Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 165Let ϕ : M → N be a C k map of manifolds. The vector–fields X ∈X k−1 (M) and Y∈ X k−1 (N) are called ϕ−related, denoted X ∼ ϕ Y , ifT ϕ ◦ X = Y ◦ ϕ.Note that if ϕ is diffeomorphism and X and Y are ϕ−related, then Y =ϕ ∗ X. However, in general, X can be ϕ−related to more than one vector–field on N. ϕ−relatedness means that the following diagram commutes:T M✻XMT ϕϕ✲ T N✻Y✲ NThe behavior of flows under these operations is as follows: Let ϕ : M →N be a C k −map of manifolds, X ∈ X k (M) and Y ∈ X k (N). Let F t and G tdenote the flows of X and Y respectively. Then X ∼ ϕ Y iff ϕ ◦ F t = G t ◦ ϕ.In particular, if ϕ is a diffeomorphism, then the equality Y = ϕ ∗ X holds iffthe flow of Y is ϕ◦F t ◦ϕ −1 (This is called the push–forward of F t by ϕ sinceit is the natural way to construct a diffeomorphism on N out of one on M).In particular, (F t ) ∗X = X. Therefore, the flow of the push–forward of avector–field is the push–forward of its flow.3.6.1.2 Dynamical Evolution and FlowAs a motivational example, consider a mechanical system that is capableof assuming various states described by points in a set U. For example, Umight be R 3 × R 3 and a state might be the positions and momenta (x i , p i )of a particle moving under the influence of the central force field, withi = 1, 2, 3. As time passes, the state evolves. If the state is γ 0 ∈ U at times and this changes to γ at a later time t, we setF t,s (γ 0 ) = γ,and call F t,s the evolution operator; it maps a state at time s to what thestate would be at time t; that is, after time t−s. has elapsed. Determinismis expressed by the Chapman–Kolmogorov law [Abraham et al. (1988)]:F τ,t ◦ F t,s = F τ,s, F t,t = identity. (3.30)The evolution laws are called time independent, or autonomous, whenF t,s depends only on t − s. In this case the preceding law (3.30) becomes