The Poincaré variational principle in the Lagrange–Poincaré reduction of mechanical systems with symmetry

2019 ◽  
Vol 16 (05) ◽  
pp. 1950068
Author(s):  
S. N. Storchak
Keyword(s):  
Riemannian Manifold ◽  
Variational Principle ◽  
Mechanical Systems ◽  
Scalar Particle ◽  
Orbit Space ◽  
Local Dynamic ◽  
Lagrange Equations ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Dependent Coordinates

The local Lagrange–Poincaré equations (the reduced Euler–Lagrange equations) for the mechanical system describing the motion of a scalar particle on a finite-dimensional Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are obtained. The equations are written in terms of dependent coordinates which are used to represent the local dynamic given on the orbit space of the principal fiber bundle. The derivation of the equations is performed with the help of the variational principle developed by Poincaré for mechanical systems with symmetry.

scholarly journals Wong’s equations in Yang-Mills theory

Open Physics ◽  
2014 ◽  
Vol 12 (4) ◽  
Author(s):  
Sergey Storchak
Keyword(s):  
Dynamical System ◽  
Compact Riemannian Manifold ◽  
Scalar Particle ◽  
Orbit Space ◽  
Yang Mills ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Gauge Fixing ◽  
Dimensional Dynamical System ◽  
Dependent Coordinates

AbstractWong’s equations for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are derived. The equations obtained are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these equations, we obtain Wong’s equations in a pure Yang-Mills gauge theory with Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures performed in a finite-dimensional dynamical system and the reduction procedure in Yang-Mills gauge fields.

Dynamical systems: Approximate Lagrangians and Noether symmetries

2019 ◽  
Vol 16 (10) ◽  
pp. 1950160 ◽  
Author(s):  
Sameerah Jamal
Keyword(s):  
Riemannian Manifold ◽  
Dynamical Systems ◽  
Variational Principle ◽  
Kinetic Energy ◽  
Equations Of Motion ◽  
Noether Symmetry ◽  
Noether Symmetries ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Second Order Equations

We determine the approximate Noether point symmetries of the variational principle characterizing second-order equations of motion of a particle in a (finite-dimensional) Riemannian manifold. In particular, the Lagrangian comprises of kinetic energy and a potential [Formula: see text], perturbed to [Formula: see text]. We establish a convenient system of approximate geometric conditions that suffices for the computation of approximate Noether symmetry vectors and moreover, simplifies the problem of the effect of higher orders of the perturbation. The general results are applied to several practical problems of interest and we find extra Noether symmetries at [Formula: see text].

FRACTIONAL TIME ACTION AND PERTURBED GRAVITY

Fractals ◽  
2011 ◽  
Vol 19 (02) ◽  
pp. 243-247 ◽  
Author(s):  
MADHAT SADALLAH ◽  
SAMI I. MUSLIH ◽  
DUMITRU BALEANU ◽  
EQAB RABEI
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Variational Problems ◽  
Integral Function ◽  
Lagrange Equations ◽  
Action Integral

In this paper, we used the scaling concepts of Mandelbrot of fractals in variational problems of mechanical systems in order to re-write the action integral function as an integration over the fractional time. In addition, by applying the variational principle to this new fractional action, we obtained the modified Euler-Lagrange equations of motion in any fractional time of order 0 < α ≤ 1. Two examples are investigated in detail.

scholarly journals Solutions of a Certain Nonlinear Elliptic Equation on Riemannian Manifolds

2001 ◽  
Vol 162 ◽  
pp. 149-167
Author(s):  
Yong Hah Lee
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Nonlinear Elliptic Equation ◽  
Complete Riemannian Manifold ◽  
Doubling Condition ◽  
Finite Covering ◽  
Nonlinear Elliptic ◽  
Finite Dimensional ◽  
Volume Doubling

In this paper, we prove that if a complete Riemannian manifold M has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded A-harmonic functions on M is one to one corresponding to Rl, where A is a nonlinear elliptic operator of type p on M and l is the number of p-nonparabolic ends of M. We also prove that if a complete Riemannian manifold M is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded A-harmonic functions on M is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor’yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands.

scholarly journals Alternative Lagrangians obtained by scalar deformations

2020 ◽  
Vol 17 (04) ◽  
pp. 2050050
Author(s):  
Oana A. Constantinescu ◽  
Ebtsam H. Taha
Keyword(s):  
Force Field ◽  
Differentiable Function ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Initial System ◽  
Lagrange Equations ◽  
Necessary And Sufficient ◽  
Equations Of Motions

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 [Formula: see text], including a covariant force field. We find necessary and sufficient conditions for the existence of a differentiable function [Formula: see text] such that the initial system is equivalent to the system of Euler–Lagrange equations of the deformed Lagrangian [Formula: see text].

Automatic Generation of Equations of Motion of Mechanical Systems

2014 ◽  
Vol 611 ◽  
pp. 40-45
Author(s):  
Darina Hroncová ◽  
Jozef Filas
Keyword(s):  
Computer Simulation ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Automatic Generation ◽  
Lagrange Equations ◽  
Kinematic Chains ◽  
Transformation Matrices

The paper describes an algorithm for automatic compilation of equations of motion. Lagrange equations of the second kind and the transformation matrices of basic movements are used by this algorithm. This approach is useful for computer simulation of open kinematic chains with any number of degrees of freedom as well as any combination of bonds.

scholarly journals Discrete Schrödinger operators on a graph

1992 ◽  
Vol 125 ◽  
pp. 141-150 ◽  
Author(s):  
Polly Wee Sy ◽  
Toshikazu Sunada
Keyword(s):  
Riemannian Manifold ◽  
Automorphism Group ◽  
Connected Graph ◽  
Spectral Properties ◽  
Orbit Space ◽  
Finite Graph ◽  
Discrete Analogue ◽  
Discrete Laplacian ◽  
Locally Finite ◽  
Discrete Schrödinger Operators

In this paper, we study some spectral properties of the discrete Schrödinger operator = Δ + q defined on a locally finite connected graph with an automorphism group whose orbit space is a finite graph.The discrete Laplacian and its generalization have been explored from many different viewpoints (for instance, see [2] [4]). Our paper discusses the discrete analogue of the results on the bottom of the spectrum established by T. Kobayashi, K. Ono and T. Sunada [3] in the Riemannian-manifold-setting.

scholarly journals Covariant brackets for particles and fields

2017 ◽  
Vol 32 (19) ◽  
pp. 1750100 ◽  
Author(s):  
M. Asorey ◽  
M. Ciaglia ◽  
F. Di Cosmo ◽  
A. Ibort
Keyword(s):  
Variational Principle ◽  
Geometrical Approach ◽  
Covariant Formulation ◽  
Lagrange Equations ◽  
Relativistic Systems

A geometrical approach to the covariant formulation of the dynamics of relativistic systems is introduced. A realization of Peierls brackets by means of a bivector field over the space of solutions of the Euler–Lagrange equations of a variational principle is presented. The method is illustrated with some relevant examples.

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