# What was Liouville's contribution to Liouville's theorem?

Liouville's theorem asserts that the volume of phase space is incompressible. This was first shown by Gibbs 1903 but his name is mostly forgotten. The Wikipedia article says

It is referred to as the Liouville equation because its derivation for non-canonical systems utilises an identity first derived by Liouville in 1838.

What identity does this refer to?

Liouville shows (1838, pp. 347-349) that if $x=\phi_t(a)$ is the "complete integral" of an ODE $$\frac{dx}{dt}=P(t,x) \tag{*}$$ on $\mathbf R^n$ (meaning that $a\in\mathbf R^n$ are $n$ otherwise arbitrary parameters for the general solution), then the jacobian $u=\det\bigl(\frac{\partial x_i}{\partial a_j}\bigr)$ satisfies (the identity you are after): $$\frac{du}{dt}=\operatorname{div}(P)u \tag{**}$$ where $\operatorname{div}(P)=\sum\frac{\partial P_i}{\partial x_i}$. In particular $u$ is constant if $\operatorname{div}(P)=0$, e.g. for $\frac{dp_i}{dt}=-\frac{\partial H}{\partial q_i}$, $\frac{dq_i}{dt}=\frac{\partial H}{\partial p_i}$ (Hamilton equations).
But as observed by Prange (1935, p. 674) or Lützen (1990, pp. 51, 661), Liouville did not apply his result to Hamilton equations, nor indeed speak of "phase space volume", as his arbitrary constants $a$ need not be the initial conditions $\phi_0(a)$. That application came with Jacobi (1844, §24), and the statistical or volumic interpretation with (before Gibbs) Maxwell, Boltzmann, and Kirchhoff: