Jacobian of the flow

Let $X$ be a vector field on a manifold $M$ and $ϕ_{t} = ϕ_{X} (t, -)$ its flow. Each $ϕ_{t}$ is a diffeomorphism, so it has a Jacobian (a differential map on the tangent spaces). We first describe the Jacobian matrix $d (ϕ_{t})_{x}$ and then the formula for its determinant $J (t, x) = det (d (ϕ_{t})_{x})$ .

Jacobian matrix of the flow

A useful way to picture this is to perturb the initial condition, $x_{0} \mapsto x_{0} + ϵ v$ , and track the resulting infinitesimal displacement

Y_{v} (t) = {\frac{d}{d ϵ} ϕ_{t} (x_{0} + ϵ v) |}_{ϵ = 0} .

This displacement is the object classical mechanics calls a variation of the trajectory. It measures how the flow separates nearby initial data and satisfies the linear equation

{\dot{Y}}_{v} = d X_{ϕ_{t} (x_{0})} Y_{v},

we only have to differentiate the flow equation ${\dot{ϕ}}_{t} (x) = X (ϕ_{t} (x))$ with respect to the initial point and evaluate at $ϵ = 0$ .

Applied to a full basis of variations $v = e_{1}, \dots, e_{n}$ , we construct the matrix $Y (t)$ , which is nothing but the Jacobian of the flow, since $(d ϕ_{t})_{x_{0}} v$ is exactly this derivative of the curve $t \mapsto ϕ_{t} (x_{0} + ϵ v)$ at $ϵ = 0$ ; taking $v = e_{1}, \dots, e_{n}$ therefore produces the columns of the matrix $d (ϕ_{t})_{x_{0}}$ .

So the flow satisfies

\frac{d}{d t} ϕ_{t} (x) = X (ϕ_{t} (x)) .

and we have the so called variational equation

\frac{d}{d t} d (ϕ_{t})_{x} = d X_{ϕ_{t} (x)} d (ϕ_{t})_{x} .

In other words, $d (ϕ_{t})_{x}$ (its columns) solves the linear system

\dot{y} = d X_{ϕ_{t} (x)} y, y : R \to R^{n},

whose coefficients are evaluated along the orbit $ϕ_{t} (x)$ : the columns of $d (ϕ_{t})_{x}$ are the solutions with initial conditions $y (0) = e_{1}, \dots, e_{n}$ (the standard basis of $R^{n}$ ). Thus $d (ϕ_{t})_{x}$ is the fundamental matrix of the linearized flow, uniquely determined by $d (ϕ_{0})_{x} = Id$ .

Why "variational"?

The adjective is inherited from the calculus of variations. Euler, Lagrange and later Poincaré studied families of neighbouring trajectories and wrote $δ x (t)$ for the infinitesimal displacement between a solution $x (t)$ and an adjacent one. Substituting $x (t) + δ x (t)$ into $\dot{x} = X (x)$ and expanding to first order,

\dot{x} + \frac{d}{d t} (δ x) = X (x) + d X_{x} δ x + O (‖ δ x ‖^{2}),

and cancelling $\dot{x} = X (x)$ leaves the linear equation for the first variation

\frac{d}{d t} (δ x) = d X_{x (t)} δ x,

which is the matrix equation above for a single column. Poincaré coined the name équation aux variations for this linear equation, whence the modern term.

The same idea in the Riemannian setting produces the Jacobi field: the variation vector of a family of geodesics, $J (t) = \partial_{s} γ (t, s)$ , obeys the Jacobi equation, i.e. the variational equation of the geodesic flow.

Determinant of the Jacobian

Proposition
Let $X$ be a vector field and $ϕ_{t} (p)$ its flow. For every $t$ the map $ϕ_{t} (-)$ is a diffeomorphism, so we can consider its Jacobian determinant

J (t, x) := det (D_{x} ϕ_{t} (x)) .

It is satisfied that

\frac{d}{d t} |_{t = 0} J (t, x) = div X (x)

where $div$ is the divergence.

Moreover

\frac{d}{d t} J (t, x) = J (t, x) div X (ϕ_{t} (x)) .

Proof
(See this answer in Mathstackexchange)
The equation for the Jacobian matrix above, together with Liouville's formula applied to the linear system $\dot{y} = d X_{ϕ_{t} (x)} y$ (whose coefficient matrix has trace $tr d X_{ϕ_{t} (x)} = div X (ϕ_{t} (x))$ ), gives

J (t, x) = J (t_{0}, x) e^{\int_{t_{0}}^{t} tr d X_{ϕ_{s} (x)} d s}

and deriving with respect to $t$ :

\frac{d}{d t} J (t, x) = J (t, x) div X (ϕ_{t} (x)) .

Since $J (0, x) = 1$ , evaluating at $t = 0$ yields $\frac{d}{d t} |_{t = 0} J (t, x) = div X (x)$ , and integrating the equation we get the closed form

J (t, x) = \exp (\int_{0}^{t} div X (ϕ_{s} (x)) d s) .

$◼$

Geometric intuition: population growth with varying rate. The ODE $\frac{d}{d t} J = J \cdot div X (ϕ_{t} (x))$ has exactly the structure of a population growth equation $\frac{d P}{d t} = r (t) \cdot P$ , but with a non-constant per-capita rate $r (t) = div X (ϕ_{t} (x))$ that changes at each instant because the orbit $ϕ_{t} (x)$ moves through regions of space where $div X$ takes different values. Think of a population living in a landscape where fertility depends on location: as the population migrates along the orbit, the birth rate changes according to the terrain it currently occupies. The solution is $J (t, x) = \exp (\int_{0}^{t} div X (ϕ_{s} (x)) d s)$ , where the exponent is the Malthusian parameter accumulated along the trajectory — the time-integral of the local rate, not the net new volume.