Integral Curves
A smooth curve \(\gamma : J \to M\) on a smooth manifold has, at each parameter value
\(t \in J\), a well-defined
velocity vector
\(\gamma'(t) \in T_{\gamma(t)} M\). The construction so far runs in one direction: starting
from a curve, we read off a tangent vector at each of its points. In the present chapter
we run the construction in reverse. Given a
vector field
\(V\) on \(M\) — a tangent vector \(V_p \in T_p M\) prescribed at every point — we seek
a curve whose velocity at each of its points matches the prescribed vector there. Such a
curve, when it exists, threads through the manifold along the direction the vector field
indicates, and the family of all such curves encodes the dynamical content of the vector
field.
Definition: Integral Curve
Let \(V\) be a vector field on a smooth manifold \(M\). An integral curve
of \(V\) is a differentiable curve \(\gamma : J \to M\), defined on an open interval
\(J \subseteq \mathbb{R}\), whose velocity at each point equals the value of \(V\) at
that point:
\[
\gamma'(t) = V_{\gamma(t)} \qquad \text{for all } t \in J.
\]
Definition: Starting Point
If \(0 \in J\), the point \(\gamma(0) \in M\) is called the starting point
of \(\gamma\).
The differentiability hypothesis is essential to the definition: the equation
\(\gamma'(t) = V_{\gamma(t)}\) makes sense only when the left-hand side, the velocity
vector, exists. A merely continuous curve cannot be an integral curve in any meaningful
sense, because the condition that defines the notion would have nothing to evaluate.
Examples in the Plane
Two small examples on \(\mathbb{R}^2\) — both fundamental for what follows —
show how the abstract definition unwinds into a concrete computation.
Example (The Horizontal Vector Field on \(\mathbb{R}^2\)):
Let \((x, y)\) be standard coordinates on \(\mathbb{R}^2\), and let
\(V = \partial / \partial x\), regarded as the vector field on \(\mathbb{R}^2\) whose
value at every point is the corresponding coordinate basis vector.
A curve of the form \(\gamma(t) = (a + t, \, b)\), for constants \(a, b \in \mathbb{R}\),
has velocity \(\gamma'(t) = \partial / \partial x|_{\gamma(t)}\), so it is an integral
curve of \(V\). Conversely, any integral curve of \(V\) satisfies the system
\(\dot x(t) = 1\), \(\dot y(t) = 0\), whose general solution is \(x(t) = a + t\),
\(y(t) = b\). The integral curves of \(V\) are precisely the straight lines parallel
to the \(x\)-axis, and through each point \((a, b) \in \mathbb{R}^2\) there is exactly
one such curve with \(\gamma(0) = (a, b)\).
Example (The Rotation Vector Field on \(\mathbb{R}^2\)):
Let \(W = x \, \partial / \partial y - y \, \partial / \partial x\) on \(\mathbb{R}^2\).
Writing a candidate integral curve in coordinates as \(\gamma(t) = (x(t), y(t))\), the
defining condition \(\gamma'(t) = W_{\gamma(t)}\) reduces to the linear system
\[
\dot x(t) = -y(t), \qquad \dot y(t) = x(t).
\]
The general solution is
\[
x(t) = a \cos t - b \sin t, \qquad y(t) = a \sin t + b \cos t,
\]
with \(\gamma(0) = (a, b)\). For \((a, b) = (0, 0)\) the curve is the constant curve
at the origin; for any other starting point, the integral curve traces a circle
centered at the origin, traversed counterclockwise with unit angular speed.
Local Coordinates and the Underlying ODE System
The two examples above are direct because the manifold is \(\mathbb{R}^2\) and the vector
field has constant-coefficient expressions in the global standard coordinates. On a
general smooth manifold the same idea applies inside any
smooth chart:
finding an integral curve reduces, locally, to solving a system of ordinary differential
equations.
Let \(V\) be a smooth vector field on \(M\), and let \(\gamma : J \to M\) be a smooth
curve passing through a coordinate domain \(U \subseteq M\) with coordinates
\((x^1, \ldots, x^n)\). Writing \(\gamma\) in coordinates as
\(\gamma(t) = (\gamma^1(t), \ldots, \gamma^n(t))\) and \(V\) as
\(V = V^i (\partial / \partial x^i)\) (summation convention), the velocity vector is
\[
\gamma'(t) = \dot \gamma^i(t) \, \frac{\partial}{\partial x^i} \bigg|_{\gamma(t)} ,
\]
while the prescribed vector at \(\gamma(t)\) is
\[
V_{\gamma(t)} = V^i(\gamma(t)) \, \frac{\partial}{\partial x^i} \bigg|_{\gamma(t)} .
\]
The condition \(\gamma'(t) = V_{\gamma(t)}\) holds if and only if the coefficients of each
coordinate basis vector match. We therefore obtain the autonomous system of ordinary
differential equations
\[
\dot \gamma^i(t) = V^i \bigl( \gamma^1(t), \ldots, \gamma^n(t) \bigr) ,
\qquad i = 1, \ldots, n.
\]
The terminology now reveals itself. An integral curve of \(V\) is precisely a solution of
this system of ordinary differential equations; finding such a curve is what is classically
meant by integrating
the system of ODEs. The geometric definition and the analytical system are two views of
the same object — coordinate-free on one side, coordinate-dependent on the other.
Local Existence
The classical theory of ordinary differential equations guarantees that smooth autonomous
systems of the kind just derived admit local solutions through any initial condition. Transported
to the manifold setting through a chart, this analytical fact becomes a geometric
existence statement for integral curves.
Theorem: Local Existence of Integral Curves
Let \(V\) be a smooth vector field on a smooth manifold \(M\). For each point
\(p \in M\), there exist \(\varepsilon > 0\) and a smooth curve
\(\gamma : (-\varepsilon, \varepsilon) \to M\) that is an integral curve of \(V\)
starting at \(p\).
A formal proof reduces, by a choice of smooth chart centered at \(p\), to the local
existence statement for autonomous systems of ordinary differential equations with
smooth right-hand side. That existence statement — together with the corresponding
uniqueness and smooth-dependence-on-initial-conditions results that we shall invoke
in the next section — ultimately rests on the contraction mapping principle applied
to an integral-equation reformulation of the system, and we take the resulting
existence-uniqueness-smoothness package as a foundational fact from the classical
theory of ordinary differential equations. The present theorem
asserts neither uniqueness of the integral curve through \(p\) nor any lower bound on
the existence interval beyond positivity; both will be addressed in the next section.
Local Existence and Reparametrization
The local existence statement of the previous section is the entry point to a small but
central body of structural results about integral curves. Two of these — the rescaling
and translation lemmas — describe how an integral curve transforms under affine
reparametrizations of its time variable. Both are elementary applications of the chain
rule, yet they encode the algebraic content that will, in the next section, organize the
family of integral curves of a vector field into a one-parameter group of
transformations. A third result, the naturality proposition, describes how integral
curves behave under smooth maps between manifolds; it provides the geometric counterpart
of the algebraic characterization of \(F\)-relatedness obtained earlier.
Rescaling and Translation
Lemma (Rescaling)
Let \(V\) be a smooth vector field on a smooth manifold \(M\), let \(J \subseteq \mathbb{R}\)
be an open interval, and let \(\gamma : J \to M\) be an integral curve of \(V\). For any
\(a \in \mathbb{R}\), the curve \(\tilde \gamma : \tilde J \to M\) defined by
\(\tilde \gamma(t) = \gamma(at)\), where \(\tilde J = \{ t \in \mathbb{R} : a t \in J \}\),
is an integral curve of the rescaled vector field \(aV\).
Proof:
We evaluate \(\tilde\gamma'(t_0)\) on a smooth function \(f\) defined in a neighborhood
of \(\tilde\gamma(t_0)\). The chain rule, together with the fact that \(\gamma\) is an
integral curve of \(V\), gives
\[
\begin{align*}
\tilde\gamma'(t_0)\, f
&= \frac{d}{dt}\bigg|_{t = t_0} (f \circ \tilde\gamma)(t) \\\\
&= \frac{d}{dt}\bigg|_{t = t_0} (f \circ \gamma)(at) \\\\
&= a \, (f \circ \gamma)'(a t_0) \\\\
&= a \, \gamma'(a t_0)\, f \\\\
&= a \, V_{\gamma(a t_0)}\, f \\\\
&= (a V)_{\tilde\gamma(t_0)}\, f.
\end{align*}
\]
Since this holds for every test function \(f\), we conclude
\(\tilde\gamma'(t_0) = (a V)_{\tilde\gamma(t_0)}\), as required.
Lemma (Translation)
Let \(V\), \(M\), \(J\), and \(\gamma\) be as above. For any \(b \in \mathbb{R}\), the
curve \(\hat\gamma : \hat J \to M\) defined by \(\hat\gamma(t) = \gamma(t + b)\), where
\(\hat J = \{ t \in \mathbb{R} : t + b \in J \}\), is also an integral curve of \(V\).
Proof:
The chain rule gives
\[
\begin{align*}
\hat\gamma'(t_0)\, f
&= \frac{d}{dt}\bigg|_{t = t_0} (f \circ \gamma)(t + b) \\\\
&= (f \circ \gamma)'(t_0 + b) \\\\
&= \gamma'(t_0 + b)\, f \\\\
&= V_{\gamma(t_0 + b)}\, f \\\\
&= V_{\hat\gamma(t_0)}\, f
\end{align*}
\]
for every test function \(f\), so \(\hat\gamma'(t_0) = V_{\hat\gamma(t_0)}\).
The translation lemma has a transparent dynamical interpretation: shifting the time
parameter of an integral curve by a constant amount produces another integral curve of
the same vector field. This invariance — that the law of motion does not change
when we reset the clock — is exactly what is meant by saying that the system of ordinary
differential equations defining \(V\) is autonomous. The structural payoff of
this innocuous observation will appear in the next section, where the same algebraic
identity \(t_0 \mapsto t_0 + b\) reappears as the group law \(\theta_t \circ \theta_s = \theta_{t+s}\)
of the flow generated by \(V\).
Naturality Under Smooth Maps
The remaining structural result connects integral curves to the action of smooth maps.
The notion of
\(F\)-related vector fields
was introduced as the algebraic condition \(dF_p(X_p) = Y_{F(p)}\) compatible with a
smooth map \(F : M \to N\); the present proposition recasts that condition as a
statement about the trajectories of \(X\) and \(Y\).
Proposition (Naturality of Integral Curves)
Let \(F : M \to N\) be a smooth map between smooth manifolds, and let
\(X \in \mathfrak{X}(M)\), \(Y \in \mathfrak{X}(N)\). Then \(X\) and \(Y\) are
\(F\)-related if and only if \(F\) carries integral curves of \(X\) to integral curves
of \(Y\): for every integral curve \(\gamma : J \to M\) of \(X\), the composition
\(F \circ \gamma : J \to N\) is an integral curve of \(Y\).
Proof:
Suppose first that \(X\) and \(Y\) are \(F\)-related, and let \(\gamma : J \to M\) be
an integral curve of \(X\). Using the
chain rule for velocities of composite curves
and the \(F\)-relatedness condition, we compute
\[
\begin{align*}
(F \circ \gamma)'(t)
&= dF_{\gamma(t)} \bigl( \gamma'(t) \bigr) \\\\
&= dF_{\gamma(t)} \bigl( X_{\gamma(t)} \bigr) \\\\
&= Y_{F(\gamma(t))} ,
\end{align*}
\]
which is exactly the condition for \(F \circ \gamma\) to be an integral curve of
\(Y\).
Conversely, suppose \(F\) sends integral curves of \(X\) to integral curves of \(Y\).
Fix a point \(p \in M\). The local existence theorem provides an integral curve
\(\gamma : (-\varepsilon, \varepsilon) \to M\) of \(X\) with \(\gamma(0) = p\); by
the integral-curve condition, its starting velocity is
\(\gamma'(0) = X_{\gamma(0)} = X_p\). By hypothesis, \(F \circ \gamma\) is an
integral curve of \(Y\) starting at \(F(p)\), so its starting velocity is
\((F \circ \gamma)'(0) = Y_{F(p)}\). Equating these two velocities through the
chain rule yields
\[
\begin{align*}
Y_{F(p)}
&= (F \circ \gamma)'(0) \\\\
&= dF_p \bigl( \gamma'(0) \bigr) \\\\
&= dF_p ( X_p ).
\end{align*}
\]
Since \(p \in M\) was arbitrary, \(X\) and \(Y\) are \(F\)-related.
Two Characterizations of \(F\)-Relatedness
The condition that two vector fields be \(F\)-related now admits two complementary
descriptions on the same footing. The algebraic version, established earlier, asserts
that
\(F\)-relatedness is equivalent to the identity \(X(f \circ F) = (Yf) \circ F\)
on the action of \(X\) and \(Y\) on smooth functions. The geometric version, just
proved, asserts that \(F\)-relatedness is equivalent to \(F\) sending the trajectories
of \(X\) into trajectories of \(Y\).
The two characterizations are not alternatives: they are two faces of the same
relation, each suited to a different style of reasoning. The function-action
characterization is the natural tool for algebraic manipulations involving brackets
and derivations, while the integral-curve characterization is the natural tool for
arguments about flows. The next section will exploit the geometric face directly:
passing from the trajectory \(\gamma\) to the family of diffeomorphisms it generates
is what turns the trajectory-level statement of naturality into a statement about
commuting one-parameter groups.
From Integral Curves to Flows
The picture so far is curve-by-curve: each point of \(M\) is the starting point of some
integral curve, and the local existence theorem guarantees that one exists. To extract
the dynamical structure of a vector field, we collect all of these curves into a single
map. Fix a smooth vector field \(V\) on \(M\), and suppose, for the moment, that through
every point \(p \in M\) there passes a unique integral curve \(\theta^{(p)} : \mathbb{R} \to M\)
of \(V\), defined for all real \(t\) and starting at \(p\). (This idealization will fail
in general for vector fields whose integral curves are not all defined for all real
time — a phenomenon that occupies the next chapter; the next section of the present
development exhibits two vector fields for which the idealization happens to hold, and
here we use the assumption only to motivate the definition.) The data of these curves
can be repackaged as a single map
\[
\theta : \mathbb{R} \times M \to M , \qquad \theta(t, p) = \theta^{(p)}(t) .
\]
For each \(t \in \mathbb{R}\), the partial map \(\theta_t : M \to M\) defined by
\(\theta_t(p) = \theta(t, p)\) sends each point of \(M\) to the position reached by its
integral curve after time \(t\) — it slides the manifold along the trajectories of
\(V\).
The transformations \(\theta_t\) are not independent of one another. The translation
lemma of the previous section asserted that shifting the time parameter of an integral
curve produces another integral curve of the same vector field; combined with the
assumed uniqueness, this forces an algebraic identity on the maps \(\theta_t\). Indeed,
setting \(q = \theta_s(p)\), the curve \(t \mapsto \theta^{(p)}(t + s)\) is, by the
translation lemma, an integral curve of \(V\) with value \(q\) at \(t = 0\); uniqueness
then identifies it with \(\theta^{(q)}\), so \(\theta^{(p)}(t + s) = \theta^{(q)}(t)\),
which translates to
\[
\theta_t \circ \theta_s (p) = \theta_{t + s}(p) .
\]
Together with \(\theta_0(p) = \theta^{(p)}(0) = p\), this is exactly the statement that
the family \(\{\theta_t\}_{t \in \mathbb{R}}\) is a left action of the additive group
\(\mathbb{R}\) on \(M\). The structural identity that earlier expressed the autonomy of
the underlying differential equation now reappears as a group law on a family of
transformations.
Global Flows
The structure these observations isolate is the starting point of the formal theory.
Definition: Global Flow
A global flow on a smooth manifold \(M\) (also called a
one-parameter group action) is a continuous map
\(\theta : \mathbb{R} \times M \to M\) satisfying, for all \(s, t \in \mathbb{R}\)
and all \(p \in M\),
\[
\theta(t, \theta(s, p)) = \theta(t + s, p) , \qquad \theta(0, p) = p .
\]
Writing \(\theta_t : M \to M\) for the map \(\theta_t(p) = \theta(t, p)\), the
defining identities take the equivalent form of the group laws
\[
\theta_t \circ \theta_s = \theta_{t + s} , \qquad \theta_0 = \mathrm{Id}_M .
\]
The group laws have an immediate consequence: each \(\theta_t\) is a continuous bijection
with continuous inverse \(\theta_{-t}\), so every map \(\theta_t\) is a homeomorphism of
\(M\). When \(\theta\) is in addition smooth as a map \(\mathbb{R} \times M \to M\), each
\(\theta_t\) is a diffeomorphism, with smooth inverse \(\theta_{-t}\). A smooth global
flow is, in this sense, a one-parameter family of diffeomorphisms of \(M\), parametrized
by the real line and composing according to the additive structure of \(\mathbb{R}\).
For each \(p \in M\), the partial map \(\theta^{(p)} : \mathbb{R} \to M\) defined by
\(\theta^{(p)}(t) = \theta(t, p)\) traces out the trajectory of \(p\) under the flow.
The image of this map — the set of points reached from \(p\) at some time — is the
orbit of \(p\) under the \(\mathbb{R}\)-action. Smoothness of \(\theta\) makes each
\(\theta^{(p)}\) a smooth curve in \(M\).
The Infinitesimal Generator
Given a smooth global flow, we can recover a vector field from it by differentiating
each trajectory at its starting time. The construction reverses the passage from
integral curves to flows: instead of integrating a prescribed vector field, we
differentiate the trajectories of a given flow.
Definition: Infinitesimal Generator
Let \(\theta : \mathbb{R} \times M \to M\) be a smooth global flow. The
infinitesimal generator of \(\theta\) is the assignment
\(p \mapsto V_p\), where
\[
V_p = (\theta^{(p)})'(0) \in T_p M
\]
is the velocity at \(t = 0\) of the trajectory through \(p\).
The terminology anticipates the next result: \(V\) is the vector field whose
"infinitesimal" data — the time-zero velocities of all trajectories — encode the
"finite" data of the entire one-parameter family \(\{\theta_t\}\). At this stage the
assignment \(p \mapsto V_p\) is only a
rough vector field,
in the sense that no smoothness has yet been established and the construction is
purely pointwise.
The Flow Determines a Smooth Vector Field
The central proposition of this section says that the rough vector field obtained by
differentiating a smooth flow at time zero is in fact a smooth vector field on \(M\),
and that the trajectories of the flow recover the integral curves of this vector field.
Smooth global flows and the smooth vector fields they generate are tied together by a
perfect correspondence.
Proposition (Infinitesimal Generator of a Smooth Flow)
Let \(\theta : \mathbb{R} \times M \to M\) be a smooth global flow on a smooth
manifold \(M\), and let \(V\) be its infinitesimal generator. Then \(V\) is a smooth
vector field on \(M\), and for each \(p \in M\) the trajectory \(\theta^{(p)}\) is
an integral curve of \(V\) defined on all of \(\mathbb{R}\).
Proof:
We verify the two assertions in turn. For smoothness, we use the
smoothness criterion via the action on functions:
it suffices to check that, for every open \(U \subseteq M\) and every
\(f \in C^\infty(U)\), the function \(Vf : U \to \mathbb{R}\) defined by
\((Vf)(p) = V_p f\) is smooth. For \(p \in U\), the identity
\(\theta^{(p)}(t) = \theta(t, p)\) lets us pass from the one-variable trajectory
\(t \mapsto \theta^{(p)}(t)\) to the two-variable flow map \((t, p) \mapsto
\theta(t, p)\): the one-variable derivative at \(t = 0\) of the trajectory
through \(p\) is the partial \(t\)-derivative of \(\theta\) at \((0, p)\). So
\[
\begin{align*}
(Vf)(p)
&= V_p f \\\\
&= (\theta^{(p)})'(0)\, f \\\\
&= \frac{d}{dt}\bigg|_{t = 0} f \bigl( \theta^{(p)}(t) \bigr) \\\\
&= \frac{\partial}{\partial t}\bigg|_{(0, p)} f \bigl( \theta(t, p) \bigr) .
\end{align*}
\]
The composition \(f \circ \theta\) is smooth on an open neighborhood of
\(\{0\} \times U\) in \(\mathbb{R} \times M\), so its partial derivative with
respect to \(t\) at \(t = 0\) is a smooth function of \(p\). Hence \(Vf\) is
smooth on \(U\), and the cited criterion identifies \(V\) as a smooth vector field
on \(M\).
For the integral-curve assertion, fix \(p \in M\) and \(t_0 \in \mathbb{R}\), and
set \(q = \theta^{(p)}(t_0) = \theta_{t_0}(p)\). The goal is to show
\((\theta^{(p)})'(t_0) = V_q\). The group law of the flow gives, for all
\(t \in \mathbb{R}\),
\[
\theta^{(q)}(t)
= \theta_t(q)
= \theta_t \bigl( \theta_{t_0}(p) \bigr)
= \theta_{t + t_0}(p)
= \theta^{(p)}(t + t_0) .
\]
Differentiating at \(t = 0\) and applying the chain rule together with the
definition of \(V\) yields
\[
\begin{align*}
V_q
&= (\theta^{(q)})'(0) \\\\
&= \frac{d}{dt}\bigg|_{t = 0} \theta^{(p)}(t + t_0) \\\\
&= (\theta^{(p)})'(t_0) ,
\end{align*}
\]
which is the integral-curve condition for \(\theta^{(p)}\) at the point \(t_0\).
Since \(t_0\) and \(p\) were arbitrary, each trajectory \(\theta^{(p)}\) is an
integral curve of \(V\) defined on all of \(\mathbb{R}\).
The proposition completes the picture in one direction. Every smooth global flow is
determined by — and in fact is the family of integral curves of — a single smooth vector
field. The vector field is recovered from the flow by an infinitesimal operation
(differentiation at \(t = 0\)), and the flow is recovered from the vector field by
integration along trajectories. The opposite direction — whether every smooth vector
field is the infinitesimal generator of some smooth global flow — is more delicate, and
will be the central question of the following development. The next section examines a
case in which the answer is straightforwardly yes: two vector fields on the plane whose
flows can be written down by inspection, and whose explicit form will reveal a first
point of contact between the present theory and the theory of one-parameter subgroups
of Lie groups.
Translation and Rotation: \(SO(2)\) as a One-Parameter Group
The two vector fields on \(\mathbb{R}^2\) examined at the start of this development —
the constant horizontal field \(V = \partial / \partial x\) and the rotation generator
\(W = x \, \partial / \partial y - y \, \partial / \partial x\) — both have integral
curves defined for all real time. By the proposition of the previous section, each is
therefore the infinitesimal generator of a smooth global flow on \(\mathbb{R}^2\), and
the integral-curve formulas computed earlier let us write those flows down explicitly.
The Horizontal Translation Flow
For \(V = \partial / \partial x\), the integral curve starting at \((a, b)\) is
\(\gamma(t) = (a + t, b)\), so the global flow generated by \(V\) is the map
\(\tau : \mathbb{R} \times \mathbb{R}^2 \to \mathbb{R}^2\) given by
\[
\tau(t, (x, y)) = (x + t, y) , \qquad \text{or equivalently} \qquad
\tau_t(x, y) = (x + t, y) .
\]
Each map \(\tau_t : \mathbb{R}^2 \to \mathbb{R}^2\) is the translation of the plane by
the vector \((t, 0)\). The group law \(\tau_t \circ \tau_s = \tau_{t + s}\) and the
identity \(\tau_0 = \mathrm{Id}\) are immediate from the addition of real numbers; each
\(\tau_t\) is a diffeomorphism with inverse \(\tau_{-t}\), and the orbit of \((a, b)\)
under the flow is the horizontal line through that point.
The Rotation Flow
For \(W = x \, \partial / \partial y - y \, \partial / \partial x\), the integral curve
starting at \((a, b)\) was computed to be
\(\gamma(t) = (a \cos t - b \sin t, \, a \sin t + b \cos t)\). The global flow generated
by \(W\) is therefore the map
\(\theta : \mathbb{R} \times \mathbb{R}^2 \to \mathbb{R}^2\) given by
\[
\theta_t(x, y) = (x \cos t - y \sin t, \, x \sin t + y \cos t) .
\]
Writing points of \(\mathbb{R}^2\) as column vectors, this is the linear action
\[
\theta_t \begin{pmatrix} x \\\\ y \end{pmatrix}
=
\begin{pmatrix} \cos t & -\sin t \\\\ \sin t & \cos t \end{pmatrix}
\begin{pmatrix} x \\\\ y \end{pmatrix} ,
\]
so each \(\theta_t\) is the linear transformation of the plane given by the rotation
matrix
\[
R(t) = \begin{pmatrix} \cos t & -\sin t \\\\ \sin t & \cos t \end{pmatrix} .
\]
The group law \(\theta_t \circ \theta_s = \theta_{t + s}\) is the matrix identity
\(R(t) R(s) = R(t + s)\), which is the angle-addition formula for sine and cosine in
matrix form. The orbit of any nonzero point under the flow is the circle through that
point centered at the origin; the origin itself is fixed.
\(SO(2)\) as a One-Parameter Subgroup
The matrices \(R(t)\) appearing in the rotation flow are exactly the elements of
the
matrix Lie group
\(SO(2) \subseteq GL(2, \mathbb{R})\) of rotations of the plane. The assignment
\(t \mapsto R(t)\) is a continuous group homomorphism \((\mathbb{R}, +) \to SO(2)\) —
precisely a
one-parameter subgroup
of \(SO(2)\), with \(R(0) = I\) and \(R(t) R(s) = R(t + s)\). The dynamical
picture of an integral curve on \(\mathbb{R}^2\) and the algebraic picture of a
curve in a matrix Lie group coincide here on a single object.
The matrix-exponential point of view makes the coincidence concrete. Setting
\[
A = \begin{pmatrix} 0 & -1 \\\\ 1 & 0 \end{pmatrix} ,
\]
one computes directly that
\(\exp(tA) = R(t)\): every
one-parameter subgroup of a matrix Lie group is given by a matrix exponential,
and the rotation flow realizes this in a one-dimensional case with a simple
closed-form generator. This is a first instance of a broader correspondence
between flows of vector fields on a manifold and one-parameter subgroups of a
Lie group; the general form of that correspondence is the content of results we
shall reach once the flow theory is developed beyond the present existence
statement.
With the two prototypes \(\tau\) and \(\theta\) in hand, the assumption used to motivate
the definition of a global flow — that through every point passes a unique integral
curve defined for all real time — looks reasonable: both vector fields satisfy it. The
next stage of the theory begins by removing that assumption. Vector fields whose
integral curves are not defined for all \(t\) do exist, and a single point can fail to
admit a global trajectory for reasons that have nothing to do with the smoothness of
the field. Tracking what survives of the flow picture in the absence of global
existence — local flows, maximal integral curves, and the conditions under which a
vector field is genuinely complete — is the agenda of the following development.