Global Aspects of Classical Integrable Systems
This book gives a complete global geometric description of the motion of the two di mensional hannonic oscillator, the Kepler problem, the Euler top, the spherical pendulum and the Lagrange top. These classical integrable Hamiltonian systems one sees treated in almost every physics book on classical mechanics. So why is this book necessary? The answer is that the standard treatments are not complete. For instance in physics books one cannot see the monodromy in the spherical pendulum from its explicit solution in terms of elliptic functions nor can one read off from the explicit solution the fact that a tennis racket makes a near half twist when it is tossed so as to spin nearly about its intermediate axis. Modem mathematics books on mechanics do not use the symplectic geometric tools they develop to treat the qualitative features of these problems either. One reason for this is that their basic tool for removing symmetries of Hamiltonian systems, called regular reduction, is not general enough to handle removal of the symmetries which occur in the spherical pendulum or in the Lagrange top. For these symmetries one needs singular reduction. Another reason is that the obstructions to making local action angle coordinates global such as monodromy were not known when these works were written.
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2-sphere 2-torus angular momentum appendix B section argument bundle projection circle Claim coadjoint equivariant compact compute coordinates critical point critical value defined definition Delaunay vector field diffeomorphism Ehresmann connection energy momentum mapping Euler top fiber figure axis flow follows G-action G-invariant geodesic vector field gives Hamiltonian system Hamiltonian vector field harmonic oscillator Hence homeomorphism integral curve intersects isomorphic isotropy group Lagrange top lemma level set Lie algebra Lie group linear map metric momentum mapping monodromy motion obtain open neighborhood orbifold orbit space parametrization period lattice Poisson algebra Poisson bracket polynomial positively oriented principal bundle Proof pull back R3 x R3 reduced Hamiltonian regular value rotation number S1 action singular point smooth function smooth manifold spherical pendulum subgroup submanifold submersion subspace Suppose surjective symmetry symplectic form symplectic manifold tangent theorem topology trivial vector field Xh
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Normal Forms and Unfoldings for Local Dynamical Systems
No preview available - 2003