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Vector Fields and Dynamical Systems
This chapter studies vector fields and the dynamical systems they determine. The ensuing chapters will
study the related topics of tensors and differential forms. A basic operation introduced in this chapter is
the Lie derivative of a function or a vector field. It is introduced in two different ways, algebraically as a
type of directional derivative and dynamically as a rate of change along a flow. The Lie derivative formula
asserts the equivalence of these two definitions. The Lie derivative is a basic operation used extensively in
differential geometry, general relativity, Hamiltonian mechanics, and continuum mechanics.
Vector Fields and Flows
This section introduces vector fields and the flows they determine. This topic puts together and globalizes
two basic ideas we learn in undergraduate calculus: the study of vector fields on the one hand and differential
equations on the other.
Definition. Let M be a manifold. A vector field on M is a section of the tangent bundle TM of
M. The set of all Cr vector fields on M is denoted by Xr(M) and the C∞ vector fields by X∞(M) or X(M).
Thus, a vector field X on a manifold M is a mapping X : M → TM such that X(m) ∈ TmM for all
m ∈ M. In other words, a vector field assigns to each point of M a vector based (., bound) at that point.
Example. Consider the force field determined by Newton’s law of gravitation. Here the manifold is
R3 minus the origin and the vector field is
mMG
F(x, y, x)=− r,
r3
where m is the mass of a test body, M is the mass of the central body, G is the constant of gravitation, r
is the vector from the origin to (x, y, z), and r =(x2 + y2 + z2)1/2; see Figure .
The study of dynamical systems, also called flows, may be motivated as follows. Consider a physical
system that is capable of assuming various “states” described by points in a set S. For example, S might
be R3 × R3 and a state might be the position and momentum (q,