Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 221momentum p i = m ˙q i , and the energyThenH(q, p) = 12m3∑p 2 i + V (q).i=1∂H∂q i = ∂V∂q i = −m¨qi = −ṗ i , and∂H= 1 ∂p i m p i = ˙q i , (i = 1, 2, 3),and hence Newtonian law F = m¨q i is equivalent to Hamiltonian equations˙q i = ∂H∂p i,ṗ i = − ∂H∂q i .Now, writing z = (q i , p i ) [Marsden and Ratiu (1999)],J grad H(z) =so Hamiltonian equations read( ) ( )0 I∂H∂q i∂H = ( ˙q i ), ṗ i = ż,−I 0∂p iż = J grad H(z). (3.57)Now let f : R 3 × R 3 → R 3 × R 3 and write w = f(z). If z(t) satisfiesHamiltonian equations (3.57) then w(t) = f(z(t)) satisfies ẇ = A T ż, whereA T = [∂w i /∂z j ] is the Jacobian matrix of f. By the chain rule,ẇ = A T J gradzH(z) = A T J A grad H(z(w)).wThus, the equations for w(t) have the form of Hamiltonian equations withenergy K(w) = H(z(w)) iff A T J A = J, that is, iff A is symplectic. Anonlinear transformation f is canonical iff its Jacobian matrix is symplectic.Sp(2n, R) is the linear invariance group of classical mechanics [Marsden andRatiu (1999)].