Mathematical Methods of Classical Mechanics
Many different mathematical methods and concepts are used in classical mechanics: differential equations and phase ftows, smooth mappings and manifolds, Lie groups and Lie algebras, symplectic geometry and ergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them so hard to study. In this book we construct the mathematical apparatus of classical mechanics from the very beginning; thus, the reader is not assumed to have any previous knowledge beyond standard courses in analysis (differential and integral calculus, differential equations), geometry (vector spaces, vectors) and linear algebra (linear operators, quadratic forms). With the help of this apparatus, we examine all the basic problems in dynamics, including the theory of oscillations, the theory of rigid body motion, and the hamiltonian formalism. The author has tried to show the geometric, qualitative aspect of phenomena. In this respect the book is closer to courses in theoretical mechanics for theoretical physicists than to traditional courses in theoretical mechanics as taught by mathematicians.
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action angular axis body called canonical characteristic closed condition consider constructed coordinate system coordinates corresponding curvature curve defined Definition denote depend derivative determined diffeomorphism differential dimension direction eigenvalues elements ellipsoid energy equal equations euclidean example exterior derivative fact Figure fixed force formula frequencies geodesic given gives hamiltonian function identity independent initial integral invariant lagrangian Lie algebra linear lines manifold mapping mechanics metric momentum motion multiplication n-dimensional neighborhood normal obtain operator orbit oriented original oscillations parameter particular periodic phase flow phase space plane position potential preserves PROBLEM projection PROOF proved reduced region resonance respect riemannian riemannian curvature rotation Show singularities smooth solution stable stationary surface symplectic symplectic structure tangent space theorem torus trajectory transformation translation variables vector field velocity zero