Liouville's equation explained

For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).

For Liouville's equation in quantum mechanics, see Von Neumann equation.

For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation.

In differential geometry, Liouville's equation, named after Joseph Liouville,^{[1] [2]} is the nonlinear partial differential equation satisfied by the conformal factor of a metric on a surface of constant Gaussian curvature :

\Delta _{0log} f = - K f ^{2,}

where is the flat Laplace operator

\Delta _{0} = (\partial ^{2})/(+ \partial x ^{2}) (\partial ^{2})/(\partial y ^{2}) = 4 (\partial)/(\partial z) (\partial)/(\partial \bar z) .

Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables are the coordinates, while can be described as the conformal factor with respect to the flat metric. Occasionally it is the square that is referred to as the conformal factor, instead of itself.

Liouville's equation was also taken as an example by David Hilbert in the formulation of his nineteenth problem.^[3]

Other common forms of Liouville's equation

By using the change of variables, another commonly found form of Liouville's equation is obtained:

\Delta _{0} u = - K e ^{2u} .

Other two forms of the equation, commonly found in the literature,^[4] are obtained by using the slight variant of the previous change of variables and Wirtinger calculus:^[5]

\Delta _{0} u = - 2 K e ^{u} \Longleftrightarrow (\partial ^{2} u)/({\partial z) {\partial \bar z}} = - (K)/(2) e ^{u} .

Note that it is exactly in the first one of the preceding two forms that Liouville's equation was cited by David Hilbert in the formulation of his nineteenth problem.

Notes and References

1. Liouville. Joseph. 1838. Sur la Theorie de la Variation des constantes arbitraires. Journal de mathématiques pures et appliquées. 3. 342–349.
2. Book: Ehrendorfer, Martin. The Liouville Equation in Atmospheric Predictability. 48–49. The Liouville Equation: Background - Historical Background.
3. See : Hilbert does not cite explicitly Joseph Liouville.
4. See and .
5. See .