Whether time is discrete at the fundamental level is still unknown. But for computational purposes, it is very useful to have a form of mechanics in which time increases in finite steps. Each step ...
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Whether time is discrete at the fundamental level is still unknown. But for computational purposes, it is very useful to have a form of mechanics in which time increases in finite steps. Each step must be a canonical transformation. Using the Poincare-Birkhoff-Witt lemma (also called the Baker-Campbell-Hausdorff formula) this chapter derives first and second order symplectic integrators that allow a discrete time formulation of mechanics. The example of the standard map of Chiriikov (discrete time form of the pendulum) gives the first example of chaos. Arnold's cat map is introduced in an exercise.Less

Discrete time

S. G. Rajeev

Published in print: 2013-07-25

Whether time is discrete at the fundamental level is still unknown. But for computational purposes, it is very useful to have a form of mechanics in which time increases in finite steps. Each step must be a canonical transformation. Using the Poincare-Birkhoff-Witt lemma (also called the Baker-Campbell-Hausdorff formula) this chapter derives first and second order symplectic integrators that allow a discrete time formulation of mechanics. The example of the standard map of Chiriikov (discrete time form of the pendulum) gives the first example of chaos. Arnold's cat map is introduced in an exercise.

This book begins with the ancient parts of classical mechanics: the variational principle, Lagrangian and Hamiltonian formalisms, and Poisson brackets. The simple pendulum provides a glimpse of the ...
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This book begins with the ancient parts of classical mechanics: the variational principle, Lagrangian and Hamiltonian formalisms, and Poisson brackets. The simple pendulum provides a glimpse of the beauty of elliptic curves, which will also appear later in rigid body mechanics. Geodesics in Riemannian geometry are presented as an example of a Hamiltonian system. Conversely, the path of a non-relativistic particle is a geodesic in a metric that depends on the potential. Orbits around a black hole are found. Hamilton-Jacobi theory is discussed, showing a path towards quantum mechanics and a connection to the eikonal of optics. The three body problem is studied in detail, including small orbits around the Lagrange points. The dynamics of a charged particle in a magnetic field, especially a magnetic monopole, is studied in the Hamiltonian formalism. Spin is shown to be a classical phenomenon. Symplectic integrators that allow numerical solutions of mechanical systems are derived. A simplified version of Feigenbaum's theory of period doubling introduces chaos. Following a classification of Mobius transformations, this book studies chaos on the complex plane: Julia sets, Fatou sets, and the Mandelblot are explained. Newton's method for solution of non-linear equations is viewed as a dynamical system, allowing a novel approach to the reduction of matrices to canonical form. This is used as a stepping stone to the KAM theory of maps of a circle to itself, unravelling a connection to the Diophantine problem of number theory. KAM theory of the solution of the Hamilton-Jacobi equation using Newton's iteration concludes the book.Less

Advanced Mechanics : From Euler's Determinism to Arnold's Chaos

S. G. Rajeev

Published in print: 2013-07-25

This book begins with the ancient parts of classical mechanics: the variational principle, Lagrangian and Hamiltonian formalisms, and Poisson brackets. The simple pendulum provides a glimpse of the beauty of elliptic curves, which will also appear later in rigid body mechanics. Geodesics in Riemannian geometry are presented as an example of a Hamiltonian system. Conversely, the path of a non-relativistic particle is a geodesic in a metric that depends on the potential. Orbits around a black hole are found. Hamilton-Jacobi theory is discussed, showing a path towards quantum mechanics and a connection to the eikonal of optics. The three body problem is studied in detail, including small orbits around the Lagrange points. The dynamics of a charged particle in a magnetic field, especially a magnetic monopole, is studied in the Hamiltonian formalism. Spin is shown to be a classical phenomenon. Symplectic integrators that allow numerical solutions of mechanical systems are derived. A simplified version of Feigenbaum's theory of period doubling introduces chaos. Following a classification of Mobius transformations, this book studies chaos on the complex plane: Julia sets, Fatou sets, and the Mandelblot are explained. Newton's method for solution of non-linear equations is viewed as a dynamical system, allowing a novel approach to the reduction of matrices to canonical form. This is used as a stepping stone to the KAM theory of maps of a circle to itself, unravelling a connection to the Diophantine problem of number theory. KAM theory of the solution of the Hamilton-Jacobi equation using Newton's iteration concludes the book.