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For a smooth autonomous differential equation and a hyperbolic equilibrium point $p$, the unstable manifold consists of points whose backward trajectories approach $p$:where the backward solution must exist for all negative time. The local unstable manifold additionally requires that the backward trajectory remain in a chosen small neighborhood. The global unstable manifold is obtained by following this local set forward as far as the flow map exists.

Its tangent space at $p$ is the real sum of the generalized eigenspaces of the Jacobian matrix with eigenvalues of positive real part. Generalized eigenspaces are needed when the matrix is not diagonalizable. Nonreal eigenvalues occur in conjugate pairs, together contributing a real invariant plane. The stable manifold uses negative real parts and convergence in forward time.

A nonlinear example shows why the tangent space alone is insufficient. For $\dot{x}=x$, $\dot{y}=-y+x^2$, the curve $y=x^2/3$ is invariant because its time derivative is $2x^2/3=-x^2/3+x^2$. On this curve $x(t)=x_0{e}^t$ and $y(t)=x_0^2{e}^{2t}/3$ approach zero backward, so it is the unstable manifold. It is tangent to the $x$-axis at the origin, but it is not that axis. In fact $y-x^2/3$ obeys $\dot{w}=-w$, so every point off the curve fails to approach zero backward.

The proof is a single time derivative. Apply the product rule to each dot product and use Newton's second law:Integrating from $0$ to $T$ gives the endpoint identity; the stated sublinear condition removes the endpoint term only in the infinite-time limit. Equivalently, for the scalar inertia $I=\frac{1}{2}{\sum }_i{m}_i|{\mathbf{r}}_i{|}^2$ used in the pairwise moment-of-inertia identity, $\dot{I}=G$ and $\ddot{I}=2K+{\sum }_i{\mathbf{r}}_i\cdot {\mathbf{F}}_i$. Specifying this factor of one half avoids confusion with another common inertia convention.

In On a Mechanical Theorem applicable to Heat by R. Clausius (1870, pp. 123-124), the author defines stationary motion through bounded positions and velocities, includes planetary orbits and elastic vibrations, and names the average negative half of the force-position sum the “virial”. That historical convention differs from the auxiliary $G(t)$ defined above. His proof on pp. 126-127 keeps an endpoint term, then explains why it vanishes over a complete period or becomes small in a long bounded irregular motion. The same paper separates internal and external forces and includes uniform boundary pressure on pp. 125-126: confinement was already part of the formulation, rather than a reason to omit external stresses.^[ref]