Reduction of a System on One Isolated Level Surface

Consider the results obtained in 1 from a more general point of view. Let us single out in the pure form the most general elements of the constructions described above. Recall that a smooth function / on a manifold Af is called the integral of a vector field V if it is constant along all integral trajectories of the field. In other words, an arbitrary function / in the direction of a field v is equal to zero, that is, v f) = 0. In this case, integral trajectories of the field lie on the levels Me = (/ = c) of the function /. 

A smooth function / is called a particular integral of a field t if one ( ) of its level surfaces is invariant with respect to the field t (in particular, with respect to the flow induced by the field). If this level has the form Mq = (/ = 0), then the indicated condition will be written as v = (/) (/=o) = 0. In the case of the general position, Af is an invariant submanifold of codimension 1. More generally, a submanifold L of codimension A in Af given by the equations L = (/i = 0. / . = 0) is called invariant (with respect to the field v) if u(/ ) l = 0. In particular, if several functionally independent first integrals /i. /jb of the system v are given, then each submanifold of the form 

Proposition 3.2.1. Let v = sgradlT be a Hamiltonian Geld on a symplectic manifold and f be a smooth function commuting (in the sense of the usual Poisson bracket) with the Hamiltonian H of this Geld, that is, H, / = 0 on Af. Then the function f is an integral of the Geld v. Next, a submanifold = 

Ifi = 0. fk = 0) is invariant with respect to the Geld v under the condition that 

Remark We may say that the functions / must commute with the function H on the surface L only (and generally, they do not already commute) outside this surface. 

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