Mathematical Methods of Classical Mechanics
In this text, the author constructs the mathematical apparatus of classical mechanics from the beginning, examining all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the Hamiltonian formalism. This modern approch, based on the theory of the geometry of manifolds, distinguishes iteself from the traditional approach of standard textbooks. Geometrical considerations are emphasized throughout and include phase spaces and flows, vector fields, and Lie groups. The work includes a detailed discussion of qualitative methods of the theory of dynamical systems and of asymptotic methods like perturbation techniques, averaging, and adiabatic invariance.
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angular momentum angular velocity axis called canonical transformation characteristic frequencies characteristic oscillations closed configuration space consider contact elements contact manifold contact structure coordinate system Corollary corresponding cotangent bundle curvature defined Definition degrees of freedom denote diffeomorphism differential equations dimension dimensional direction eigenvalues ellipse ellipsoid equal to zero equilibrium position euclidean space example Figure formula function H geodesic given hamiltonian function hyperplane hypersurface inertia inertia ellipsoid initial conditions integral intersection invariant tori kinetic energy lagrangian manifold Lemma Lie algebra linear mapping metric multiplication n-dimensional neighborhood nondegeneracy nondegenerate normal form obtain one-parameter orbit oriented parameter pendulum perturbation phase curves phase flow phase space plane Poisson bracket Poisson structure polynomials potential energy PROBLEM projection PROOF quadratic form resonance rigid body rotation Show smooth solution stable stationary submanifold surface symmetric symplectic manifold symplectic structure tangent space theorem theory three-dimensional torus trajectory two-dimensional vector field vector space