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Advanced mechanics : from Euler's determinism to Arnold's chaos /
Classical Mechanics is the oldest and best understood part of physics. This does not mean that it is cast in marble yet, a museum piece to be admired from a distance. Instead, mechanics continues to be an active area of research by physicists and mathematicians. Every few years, we need to re-evalua...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Oxford, United Kingdom :
Oxford University Press,
2013.
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| Edición: | First edition. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Contents; List of Figures; 1 The variational principle; 1.1 Euler-Lagrange equations; 1.2 The Variational principle of mechanics; 1.3 Deduction from quantum mechanics*; 2 Conservation laws; 2.1 Generalized momenta; 2.2 Conservation laws; 2.3 Conservation of energy; 2.4 Minimal surface of revolution; 3 The simple pendulum; 3.1 Algebraic formulation; 3.2 Primer on Jacobi functions; 3.3 Elliptic curves*; 3.4 Imaginary time; 3.5 The arithmetic-geometric mean*; 3.6 Doubly periodic functions*; 4 The Kepler problem; 4.1 The orbit of a planet lies on a plane which contains the Sun.
- 4.2 The line connecting the planet to the Sun sweeps equal areas in equal times4.3 Planets move along elliptical orbits with the Sun at a focus; 4.4 The ratio of the cube of the semi-major axis to the square of the period is the same for all planets; 4.5 The shape of the orbit; 5 The rigid body; 5.1 The moment of inertia; 5.2 Angular momentum; 5.3 Euler's equations; 5.4 Jacobi's solution; 6 Geometric theory of ordinary differential equations; 6.1 Phase space; 6.2 Differential manifolds; 6.3 Vector fields as derivations; 6.4 Fixed points; 7 Hamilton's principle; 7.1 Generalized momenta.
- 7.2 Poisson brackets7.3 The star product*; 7.4 Canonical transformation; 7.5 Infinitesimal canonical transformations; 7.6 Symmetries and conservation laws; 7.7 Generating function; 8 Geodesics; 8.1 The metric; 8.2 The variational principle; 8.3 The sphere; 8.4 Hyperbolic space; 8.5 Hamiltonian formulation of geodesics; 8.6 Geodesic formulation of Newtonian mechanics*; 8.7 Geodesics in general relativity*; 9 Hamilton-Jacobi theory; 9.1 Conjugate variables; 9.2 The Hamilton-Jacobi equation; 9.3 The Euler problem; 9.4 The classical limit of the Schrödinger equation*
- 9.5 Hamilton-Jacobi equation in Riemannian manifolds*9.6 Analogy to optics*; 10 Integrable systems; 10.1 The simple harmonic oscillator; 10.2 The general one-dimensional system; 10.3 Bohr-Sommerfeld quantization; 10.4 The Kepler problem; 10.5 The relativistic Kepler problem*; 10.6 Several degrees of freedom; 10.7 The heavy top; 11 The three body problem; 11.1 Preliminaries; 11.2 Scale invariance; 11.3 Jacobi co-ordinates; 11.4 The 1/r[Sup(2)] potential; 11.5 Montgomery's pair of pants; 12 The restricted three body problem; 12.1 The motion of the primaries; 12.2 The Lagrangian.
- 12.3 A useful identity12.4 Equilibrium points; 12.5 Hill's regions; 12.6 The second derivative of the potential; 12.7 Stability theory; 13 Magnetic fields; 13.1 The equations of motion; 13.2 Hamiltonian formalism; 13.3 Canonical momentum; 13.4 The Lagrangian; 13.5 The magnetic monopole*; 13.6 The Penning trap; 14 Poisson and symplectic manifolds; 14.1 Poisson brackets on the sphere; 14.2 Equations of motion; 14.3 Poisson manifolds; 14.4 Liouville's theorem; 15 Discrete time; 15.1 First order symplectic integrators; 15.2 Second order symplectic integrator; 15.3 Chaos with one degree of freedom.