Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 353This is a Hamiltonian system with {q i , p i } as canonical coordinates on R 6 ,M = T ∗ R 3 ≃ R 6 , ω = dp i ∧ dq i ,H = 1 (p22 1 + p 2 2 + p 2 )3 + U.The Toda molecule (3.12.3.2) is an integrable Hamiltonian system in anopen set of T ∗ R 3 with:K 1 = H, K 2 = p 1 + p 2 + p 3 ,K 3 = 1 9 (p 1 + p 2 + p 3 ) (p 2 + p 3 − 2p 1 ) (p 3 + p 1 − 2p 2 ) − (p 1 + p 2 − 2p 3 ) e q1 −q 2− (p 2 + p 3 − 2p 1 ) e q2 −q 3 − (p 3 + p 1 − 2p 2 ) e q3 −q 1 .3–Point Vortex ProblemThe motion of three–point vortices for an ideal incompressible fluid inthe plane is given by the equations:˙q j = − 12πṗ j = 12π∑Γ i (p j − p i ) /rij,2i≠j∑ (Γ i q i − q j) /rij,2i≠jr 2 ij = ( q i − q j) 2+ (pj − p i ) 2 ,where i, j = 1, 2, 3, and Γ i are three nonzero constants. This mechanicalsystem is Hamiltonian if we take:M = T ∗ R 3 ≃ R 6 , ω = dp i ∧ dq i , (i = 1, ..., 3),H = − 1 3∑Γ i Γ i ln (r ij ) .4πi,j=1Moreover, it is integrable in an open set of T ∗ R 3 with:K 1 = H, K 2 =3∑i=1( 3∑) 2K 3 = Γ i q i + K2.2i=1Γ i[ (qi ) 2+ p2i],The Newton’s Second Law as a Hamiltonian System