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 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.

scholarly journals ON THE JACOBI EQUATION AND MANIFOLDS WITH MULTIPLE CONJUGATE POINTS

2013 ◽  
Vol 113A (1) ◽  
pp. 19-30
Author(s):  
J.M. Burns ◽  
E. Staunton ◽  
D.J. Wraith
Keyword(s):  
Jacobi Equation ◽  
Conjugate Points

scholarly journals Gradient vector fields which characterize warped products

2001 ◽  
Vol 88 (2) ◽  
pp. 182
Author(s):  
Nobuhiro Innami ◽  
Byung Hak Kim
Keyword(s):  
Vector Fields ◽  
Conjugate Points ◽  
Warped Products ◽  
Gradient Vector ◽  
Jacobi Equations

We find what condition on gradient vector fields characterizes warped products, Riemannian products and round spheres. To do this we apply the theory of Jacobi equations without conjugate points to the differential maps of the local one-parameter groups generated by gradient vector fields.

scholarly journals The Hamilton–Jacobi Theory for Contact Hamiltonian Systems

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1993
Author(s):  
Manuel de León ◽  
Manuel Lainz ◽  
Álvaro Muñiz-Brea
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Jacobi Equation ◽  
Vector Fields ◽  
Hamiltonian Function ◽  
Hamilton Jacobi Equation

The aim of this paper is to develop a Hamilton–Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton–Jacobi equation accordingly to the Hamiltonian and the evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.

Three dimensional equilibria in the magnetohydrodynamic approximation

1976 ◽  
Vol 15 (3) ◽  
pp. 423-430 ◽  
Author(s):  
M. L. Woolley
Keyword(s):  
Current Density ◽  
Variational Principles ◽  
Three Dimensional ◽  
The Other ◽  
Plane Symmetry ◽  
Equilibrium System ◽  
Current Line ◽  
Lagrangian Functions ◽  
New Class ◽  
Cartesian Coordinate

By introducing field line potentials for the magnetic induction and current line potentials for the current density, it is shown that the equations which describe a static equilibrium system, in the ideally conducting magnetohydrodynamic approximation, may be derived from two equivalent variational principles. The problem of integration is essentially that of finding a transformation of the potentials which changes one of the associated Lagrangian functions into the other. The assumption that the current density has no component in one direction of a rectangular Cartesian coordinate system leads to a new class of fully three dimensional equilibria having a plane symmetry.

scholarly journals THE RELATIVE ENERGY OF HOMOGENEOUS AND ISOTROPIC UNIVERSES FROM VARIATIONAL PRINCIPLES

2009 ◽  
Vol 06 (07) ◽  
pp. 1193-1205 ◽  
Author(s):  
ENRICO BIBBONA ◽  
LORENZO FATIBENE ◽  
MAURO FRANCAVIGLIA
Keyword(s):  
Vector Fields ◽  
Variational Principles ◽  
Cosmic Time ◽  
Matching Condition ◽  
Relative Energy ◽  
Conserved Quantity ◽  
Conserved Quantities ◽  
Time Coordinate ◽  
Poincaré Algebra ◽  
Conserved Currents

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular, we prove that their relative "energy" (defined as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quantities related to a Lie subalgebra of vector fields isomorphic to the Poincaré algebra. These quantities are also conserved in time. We also find a relative conserved quantity for such a kind of solution which is conserved in time though non-vanishing. This example provides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter fluid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our superpotential (that happens to coincide with the so-called KBL potential although it is generated differently) provides strong conservation laws under much weaker hypotheses than the ones usually required. In particular, the symmetry generator is not needed to be Killing (nor Killing of the background, nor asymptotically Killing), the prescription is quasi-local and it works fine in a finite region too and no matching condition on the boundary is required.

scholarly journals On the Jacobi Equation and Manifolds with Multiple Conjugate Points

2013 ◽  
Vol 113 (1) ◽  
pp. 19-30
Author(s):  
J.M. Burns ◽  
E. Staunton ◽  
D.J. Wraith
Keyword(s):  
Jacobi Equation ◽  
Conjugate Points
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