Vector Fields
In the first four sections, we discuss elementary aspects of vector fields. We show that it is fruitful to view vector fields as derivations of the algebra of smooth functions, discuss the notions of integral curve and flow, introduce the Lie derivative a
- PDF / 952,614 Bytes
- 72 Pages / 439.37 x 666.142 pts Page_size
- 75 Downloads / 177 Views
Vector Fields
In the first four sections, we discuss elementary aspects of the theory of vector fields. In Sect. 3.1 we show that it is fruitful to view vector fields as derivations of the algebra of functions on the manifold. Next, in Sect. 3.2, we discuss in detail the notions of integral curve and flow and, in Sect. 3.3 we introduce the Lie derivative of a tensor field with respect to a given vector field. Finally, in Sect. 3.4, we extend the notion of an ordinary vector field to that of a time-dependent vector field. Next, we pass to more advanced topics. In Sect. 3.5, we give an introduction to the theory of (geometric) distributions, a notion which generalizes that of a vector field: a distribution is a subset of the tangent bundle consisting of linear subspaces of the tangent spaces. Following the theory developed by Stefan and Sussmann, we discuss the concept of integrability in some detail. The special case of a distribution of constant rank is built in here and the classical Frobenius Theorem occurs as a special case of a general theorem yielding integrability criteria. In the remaining four sections, we give an introduction to the study of the qualitative behaviour of the flows of vector fields.1 In Sect. 3.6 we collect the basic notions related to critical integral curves.2 In Sect. 3.7, we introduce the concept of a Poincaré mapping and in Sect. 3.8 we pass to the study of elementary aspects of stability. This notion comprises a variety of concepts characterizing, in effect, two aspects of the longtime behaviour of a flow, namely, returning properties and attraction properties of integral curves or, more generally, of invariant subsets. Here, we limit our attention to the concept of so-called orbital stability, to which for simplicity we refer to merely as stability. At the end of this section we briefly discuss the relation to Lyapunov stability, a notion physicists are probably more familiar with. Finally, in Sect. 3.9, we present the concept of invariant manifolds which plays a basic role in the analysis of the qualitative behaviour of flows near critical integral curves. In all of the four final sections, the reader will find a number of illustrative examples. 1 That
is, to the theory of dynamical systems.
2 Equilibrium
points or periodic integral curves.
G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_3, © Springer Science+Business Media Dordrecht 2013
93
94
3 Vector Fields
From now on, we restrict attention to smooth manifolds and smooth mappings. Let M be a smooth manifold of dimension n. Recall from Chap. 2 that a vector field on M is a section of the tangent bundle TM and that the space of vector fields is denoted by X(M). Recall, furthermore, that for every local chart (U, κ), the local sections ∂i form a local frame in TM.
3.1 Vector Fields as Derivations In this section, we will relate vector fields to derivations of the associative algebra C ∞ (M) and use this to define their commutator.
Data Loading...
