12 - FLOWS

Summary

At any particular time the state of a system is given by some collection of parameters. A deterministic mathematical model would hope to predict future states of that system. So if say a is the state at time t = 0 we may have a state Λ^t (a) at time t. After another time lapse of say s we would have a state Λ^s(Λ^t(a)). We would therefore want our model to satisfy the relation Λ^s(Λ^t(a)) = Λ^t+s(a), that is, Λ^t+s = Λ^s∘Λ^t. In other words the “time evolution” of our system would have to satisfy certain group properties. We could thus think of the “time evolution” operator Λ^t as acting on all points of a manifold of states at once and think of this as giving a “flow” on the manifold with increasing time.

In the case of a mechanical system our manifold is TQ and the flow is a collection of integral curves in TQ of a Lagrangian vector field, each such curve corresponding to a set of initial conditions.

FLOWS GENERATED BY VECTORFIELDS

Given a smooth vectorfield on a manifold, it was shown in Chapter 10 that there is a unique integral curve at each point of the manifold. Here our concern is with the overall structure of the set of all such integral curves obtained as the point ranges over the manifold.

It is helpful to regard the integral curves as the paths traced out by particles of a fluid moving with velocities which are prescribed at each point by the vectorfield.