- 132 - 6.4 Normalization of Hamiltonian systems In this section we give the classical theory of Hamiltonian normal forms using generating functions This approach is rather awkward in its actual use, due to the fact that the generating function is defined in mixed coordinates A more modern approach uses Lie-transforms, defined as Hamiltonian flows and therefore manifestly symplectic Let H be a C™-function on the cotangent bundle, i.e H:7*M—R On the cotangent bundle lives a natural symplectic form w Given x ET" M, there exist local coordinates (4,p),g €M,p €T,M, such that w can be written as w = > 4q:dp; The coordinates g and p are called position and momen- i=1 tum We say that the Hamiltonian system has n degrees of freedom, where nis the dimension of M This symplectic form can be used to define a vector field X induced by the Hamiltonian as follows: ty = dH, where ¢ is the usual inner product of a vector with a two-form In local coordinates we find, if we denote X by S(xi— + xi J 2 hô, “p Op; " yw = Sunt + xt 2 ) > 4q,d4p, &=] qs Hs, /=I lI Ms Ms : (XGdp, — X?dqr)ôu i=] lI Ms X74 aD; — Xã đạc k=] Since — 0H 0H dH = aq, A + ap, VF we find that y= OOH 8 | 9H 3 K=1 OP 99x = Gx Oy This gives the familiar Hamilton equations: _ — oH Gk = OD,” , _ oH Pk Đá Suppose now that H has a critical point (0,0), i.e a stationary point of the differential equation; Taking local coordinates around this critical point, H can be written as follows We let H(0,0) be zero, since its value does not an: of ee — 4M oo ee a ~®>
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