This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: The geometry of reduced cotangent bundles Matthew Perlmutter Departamento de Matem´ atica Instituto Superior T´ ecnico Lisboa joint work with Miguel Rodr´ ıguezOlmos and M. E. SousaDias Supported by the E.U. through the RTN MASIE “Mechanics And Symmetry In Europe” – Typeset by Foil T E X – Regular Hamiltonian G–spaces • ( P ,ω ) symplectic manifold • G Lie group acting freely and properly on P by symplectomorphims • J : P → g * equivariant momentum map i : J 1 (0) , → P π : J 1 (0) → P := J 1 (0) /G then P is a symplectic manifold (MW reduced space) with symplectic form ω defined by π * ω = i * ω – Typeset by Foil T E X – 1 Singular Hamiltonian G–spaces If the freeness assumption is removed then P is no longer a smooth manifold but a stratified Hausdorff space P = F ( H ) ∈ I P P ( H ) (disjoint union), where • I P denotes isotropy lattice of P . • P ( H ) = ( J 1 (0)) ( H ) /G are smooth manifolds • ( J 1 (0)) ( H ) = { z ∈ J 1 (0)  G z is conjugate to H } (orbit types) are smooth manifolds this is called the symplectic stratification of P , because its pieces P ( H ) are symplectic manifolds called symplectic strata , with symplectic reduced form ω ( H ) – Typeset by Foil T E X – 2 given by π ( H ) * G ω ( H ) = i * ( H ) ω ( J 1 (0)) ( H ) i ( H ) / π ( H ) G P P ( H ) The frontier conditions are: 1. P ( K ) ∩ P ( H ) 6 = ∅ ⇔ H 6 K (subconjugation) 2. P ( K ) ∩ P ( H ) 6 = ∅ ⇒ P ( K ) ⊂ P ( H ) In this case we write P ( K ) 6 P ( H ) and say that P ( K ) is incident to P ( H ) . Example: Isotropy Latt., J 1 (0) Stratification Latt., P – Typeset by Foil T E X – 3 The Canoe: S 1 × S 2 → S 2 • ( P ,ω ) = T * S 2 with its canonical symplectic form • S 1 action on S 2 by rotations around the e 3 axis and its cotangent lift on T * S 2 by symplectomorphisms. • Equivariant momentum map: if x · x = 1 and x · p = 0 , then ( x , p ) ∈ T * S 2 , and J ( x , p ) = ( x × p ) · e 3 – Typeset by Foil T E X – 4 Using Invariant Theory the symplectic quotient P is realized as the semialgebraic variety of R 3 called “The Canoe” – Typeset by Foil T E X – 5 Objective: Find finer structure related to the symplectic stratification when P = T * M and G acts by cotangent lifts • Since in the cotangent bundle case P has more structure we want to study how this is reflected in the symplectic quotient P and in the symplectic strata P ( H ) • In particular how close are these sets to cotangent bundles of certain manifolds • To what extent is the symplectic data of the...
View
Full Document
 Fall '08
 Marsden,J
 Hamiltonian mechanics, orbit types, Symplectic manifold, Symplectic geometry, symplectic, SH →L

Click to edit the document details