Global Aspects of Classical Integrable SystemsBirkhäuser, 1 juin 2015 - 477 pages This book gives a uniquely complete description of the geometry of the energy momentum mapping of five classical integrable systems: the 2-dimensional harmonic oscillator, the geodesic flow on the 3-sphere, the Euler top, the spherical pendulum and the Lagrange top. It presents for the first time in book form a general theory of symmetry reduction which allows one to reduce the symmetries in the spherical pendulum and the Lagrange top. Also the monodromy obstruction to the existence of global action angle coordinates is calculated for the spherical pendulum and the Lagrange top. The book addresses professional mathematicians and graduate students and can be used as a textbook on advanced classical mechanics or global analysis. |
Table des matières
| 1 | |
2
Geodesics on S3 | 31 |
3
The Euler Top | 79 |
4
The spherical pendulum | 139 |
5
The Lagrange top | 192 |
Part II Theory | 281 |
6
Fundamental concepts | 281 |
7
Systems with symmetry | 305 |
8
Ehresmann connections | 373 |
9
Action angle coordinates | 384 |
10
Monodromy | 391 |
11
Basic Morse theory | 423 |
Notes | 431 |
Acknowledgments | 464 |
| 465 | |
Autres éditions - Tout afficher
Global Aspects of Classical Integrable Systems Richard H. Cushman,Larry Meredith Bates Affichage d'extraits - 1997 |
Expressions et termes fréquents
2-sphere 2-torus action angular momentum bundle projection circle Claim closed compact compute coordinates critical point critical value defined definition diffeomorphism differential space energy momentum mapping Euler top fiber flow follows function f G-action G-invariant geodesic gives Hamiltonian system Hamiltonian vector field Hence homeomorphism induced integral curve intersects invariant inverse isomorphism Kepler Lagrange top lemma level set Lie algebra Lie group linear map matrix momentum mapping monodromy Morse motion nondegenerate open neighborhood open subset orbit space period lattice Poisson algebra Poisson bracket polynomial Proof reduced Hamiltonian reduced space regular value rotation number S1-action semialgebraic variety singular smooth function smooth manifold spherical pendulum submanifold subspace Suppose surjective symmetry symplectic form symplectic manifold tangent theorem topology trivial vector field XH