Mathematical Methods of Classical MechanicsIn 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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Contents
III  1 
IV  3 
V  4 
VI  11 
VII  15 
VIII  22 
IX  28 
X  30 
XL  181 
XLI  188 
XLII  201 
XLIII  204 
XLIV  208 
XLV  214 
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XI  33 
XII  42 
XIII  44 
XIV  50 
XV  53 
XVI  55 
XVII  59 
XVIII  61 
XIX  65 
XX  68 
XXI  75 
XXII  77 
XXIII  83 
XXIV  88 
XXV  91 
XXVI  98 
XXVII  103 
XXVIII  110 
XXIX  113 
XXX  123 
XXXI  129 
XXXII  133 
XXXIII  142 
XXXIV  148 
XXXV  154 
XXXVI  161 
XXXVII  163 
XXXVIII  170 
XXXIX  174 
Common terms and phrases
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 ndimensional neighborhood nondegeneracy nondegenerate normal form obtain oneparameter 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 threedimensional torus trajectory twodimensional vector field vector space