Chapter 9 - Integral Curves and Flows Flashcards
Define: integral curve, starting point
If V is a vector field of M. an INTEGRAL CURVE of V is a differentiable curve gamma: J –> M whose velocity at each point is equal to the value of V at that point:
gamma’(t) = V_gamma(t) for all t in J.
The point gamma(0) is called the STARTING POINT of gamma.
Examples of integral curves? Global flows?
What are regular points? Singular points? Put in canonical form
- V = d/dx in R^2. Integral curves of V are gamma(t) = (a + t, b). One for each point of R^2.
- W = xd/dy - yd/dx. Get system of ODE. SOLVE
In both cases, notice that we get a unique integral curve for each starting point in R^2, and the images of the various integral curves are either identical or disjoint
pgs 206-207
These both actually generate global flows
- tau_t(x,y) = (x +t, y)
- theta_t(x,y) = (x cos t - y sin t, x sin t + y cos t)
pgs 210-211
Canonical form form: 221
How does finding integral curves reduce to solving a system of ODE in a smooth chart?
If V VF on M and p in M, is there an integral curve of V starting at p?
Say V a smooth VF on M and gamma : J –> M a smooth curve. On a smooth coordinate domain U < M, we can write gamma in local coordinates as gamma(t) = (gamma^1(t), … , gamma^n(t)). Then the condition gamma’(t) = V_gamma(t) can be written:
gamma^1’(t) = V^i((gamma(t))
an autonomous system of ODE.
Yes, for each pin in M, there exists an epsilon > 0 and a smooth curve gamma: (-epsilon, epsilon) –> M that is an integral curve of V starting at p.
Pf. Apply existence statement of Existence, Uniqueness, and Smoothness theorem for ODE to the a coordinate rep of V.
How do affine reparameterizations affect integral curves?
- Rescaling: If gamma an integral curve of V, gamma tilda defined by gamma tilda(t) = gamma(at) is an integral curve of aV
- Translation: gamma hat(t) = gamma(t + b) is also an integral curve of V
pg 208
In what sense are integral curves natural? Proof?
Prop. Suppose M and N are smooth manifolds and F : M –> N is a smooth map. Then X in chi(M) and Y in chi(N) are F-related <=> F takes integral curves of X to integral curves of Y, meaning that for each integral curve gamma of X, F o gamma is an integral curve of Y.
Pf. 208-209
Define: global flow
related collections maps?
motivation?
A GLOBAL FLOW on M (also called a ONE-PARAMETER GROUP ACTION) is a continuous left R-action on M, i.e. a continuous map theta: R x M –> M satisfying standard group action axioms.
Related collections of maps:
1. For each t in R, define a continuous map theta_t : M –> M by theta_t(p) = theta(t, p) <– homeomorphisms, diffeomorphism if smooth
- For each p in M, define a curve theta^(p) : R –> M by theta^(p)(t) = theta(t,p). The image of this curve is the orbit of p under the group action.
Motivation comes from visualizing family of integral curves
pg 209
How do smooth global flows relate to smooth vector fields?
If theta : R x M –> M is a smooth global flow, for each p in M we define a tangent vector Vp in TpM by
Vp = theta^(p)’(0).
The assignment p –> Vp is a rough vector field on M called the INFINITESIMAL GENERATOR OF THETA.
Prop. The infinitesimal generator V of theta is a smooth vector field on M, and each curve theta^(p) is an integral curve of V.
Pf. pg 210
Is every smooth vector field the infinitesimal generator of a smooth global flow? Discuss.
We have seen that every smooth global flow gives rise to a smooth vector field whose integral curves are precisely the curves defined by the flow.
The reverse direction is not true, there exist smooth vector fields which are not the infinitesimal generator of a smooth global flow. (The issue is that the flows they generate are not defined for all t)
Examples
1. M = R^2 - {0} with standard coordinates (x,y) and V = d/dx. The unique integral curve starting at (-1,0) in M is gamma(t) = (t - 1, 0). This cannot be extended continuously past t = 1. <– topological constraint - not simply connected
- M = R^2, W = x^2 d/dx. Integral curve starting at (1,0) us gamma(t) = (1/1-t, 0) can’t extend past t=1, unbounded x-coordinate)
Define: flow domain, flow, infinitesimal generator, M_t
related families of maps?
A FLOW DOMAIN for M is an open subset D < R x M with the property that for each p in M, the set D^(p) = {t in R : (t,p) un D} is an open interval containing 0.
A FLOW on M is a continuous map theta: D –> M that satisfies the following group laws:
- For all p in M, theta(0,p) = p
- For all s in D^(p) and t in D^(theta(s,p)) such that s + t in D^(p), theta(t, theta(s,p)) = theta(t+s, p)
(also called LOCAL FLOW or LOCAL ONE-PARAMETER GROUP ACTION)
Again define theta_t(p) = theta^(p)(t) = theta(t,p)
Define M_t = {p in M : (t,p) in D}
If theta is smooth, the INFINITESIMAL GENERATOR V of THETA is defined by V_p = theta^(p)’(0)
As in the case of global flows, V is smooth vector field and each curve theta^(p) is an integral curve of V. <– proof almost identical - need D^(p) open intervals
Define: maximal integral curve, maximal flow, flow generated by V
What is Fundamental Theorem on Flows? Proof?
A MAXIMAL INTEGRAL CURVE is one that cannot be extended to an integral curve on any larger open interval and a MAXIMAL FLOW is a flow that admits no extension to a flow on a larger flow domain.
Theorem. Let V be a smooth vector field on a smooth manifold M. There is a unique smooth maximal flow theta: D –> M whose infinitesimal generator is V. This flow has the following properties:
- For each p in M , the curve theta^(p) : D^(p) –> M is the unique maximal integral curve of V starting at p
- If s in D^(p), then D^(theta(s,p)) is the interval D^(p) -s = {t -s : t in D^(p)}
- For each t in R, the set M_t is open in M, and theta_t : M_t –> M_-t is a diffeomorphism with inverse theta_-t
Bijection: {maximal smooth flows} <–> {smooth vector fields}
pf. 212 - 214
Naturally of flows? Proof?
Prop. Suppose M and N are smooth manifolds, F: M –> N is a smooth map, X in chi(M), Y in chi(N). Let theta be the flow of X and eta the flow of Y.
If X and Y are F-related, then for each t in R, F(M_t) < N_t and eta_t o F = F o theta_t on M_t.
pg 215
Define: complete vector field
Discuss criteria for a vector field to be complete. Proofs?
A smooth vector field is COMPLETE if it generates a global flow, or equivalently if each of its maximal integral curves is defined for all t in R.
Criteria
1. Every compactly supported smooth vector field on a smooth manifold is complete.
Corollary. On a compact smooth manifold, every smooth vector field is complete
- Every left-invariant vector field on a Lie group is complete
Key Lemma
Uniform Time Lemma. Let V be a smooth vector field on a smooth manifold M, and let theta be its flow. Suppose there is a positive number epsilon s.t. for every p in M, the domain of theta^(p) contains (-epsilon, epsilon). Then V is complete.
Idea: Can extend integral curve past any limit point
Define: flout from S along V
What is the Flowout Theorem? Proof?
Flowout Theorem. Suppose M is a smooth manifold, S < M is an embedded k-dimensional submanifold, and V in chi(M) is a smooth vector field that is no where tangent to S. Let theta : D –> M be the flow of V, let O = R x S intersect D, and let phi = theta|O.
- phi : O –> M is an immersion
- d/dt in chi(O) is phi-related to V
- There exists a smooth positive function delta : S –> R s.t. that restriction of phi to O_delta is injective, where O_delta < O is the flow domain { (t, p) in O : |t| < delta(p) }. Thus phi(O_delta) is an immersed sub manifold of M containing S, and V is tangent to this sub manifold.
- If S has codimension 1, the phi|O_delta is a diffeomorphism onto an open sub manifold of M
The submanifold phi(O_delta) < M is called a FLOWOUT FROM S ALONG V.
pgs 217-219
Define: singular point of vector field, regular point, equilibrium point of flow
Discuss the behavior of flows near singular points and regular points
If V is a vector field on M, a point p in M is said to be a SINGULAR POINT OF V if V_p = 0 and a REGULAR POINT otherwise.
Integral curves starting at regular points and singular points behave very differently:
Prop. Let theta : D –> M be the flow generated by V. If p in M is a singular point of V, then D^(p) = R and theta^(p) is the constant curve theta^(p)(t) = p. If p is a regular point, then theta^(p) is a smooth immersion.
Pf 219-220
A point p in M is called an EQUILIBRIUM POINT of THETA if theta(t, p) = p for all t in D^(p). The above prop shows that equilibrium points of a smooth flow = singular points of infinitesimal generator.
Theorem (Canonical Form Near a Regular Point). Let p in M be a regular point of V. There exist smooth coordinates (s^i) on some neighborhood of p in which V has the coordinate representation d/ds^1. If S < M is any embedded hypersurface with p in S and Vp not in TpS, then the coordinates can also be chosen so that s^1 is a local defining function for S.
Thus a flow in a neighborhood of a regular point behaves just like translation along parallel coordinate lines of R^n. All the interesting behavior of the flow is concentrated near equilibrium points
Discuss flows and flowouts on manifolds with boundary
Collar neighborhood?
In general, a smooth vector field on a manifold with boundary need not generate a flow, because, for example, the integral curves starting at some boundary points might be defined only on half-open intervals.
Boundary Flowout Theorem. Let M be a smooth manifold with boundary and V a smooth vector field on M that is inward-pointing at each point of boundary M. There exists a smooth function delta: boundary M –> R+ and a smooth embedding phi : P_delta –> M such that phi(P_delta) is a neighborhood of boundary M and for each p in boundary M, the map t –> phi(t, p) is an integral curve starting at p
A neighborhood of boundary M is called a COLLAR NEIGHBORHOOD if it is the image of a smooth embedding [0,1) x boundary M –> M that restricts to the obvious identification {0} x boundary M –> boundary M.
Thm. If M is a smooth manifold with nonempty boundary, the boundary M has a collar neighborhood.
So we use a specific vector field and flowout from boundary to get a collar neighborhood. Once we have collar neighborhoods, many constructions are possible…
- Every smooth manifold with boundary is homotopy equivalent to its interior
- a. Whitney Approximation for Manifolds with Boundary. If M and N are smooth manifolds with boundary, then every continuous map from M to N is homotopic to a smooth map.
- b. If F,G : M –> N are homotopic smooth maps, then they are smoothly homotopic
- Attaching smooth manifolds along their boundaries. Can glue smooth manifolds along diffeomorphism between boundaries to obtain a new smooth manifold. Compact if both compact, connected if both connected.
Examples: Connected sum & Double of M - The fundamental theorem on flows holds for smooth manifolds with boundary provided vector fields are tangent to boundary everywhere as does the canonical form near a regular point.
