principles ◽  
Conjugate Points ◽  
Lagrangian Systems ◽  
Lagrangian Functions ◽  
Variation Vector

AbstractThe variation vector fields through extremals of the variational principles of natural Lagrangian functions satisfy the equation of Jacobi type. By making use of the Jacobi equation we obtain the estimates of measure-theoretic entropy for natural Lagrangian systems without conjugate points.

scholarly journals Second order infinitesimal bending of curves

Filomat ◽  
2017 ◽  
Vol 31 (13) ◽  
pp. 4127-4137 ◽  
Author(s):  
Marija Najdanovic ◽  
Ljubica Velimirovic
Keyword(s):  
Vector Fields ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Second Order ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Explicit Formulas ◽  
Infinitesimal Bending ◽  
Necessary And Sufficient ◽  
The Second Variation

We investigate a second order infinitesimal bending of curves in a three-dimensional Euclidean space in this paper. We give the necessary and sufficient conditions for the vector fields to be infinitesimal bending fields of the corresponding order, as well as explicit formulas which determine these fields. We examine the first and the second variation of some geometric magnitudes which describe a curve, specially a change of the curvature. Two illustrative examples (a circle and a helix) are studied not only analytically but also by drawing curves using computer program Mathematica.

scholarly journals Regularity and Conjugacy for Constrained Variational Problems

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Tangent Cone ◽  
Constraint Qualification ◽  
Necessary Conditions ◽  
Variational Problems ◽  
Conjugate Points ◽  
Second Variation ◽  
Mixed Constraints ◽  
Second Order Necessary Conditions ◽  
The Second Variation ◽  
Extended Notion

For problems in the calculus of variations involving equality and inequality mixed constraints we characterize, in terms of an extended notion of conjugate points, the sign of a quadratic form which corresponds to the second variation of the integral to be minimized. Second order necessary conditions are then derived assuming the well-known constraint qualification of regularity in the sense that, with respect to the set of mixed constraints, both the tangent cone and the set of tangential constraints coincide.

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

scholarly journals EXTENDED ABSOLUTE PARALLELISM GEOMETRY

2008 ◽  
Vol 05 (07) ◽  
pp. 1109-1135 ◽  
Author(s):  
NABIL. L. YOUSSEF ◽  
A. M. SID-AHMED
Keyword(s):  
Tangent Bundle ◽  
Special Form ◽  
Vector Fields ◽  
A Priori ◽  
Building Blocks ◽  
Absolute Parallelism ◽  
Linearly Independent ◽  
Geometric Objects ◽  
Directional Argument ◽  
Curvature Tensors

In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection (assumed given a priori) and 2n linearly independent vector fields (of special form) defined globally on TM defining the parallelization. Four different d-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined d-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical d-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical d-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle TM of M.

Twisted Sasakian Metric on the Tangent Bundle and Harmonicity

2021 ◽  
Vol 62 ◽  
pp. 53-66
Author(s):  
Fethi Latti ◽  
◽  
Hichem Elhendi ◽  
Lakehal Belarbi
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Tangent Bundle ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Harmonic Vector ◽  
New Class ◽  
Necessary And Sufficient ◽  
Sasakian Metric ◽  
Harmonic Vector Fields

In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.

Integral criteria for closed surfaces with constant mean curvature

2007 ◽  
Vol 463 (2081) ◽  
pp. 1199-1210
Author(s):  
Luca Guzzardi ◽  
Epifanio G Virga
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Second Variation ◽  
Closed Surfaces ◽  
Area Functional ◽  
Symmetric Surface ◽  
The Second Variation ◽  
Integral Identities

We propose three integral criteria that must be satisfied by all closed surfaces with constant mean curvature immersed in the three-dimensional Euclidean space. These criteria are integral identities that follow from requiring the second variation of the area functional to be invariant under rigid displacements. We obtain from them a new proof of the old result by Delaunay, to the effect that the sphere is the only closed axis-symmetric surface.

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