# 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?

## 1 Answer

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:

Liouville's paper had fallen into oblivion when Maxwell and Boltzmann began their work, and therefore they rediscovered Liouville's theorem independently. In special cases, it was derived by Maxwell in his second discussion of the statistical distribution of the velocities of monatomic gas molecules in thermal equilibrium [1868] and by Boltzmann for other physical systems [1868].

(...) later the same year [1871b], Boltzmann derived the general theorem (**), and, moreover, showed that it was equivalent to Jacobi's principle of the last multiplier.

(...) as soon as the connection between Liouville's theorem and Jacobi's last multiplier had been established, the realization of Liouville's priority was not far away. Indeed, in Kirchhoff's Vorlesungen über die Theorie der Wärme, held five times during the years 1876-1884 in Berlin, and published posthumously by Max Planck in 1894, it is remarked that Jacobi in his Vorlesungen über Dynamik attributed the theorem to Liouville (cf. §23). Four years later, in his second part of the Vorlesungen über Gastheorie, Boltzmann [1896-1898] baptized the theorem "Liouville's theorem."