Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 859contain no critical points of first integrals F k , and let its projection N 0onto the fibre V0 ∗ M along trajectories of γ H be compact. Then the invariantmanifold h(N) of the completely integrable Hamiltonian system(T ∗ M; H ∗ , ζ ∗ F k ) has an open neighborhood U.Now, the open neighborhood U of the invariant manifold h(N) of thecompletely integrable Hamiltonian system (T ∗ M; H ∗ , ζ ∗ F k ) is isomorphicto the symplectic annulusW ′ = V ′ × (R × T m ), V ′ = (−ε, ε) × V, (5.170)provided with the generalized action–angle coordinates(I 0 , . . . , I m ; t, φ 1 , . . . , φ m ). (5.171)Moreover, we find that J 0 = r, a α 0 = δ α 0 and, as a consequence,a 0 0 = ∂I 0∂J 0= 1,a 0 i = ∂I i∂J 0= 0,i.e., the action coordinate I 0 is linear in the coordinate r, while I i areindependent of r. With respect to the coordinates (5.171), the symplecticform on W ′ readsΩ T = dI 0 ∧ dt + dI k ∧ dφ k ,the Hamiltonian H ∗ is an affine function H ∗ = I 0 + H ′ (I j ) of the actioncoordinate I 0 , while the first integrals ζ ∗ F k depends only on the actioncoordinates I i . The Hamiltonian vector–field of the Hamiltonian H ∗ isγ T = ∂ t + ∂ i H ′ ∂ i . (5.172)Since the action coordinates I i are independent on the coordinate r, thesymplectic annulus W ′ (5.170) inherits the composite fibrationW ′ → V × (R × T m ) → R. (5.173)Therefore, one can regard W = V × (R × T m ) as a momentum phase–space of the time–dependent Hamiltonian system in question around theinvariant manifold N. It is coordinated by (I i , t, φ i ), which we agree to callthe time–dependent action–angle coordinates. By the relation similar to(5.163), W can be equipped with the Poisson structure{f, f ′ } W = ∂ i f∂ i f ′ − ∂ i f∂ i f ′ ,