Classical mechanics Hamiltonian and Lagrangian formalism /
This account of the fundamentals of Hamiltonian mechanics also covers related topics such as integral invariants and the Noether theorem. With just the elementary mathematical methods used for exposition, the book is suitable for novices as well as graduates.
| Main Author: | Deriglazov, Alexei. |
|---|---|
| Other Authors: | SpringerLink (Online Service) |
| Format: | eBook |
| Language: | English |
| Published: |
Switzerland :
Springer,
2016, ©2017.
|
| Physical Description: |
1 online resource (455 pages) |
| Edition: | 2nd ed. |
| Subjects: |
Table of Contents:
- Preface to the Second Edition; Preface to the First Edition; Contents; Notation and Conventions; 1 Sketch of Lagrangian Formalism; 1.1 Newton's Equation; 1.2 Galilean Transformations: Principle of Galilean Relativity; 1.3 Poincaré and Lorentz Transformations: The Principle of Special Relativity; 1.4 Principle of Least Action; 1.4.1 Variational Analysis; 1.4.2 Generalized Coordinates, Coordinate Transformations and Symmetries of an Action; 1.5 Examples of Continuous (Field) Systems; 1.6 Action of a Constrained System; 1.6.1 The Recipe; 1.6.2 Justification of the Recipe.
- 1.6.3 Description of Constrained System by Singular Action1.6.4 Kinetic Versus Potential Energy: Forceless Mechanics of Hertz; 1.7 Electromagnetic Field in Lagrangian Formalism; 1.7.1 Maxwell Equations; 1.7.2 Nonsingular Lagrangian Action of Electrodynamics; 1.7.3 Manifestly Poincaré-Invariant Formulation in Terms of a Singular Lagrangian Action; 1.7.4 Notion of Local (Gauge) Symmetry; 1.7.5 Lorentz Transformations of Three-Dimensional Potential: Role of Gauge Symmetry; 1.7.6 Relativistic Particle in Electromagnetic Field; 1.7.7 Speed of Light and Critical Speed in External Field.
- 1.7.8 Poincaré Transformations of Electricand Magnetic Fields2 Hamiltonian Formalism; 2.1 Derivation of Hamiltonian Equations; 2.1.1 Preliminaries; 2.1.2 From Lagrangian to Hamiltonian Equations; 2.1.3 Short Prescription for Hamiltonization Procedure, Physical Interpretation of Hamiltonian; 2.1.4 Inverse Problem: From Hamiltonian to Lagrangian Formulation; 2.2 Poisson Bracket and Symplectic Matrix; 2.3 General Solution to Hamiltonian Equations; 2.4 Picture of Motion in Phase Space; 2.5 Conserved Quantities and the Poisson Bracket; 2.6 Phase Space Transformations and Hamiltonian Equations.
- 2.7 Definition of Canonical Transformation2.8 Generalized Hamiltonian Equations: Example of Non-canonical Poisson Bracket; 2.9 Hamiltonian Action Functional; 2.9.1 Schrödinger Equation as the Hamiltonian System; 2.9.2 Lagrangian Action Associated with the Schrödinger Equation. Analogies Between Quantum Mechanics and Electrodynamics; 2.9.3 Probability as a Conserved Charge via the Noether Theorem; 2.9.4 First-Order Action Functional, Routhian and All That; 2.10 Hamiltonization of a Theory with Higher-Order Derivatives; 2.10.1 First-Order Trick; 2.10.2 Ostrogradsky Method.
- 3 Canonical Transformations of Two-Dimensional Phase Space3.1 Time-Independent Canonical Transformations; 3.1.1 Time-Independent Canonical Transformations and Symplectic Matrix; 3.1.2 Generating Function; 3.2 Time-Dependent Canonical Transformations; 3.2.1 Canonical Transformations and Symplectic Matrix; 3.2.2 Generating Function; 4 Properties of Canonical Transformations; 4.1 Invariance of the Poisson Bracket (Symplectic Matrix); 4.2 Infinitesimal Canonical Transformations: Hamiltonian as a Generator of Evolution; 4.2.1 Generator of Infinitesimal Canonical Transformation.