Ivancevic_Applied-Diff-Geom

366 Applied Differential Geometry: A Modern Introductionconditions:I = I(h),∮M hdϕ = 2π. (3.181)The action variable in the system with one DOF given by the Hamiltonianfunction H(p, q) is the quantityI(h) = 12π Π(h) = 1 ∮pdq,2π M hwhich is the area bounded by the phase curve H = h. Arnold statesthe following Theorem: Set S(I, q) = ∫ qq 0pdq| H=h(I) is a generating function.Then a canonical transformation (p, q) → (I, ϕ) satisfying conditions(3.181) is given by( )∂S(I, q) ∂S(I, q) ∂S(I, q)p = , ϕ = , H , q = h(I).∂q∂I∂qWe turn now to systems with n DOF given in R 2n = {(p i , q i ), i =1, ..., n} by a Hamiltonian function H(p i , q i ) and having n first integrals ininvolution F 1 = H, F 2 ..., F n . Let γ 1 , ..., γ n be a basis for the 1D cycles onthe torus M f = T n (the increase of the coordinate ϕ i on the cycle γ j isequal to 2π if i = j and 0 if i ≠ j). We setI i (f i ) = 12π∮M hp i dq i , (i = 1, ..., n). (3.182)The n quantities I i (f i ) given by formula (3.182) are called the actionvariables [Arnold (1989)].We assume now that, for the given values f i of the n integrals F i , then quantities I i are independent, det(∂I i /∂f i )| fi ≠ 0. Then in a neighborhoodof the torus M f = T n we can take the variables I i , ϕ i as symplecticcoordinates, i.e., the transformation (p i , q i ) → (I i , ϕ i ) is canonical, i.e.,dp i ∧ dq i = dI i ∧ dϕ i ,(i = 1, ..., n).Now, let m be a point on M f , in a neighborhood of which the n variablesq i are coordinates of M f , such that the submanifold M f ⊂ R 2n is given byn equations of the form p i = p i (I i , q i ), q i (m) = q0. i In a simply–connectedneighborhood of the point q0 i a single–valued function is defined,S(I i , q i ) =∫ qq 0p i (I i , q i ) dq i ,