Geodesic and Newtonian Vector Fields and Symmetries of Mechanical Systems (original) (raw)

Symmetries and conserved quantities in geodesic motion

Journal of Mathematical Physics, 1986

Recently obtained results linking several constants of motion to one (non-Noetherian) symmetry to the problem of geodesic motion in Riemannian space-times are applied. The construction of conserved quantities in geodesic motion as well as the deduction of geometrical statements about Riemannian space-times are achieved.

Symmetries in Non-Linear Mechanics

In this paper we exploit the use of symmetries of a physical system so as to characterize the corresponding solution manifold by means of Noether invariants. This constitutes a necessary preliminary step towards the correct quantisation in non-linear cases, where the success of Canonical Quantisation is not guaranteed in general. To achieve this task "point symmetries" of the Lagrangian are generally not enough, and the notion of contact transformations is in order. The use of the Poincaré-Cartan form permits finding both the symplectic structure on the solution manifold, through the Hamilton-Jacobi transformation, and the required symmetries, realized as Hamiltonian vector fields, associated with functions on the solution manifold (thus constituting an inverse of the Noether Theorem), lifted back to the evolution space through the inverse of this Hamilton-Jacobi mapping. In this framework, solutions and symmetries are somehow identified and this correspondence is also kept at a perturbative level. We present simple non-trivial examples of this interplay between symmetries and solutions pointing out the usefulness of this mechanism in approaching the corresponding quantisation.

Geometry, Symmetries, and Classical Physics: A Mosaic

Geometry, Symmetries, and Classical Physics: A Mosaic, 2021

This book provides advanced physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume. - - Key features: -> Contains a modern, streamlined presentation of classical topics, which are normally taught separately. -> Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity. -> Focuses on the clear presentation of the mathematical notions and calculational technique. - - - Table of Contents: - Chapter 1. Manifolds and Tensors. - Chapter 2. Geometry and Integration on Manifolds. - Chapter 3. Symmetries of Manifolds. - Chapter 4. Newtonian Mechanics. - Chapter 5. Lagrangian Methods and Symmetry. - Chapter 6. Relativistic Mechanics. - Chapter 7. Lie Groups. - Chapter 8. Lie Algebras. - Chapter 9. Representations. - Chapter 10. Rotations and Euclidean Symmetry. - Chapter 11. Boosts and Galilei Symmetry. - Chapter 12. Lorentz Symmetry. - Chapter 13. Poincare Symmetry. - Chapter 14. Conformal Symmetry. - Chapter 15. Lagrangians and Noether's Theorem. - Chapter 16. Spacetime Symmetries of Fields. - Chapter 17. Gauge Symmetry. - Chapter 18. Connection and Geodesics. - Chapter 19. Riemannian Curvature. - Chapter 20. Symmetries of Riemannian Manifolds. - Chapter 21. Einstein's Gravitation. - Chapter 22. Lagrangian Formulation. - Chapter 23. Conservation Laws and Further Symmetries. - - Appendices - A) Notation and Conventions. - Physical Units and Dimensions. - Mathematical Conventions. - Abbreviations. - B) Mathematical Tools. - Tensor Algebra. - Matrix Exponential. - Pauli and Dirac Matrices. - Dirac Delta Distribution. - Poisson and Wave Equation. - Variational Calculus. - Volume Element and Hyperspheres. - Hypersurface Elements. - C) Weyl Rescaling Formulae. - D) Spaces and Symmetry Groups. - - Bibliography. - - Index.

Symmetries and currents in nonholonomic mechanics

2014

summary:In this paper we derive general equations for constraint Noethertype symmetries of a first order non-holonomic mechanical system and the corresponding currents, i.e. functions constant along trajectories of the nonholonomic system. The approach is based on a consistent and effective geometrical theory of nonholonomic constrained systems on fibred manifolds and their jet prolongations, first presented and developed by Olga Rossi. As a representative example of application of the geometrical theory and the equations of symmetries and conservation laws derived within this framework we present the Chaplygin sleigh. It is a mechanical system subject to one linear nonholonomic constraint enforcing the plane motion. We describe the trajectories of the Chaplygin sleigh and show that the usual kinetic energy conservation law holds along them, the time translation generator being the corresponding constraint symmetry and simultaneously the symmetry of nonholonomic equations of motion....

Symmetries in Motion: Geometric Foundations of Motion Control

1998

Some interesting aspects of motion and control, such as those found in biological and robotic locomotion and attitude control of spacecraft, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can rotate or move forward. This observation leads to the general idea that when one variable in a system moves in a periodic fashion, motion of the Whole object can result. This property can be used for control purposes; the position and attitude Of a satellite, for example, are often controlled by periodic motions of parts of the satellite, such as spinning rotors. One of the geometric tools that has been used to describe this phenomenon is that of connections, a notion that is used extensively in general relativity and other parts of theoretical physics. This tool, part of the general subject Of geometric mechanics, has been helpful in the study of both the stability and instability of a system and system bifurcations, that is, changes in the nat...

Geometric Mechanics and Symmetry: from finite to infinite dimensions

2009

The purpose of the book is to provide the unifying viewpoint of Lagrangian and Hamiltonian mechanics in the coordinate-free language of differential geometry in the spirit of the Marsden-Ratiu school. The book is similar in content -although less formal -to the book by J. Marsden and T. Ratiu . One can also mention the companion two-volumes book by Holm written at a more basic level, and that one can recommend as an introductory reading. The classical treatises on the subject are the books by Abraham-Marsden [1], Arnold [2] and Libermann-Marle [6].

Geometrization of Lie and Noether symmetries and applications

International Journal of Modern Physics: Conference Series, 2015

We derive the Lie and the Noether conditions for the equations of motion of a dynamical system in an n-dimensional Riemannian space. We solve these conditions in the sense that we express the symmetry generating vectors in terms of the collineation vectors of the space. More specifically we give two theorems which contain all the necessary conditions which allow one to determine the Lie and the Noether point symmetries of a specific dynamical system evolving in a given Riemannian space in terms of the projective and the homothetic collineations of space. We apply these theorems to various interesting situations in Newtonian Physics.

On Symmetry and Conserved Quantities in Classical Mechanics

The Western Ontario Series in Philosophy of Science, 2000

This paper expounds the relations between continuous symmetries and conserved quantities, i.e. Noether's "first theorem", in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechanics' grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem.

Illustrating dynamical symmetries in classical mechanics: The Laplace–Runge–Lenz vector revisited

American Journal of Physics, 2003

The inverse square force law admits a conserved vector that lies in the plane of motion. This vector has been associated with the names of Laplace, Runge, and Lenz, among others. Many workers have explored aspects of the symmetry and degeneracy associated with this vector and with analogous dynamical symmetries. We define a conserved dynamical variable α that characterizes the orientation of the orbit in two-dimensional configuration space for the Kepler problem and an analogous variable β for the isotropic harmonic oscillator. This orbit orientation variable is canonically conjugate to the angular momentum component normal to the plane of motion. We explore the canonical one-parameter group of transformations generated by α (β). Because we have an obvious pair of conserved canonically conjugate variables, it is desirable to use them as a coordinate-mometum pair. In terms of these phase space coordinates, the form of the Hamiltonian is nearly trivial because neither member of the pair can occur explicitly in the Hamiltonian. From these considerations we gain a simple picture of dynamics in phase space. The procedure we use is in the spirit of the Hamilton-Jacobi method.