The variational equation is a fundamental tool in engineering. It is the description of the sensitivity with respect to the initial conditions. Recall that the exponential map $\exp(f)$ of a vector field $f$ is the mapping sending a point $x_0$ to the point $x_1$, solution of the differential equation $x'(t) = f(x(t))$, with initial condition $x_0$, at time $t=1$. Using the exponential, the sensitivity to initial conditions is just the derivative of the exponential map $\exp(f)$.

A remarkable fact, often presented only in $\mathbf{R}^n$, is that this derivative is in fact the solution of another differential equation, sometimes called the variational equation. This is sometimes presented as follows. Suppose the differential equation at hand is

\begin{equation*} x'(t) = f(x(t)) \end{equation*}

Now, one is interested in the evolution of an infinitesimal perturbation $\delta x$. Note that $\delta x$ is now a square matrix. By differentiating, one obtains by the chain rule

\begin{equation*} \delta x'(t) = f'(x(t)) \delta x(t) \end{equation*}

where $f'$ is the Jacobian of the vector $f$, which is also a matrix. Note that $f'$ is only defined when a connexion is available, in other words, it seems that one uses the particular structure of $\RR^n$ as a flat space.

What I would like to show here is that the same holds in a general differential geometric context, i.e., that it is defined without any reference to a metric or a connexion.

So, we start with a manifold $\Man$, and a vector field $f$ on $\Man$. We consider the vector field as a section of $\Tan\Man$, that is, as a function

\begin{equation*} f \colon \Man \to \Tan\Man \end{equation*}

which is a section, i.e., it has to map a point $x \in \Man$ to a point in $\Tan\Man$ above $x$. So, if $\pi \colon \Tan\Man \to \Man$ is the projection from a point $(x,v) \in \Tan\Man$ to the base $x \in \Man$, then the condition is $\pi \circ f = \mathrm{Id}$.

In adapted coordinates this means that there is a function $\widetilde{f}$ such that

\begin{equation*} f (x) \equiv (x, \widetilde{f}(x)) \end{equation*}

The vector field $f$, now considered as a function between two manifold, can be differentiated. The result is a function

\begin{equation*} \Tan f \colon \Tan \Man \to \Tan \Tan \Man \end{equation*}

We compute that the point $(x, \delta x) \in \Tan\Man$ is sent to $\Big(x, f(x), \delta x, \frac{Df}{Dx}\delta x\Big)$. As a result, the function $\Tan f$ is not a section!

However, there is a particular structure of $\Tan\Tan\Man$ which allows us to get back to a section in the end. This special structure is that of a double tangent bundle, and is encoded by the canonical flip mapping $\tau \colon \Tan\Tan\Man \to \Tan\Tan\Man$. The map is defined in adapted coordinates on $\Tan\Tan\Man$ by

\begin{equation*} \tau (x,{\color{blue}\delta x},{\color{blue}∆x},∆\delta x) := (x,{\color{blue}∆x},{\color{blue}\delta x},∆\delta x) \end{equation*}

We thus see that $\tau \circ \Tan f$ is a section of $\Tan\Man \leftarrow \Tan\Tan\Man$.

A fundamental property of the exponential mapping is now that the solutions of the variational equation correspond to the propagation of the perturbations. It is written as follows

\begin{equation*} \Tan\big(\exp(f)\big) = \exp(\tau \circ \Tan f) \end{equation*}