Liouville equation
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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 : ''For Liouville's equation in differential geometry, see Liouville's equation.'' In mathematics, Liouville–Bratu–Gelfand equation or Liouville's equation is a non-linear Poisson equation, named after the mathematicians Joseph Liouville, G. B ...
.'' In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, 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 other ...
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
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 t ...
of constant
Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. The Gaussian radius of curvature is the reciprocal of . F ...
: :\Delta_0\log f = -K f^2, where is the flat
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
:\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 demand ...
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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
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 In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
: \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 In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental formulas which link together the induced ...
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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
, 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 pole (complex analysis), pole ...
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 Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
if ; # the
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
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 Mary Frances Winston Newson (August 7, 1869 December 5, 1959) was an American mathematician. She became the first female American to receive a PhD in mathematics from a European university, namely the University of Göttingen in Germany.Grinstein ...
as . {{refend Differential equations Differential geometry