Liouville equations
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:''For Liouville's equation in dynamical systems, see
Liouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectorie ...
.'' : ''For Liouville's equation in quantum mechanics, see
Von Neumann equation In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation.'' In differential geometry, Liouville's equation, named after
Joseph Liouville Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
, is the
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
partial differential equation satisfied by the conformal factor of a metric on a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
of constant Gaussian curvature : :\Delta_0\log f = -K f^2, where is the flat Laplace operator :\Delta_0 = \frac +\frac = 4 \frac \frac. Liouville's equation appears in the study of
isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric l ...
in differential geometry: the
independent variables Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
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.See : Hilbert does not cite explicitly Joseph Liouville.


Other common forms of Liouville's equation

By using the
change of variables Change or Changing may refer to: Alteration * Impermanence, a difference in a state of affairs at different points in time * Menopause, also referred to as "the change", the permanent cessation of the menstrual period * Metamorphosis, or change, ...
, another commonly found form of Liouville's equation is obtained: :\Delta_0 u = - K e^. Other two forms of the equation, commonly found in the literature, are obtained by using the slight variant of the previous change of variables and Wirtinger calculus: \Delta_0 u = - 2K e^\quad\Longleftrightarrow\quad \frac = - \frac e^. 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.


A formulation using the Laplace–Beltrami operator

In a more invariant fashion, the equation can be written in terms of the ''intrinsic'' Laplace–Beltrami operator : \Delta_ = \frac \Delta_0 as follows: :\Delta_\log\; f = -K.


Properties


Relation to Gauss–Codazzi equations

Liouville's equation is equivalent to the Gauss–Codazzi equations for minimal immersions into the 3-space, when the metric is written in
isothermal coordinates In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric l ...
z such that the Hopf differential is \mathrmz^2.


General solution of the equation

In a simply connected domain , the general solution of Liouville's equation can be found by using Wirtinger calculus.See . Its form is given by : u(z,\bar z) = \ln \left( 4 \frac \right) where is any
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
such that * for every . * has at most simple poles in .


Application

Liouville's equation can be used to prove the following classification results for surfaces: .See . A surface in the Euclidean 3-space with metric , and with constant scalar curvature is locally isometric to: # the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
if ; # the Euclidean plane if ; # the Lobachevskian plane if .


See also

*
Liouville field theory In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the ...
, a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation


Notes


Citations


Works cited

*. *. *, translated into English by Mary Frances Winston Newson as . {{refend Differential equations Differential geometry