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. |
Contents
III | 1 |
IV | 3 |
V | 4 |
VI | 11 |
VII | 15 |
XI | 22 |
XII | 28 |
XIII | 30 |
XLIII | 181 |
XLIV | 188 |
XLV | 201 |
XLVI | 204 |
XLVII | 208 |
XLVIII | 214 |
XLIX | 219 |
L | 225 |
XIV | 33 |
XV | 42 |
XVI | 44 |
XVII | 50 |
XVIII | 53 |
XIX | 55 |
XX | 59 |
XXI | 61 |
XXII | 65 |
XXIII | 68 |
XXIV | 75 |
XXV | 77 |
XXVI | 83 |
XXVII | 88 |
XXVIII | 91 |
XXIX | 98 |
XXX | 103 |
XXXI | 110 |
XXXII | 113 |
XXXIII | 123 |
XXXIV | 129 |
XXXV | 133 |
XXXVI | 142 |
XXXVII | 148 |
XXXVIII | 154 |
XXXIX | 161 |
XL | 163 |
XLI | 170 |
XLII | 174 |
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Common terms and phrases
angular momentum angular velocity axis called canonical transformation characteristic frequencies configuration space consider contact elements contact manifold contact structure coordinate system Corollary corresponding cotangent bundle curvature defined Definition degrees of freedom diffeomorphism differential equations differential form dimension 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 k-form lagrangian manifold Legendre Lemma Lie algebra linear mapping Math metric n-dimensional neighborhood nondegenerate normal form obtain orbit P₁ parameter perturbation phase curves phase flow phase space plane Poisson bracket Poisson structure polynomials potential energy PROBLEM projection PROOF q₁ quadratic form resonance riemannian rigid body rotation singularities smooth solution stationary submanifold surface symmetric symplectic manifold symplectic structure tangent space theorem theory three-dimensional torus trajectory two-dimensional variables vector field vector space