In
mathematics and
physics, Laplace's equation is a second-order
partial differential equation named after
Pierre-Simon Laplace, who first studied its properties. This is often written as
or
where
is the
Laplace operator,
[The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, . Its use to represent the Laplacian should not be confused with this use.] is the
divergence operator (also symbolized "div"),
is the
gradient operator (also symbolized "grad"), and
is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
If the right-hand side is specified as a given function,
, we have
This is called
Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of
elliptic partial differential equations. Laplace's equation is also a special case of the
Helmholtz equation.
The general theory of solutions to Laplace's equation is known as
potential theory. The twice continuously differentiable solutions of Laplace's equation are the
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
s, which are important in multiple branches of physics, notably electrostatics, gravitation, and
fluid dynamics. In the study of
heat conduction, the Laplace equation is the
steady-state heat equation. In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.
Forms in different coordinate systems
In
rectangular coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
,
[Griffiths, David J. ]
Introduction to Electrodynamics
'. 4th ed., Pearson, 2013. Inner front cover. .
In
cylindrical coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference d ...
,
In
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
, using the
convention,
More generally, in arbitrary
curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
,
or
where is the Euclidean
metric tensor relative to the new coordinates and denotes its
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
.
Boundary conditions
![Laplace's equation on an annulus](https://upload.wikimedia.org/wikipedia/commons/c/cd/Laplace%27s_equation_on_an_annulus.svg)
The
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
for Laplace's equation consists of finding a solution on some domain such that on the boundary of is equal to some given function. Since the Laplace operator appears in the
heat equation, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain doesn't change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.
The
Neumann boundary condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appli ...
s for Laplace's equation specify not the function itself on the boundary of but its
normal derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of alone. For the example of the heat equation it amounts to prescribing the heat flux through the boundary. In particular, at an adiabatic boundary, the normal derivative of is zero.
Solutions of Laplace's equation are called
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
s; they are all
analytic within the domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the
principle of superposition, is very useful. For example, solutions to complex problems can be constructed by summing simple solutions.
In two dimensions
Laplace's equation in two independent variables in rectangular coordinates has the form
Analytic functions
The real and imaginary parts of a complex
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
both satisfy the Laplace equation. That is, if , and if
then the necessary condition that be analytic is that and be differentiable and that the
Cauchy–Riemann equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differenti ...
be satisfied:
where is the first partial derivative of with respect to .
It follows that
Therefore satisfies the Laplace equation. A similar calculation shows that also satisfies the Laplace equation.
Conversely, given a harmonic function, it is the real part of an analytic function, (at least locally). If a trial form is
then the Cauchy–Riemann equations will be satisfied if we set
This relation does not determine , but only its increments:
The Laplace equation for implies that the integrability condition for is satisfied:
and thus may be defined by a line integral. The integrability condition and
Stokes' theorem implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of the Laplace equation are called conjugate harmonic functions. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if and are polar coordinates and
then a corresponding analytic function is
However, the angle is single-valued only in a region that does not enclose the origin.
The close connection between the Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a power series, at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seism ...
, which generally have less regularity.
There is an intimate connection between power series and
Fourier series. If we expand a function in a power series inside a circle of radius , this means that
with suitably defined coefficients whose real and imaginary parts are given by
Therefore
which is a Fourier series for . These trigonometric functions can themselves be expanded, using
multiple angle formulae.
Fluid flow
Let the quantities and be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The continuity condition for an incompressible flow is that
and the condition that the flow be irrotational is that
If we define the differential of a function by
then the continuity condition is the integrability condition for this differential: the resulting function is called the
stream function
The stream function is defined for incompressible ( divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. T ...
because it is constant along
flow lines
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psych ...
. The first derivatives of are given by
and the irrotationality condition implies that satisfies the Laplace equation. The harmonic function that is conjugate to is called the
velocity potential. The Cauchy–Riemann equations imply that
Thus every analytic function corresponds to a steady incompressible, irrotational, inviscid fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function.
Electrostatics
According to
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
, an electric field in two space dimensions that is independent of time satisfies
and
where is the charge density. The first Maxwell equation is the integrability condition for the differential
so the electric potential may be constructed to satisfy
The second of Maxwell's equations then implies that
which is the
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.
In three dimensions
Fundamental solution
A
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
of Laplace's equation satisfies
where the
Dirac delta function denotes a unit source concentrated at the point . No function has this property: in fact it is a
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
rather than a function; but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see
weak solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precis ...
). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a
positive operator In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \l ...
. The definition of the fundamental solution thus implies that, if the Laplacian of is integrated over any volume that encloses the source point, then
The Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance from the source point. If we choose the volume to be a ball of radius around the source point, then
Gauss' divergence theorem
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface (mathematics), surface to the ''divergence'' o ...
implies that
It follows that
on a sphere of radius that is centered on the source point, and hence
Note that, with the opposite sign convention (used in
physics), this is the
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
generated by a
point particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
, for an
inverse-square law force, arising in the solution of
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
. A similar argument shows that in two dimensions
where denotes the
natural logarithm. Note that, with the opposite sign convention, this is the
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
generated by a pointlike
sink
A sink is a bowl-shaped plumbing fixture for washing hands, dishwashing, and other purposes. Sinks have a tap (faucet) that supply hot and cold water and may include a spray feature to be used for faster rinsing. They also include a drain t ...
(see
point particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
), which is the solution of the
Euler equations
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
in two-dimensional
incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. A ...
.
Green's function
A
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
is a fundamental solution that also satisfies a suitable condition on the boundary of a volume . For instance,
may satisfy
Now if is any solution of the Poisson equation in :
and assumes the boundary values on , then we may apply
Green's identity, (a consequence of the divergence theorem) which states that
The notations ''u
n'' and ''G
n'' denote normal derivatives on . In view of the conditions satisfied by and , this result simplifies to
Thus the Green's function describes the influence at of the data and . For the case of the interior of a sphere of radius , the Green's function may be obtained by means of a reflection : the source point at distance from the center of the sphere is reflected along its radial line to a point ''P that is at a distance
Note that if is inside the sphere, then ''P′'' will be outside the sphere. The Green's function is then given by
where denotes the distance to the source point and denotes the distance to the reflected point ''P''′. A consequence of this expression for the Green's function is the
Poisson integral formula
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the deriva ...
. Let , , and be
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
for the source point . Here denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice. Then the solution of the Laplace equation with Dirichlet boundary values inside the sphere is given by
where
is the cosine of the angle between and . A simple consequence of this formula is that if is a harmonic function, then the value of at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point.
Laplace's spherical harmonics
![Rotating_spherical_harmonics](https://upload.wikimedia.org/wikipedia/commons/1/12/Rotating_spherical_harmonics.gif)
Laplace's equation in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
is:
Consider the problem of finding solutions of the form . By
separation of variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
, two differential equations result by imposing Laplace's equation:
The second equation can be simplified under the assumption that has the form . Applying separation of variables again to the second equation gives way to the pair of differential equations
for some number . A priori, is a complex constant, but because must be a
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
whose period evenly divides , is necessarily an integer and is a linear combination of the complex exponentials . The solution function is regular at the poles of the sphere, where . Imposing this regularity in the solution of the second equation at the boundary points of the domain is a
Sturm–Liouville problem that forces the parameter to be of the form for some non-negative integer with ; this is also explained
below in terms of the
orbital angular momentum. Furthermore, a change of variables transforms this equation into the
Legendre equation, whose solution is a multiple of the
associated Legendre polynomial
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
\left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0,
or equivalently ...
. Finally, the equation for has solutions of the form ; requiring the solution to be regular throughout forces .
[Physical applications often take the solution that vanishes at infinity, making . This does not affect the angular portion of the spherical harmonics.]
Here the solution was assumed to have the special form . For a given value of , there are independent solutions of this form, one for each integer with . These angular solutions are a product of
trigonometric function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s, here represented as a
complex exponential
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
, and associated Legendre polynomials:
which fulfill
Here is called a spherical harmonic function of degree and order , is an
associated Legendre polynomial
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
\left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0,
or equivalently ...
, is a normalization constant, and and represent colatitude and longitude, respectively. In particular, the
colatitude , or polar angle, ranges from at the North Pole, to at the Equator, to at the South Pole, and the
longitude
Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...
, or
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematical ...
, may assume all values with . For a fixed integer , every solution of the eigenvalue problem
is a
linear combination of . In fact, for any such solution, is the expression in spherical coordinates of a
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
that is harmonic (see
below), and so counting dimensions shows that there are linearly independent such polynomials.
The general solution to Laplace's equation in a ball centered at the origin is a
linear combination of the spherical harmonic functions multiplied by the appropriate scale factor ,
where the are constants and the factors are known as
solid harmonics In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions \mathbb^3 \to \mathbb. There are two kinds: the ''regular solid harmonics'' R^m_\ell(\mathbf), wh ...
. Such an expansion is valid in the
ball
For
, the solid harmonics with negative powers of
are chosen instead. In that case, one needs to expand the solution of known regions in
Laurent series (about
), instead of
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
(about
), to match the terms and find
.
Electrostatics
Let
be the electric field,
be the electric charge density, and
be the permittivity of free space. Then
Gauss's law
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it sta ...
for electricity (Maxwell's first equation) in differential form states
[Griffiths, David J. ''Introduction to Electrodynamics''. Fourth ed., Pearson, 2013. Chapter 2: Electrostatics. p. 83-4. .]
Now, the electric field can be expressed as the negative gradient of the electric potential
,
if the field is irrotational,
. The irrotationality of
is also known as the electrostatic condition.
Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity,
In the particular case of a source-free region,
and Poisson's equation reduces to Laplace's equation for the electric potential.
If the electrostatic potential
is specified on the boundary of a region
, then it is uniquely determined. If
is surrounded by a conducting material with a specified charge density
, and if the total charge
is known, then
is also unique.
[Griffiths, David J. ''Introduction to Electrodynamics''. Fourth ed., Pearson, 2013. Chapter 3: Potentials. p. 119-121. .]
A potential that doesn't satisfy Laplace's equation together with the boundary condition is an invalid electrostatic potential.
Gravitation
Let
be the gravitational field,
the mass density, and
the gravitational constant. Then Gauss's law for gravitation in differential form is
The gravitational field is conservative and can therefore be expressed as the negative gradient of the gravitational potential:
Using the differential form of Gauss's law of gravitation, we have
which is Poisson's equation for gravitational fields.
In empty space,
and we have
which is Laplace's equation for gravitational fields.
In the Schwarzschild metric
S. Persides
solved the Laplace equation in
Schwarzschild spacetime on hypersurfaces of constant . Using the canonical variables , , the solution is
where is a
spherical harmonic function, and
Here and are
Legendre functions
In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated ...
of the first and second kind, respectively, while is the
Schwarzschild radius
The Schwarzschild radius or the gravitational radius is a physical parameter in the Schwarzschild solution to Einstein's field equations that corresponds to the radius defining the event horizon of a Schwarzschild black hole. It is a characteris ...
. The parameter is an arbitrary non-negative integer.
See also
*
6-sphere coordinates
In mathematics, 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the 3D Cartesian coordinates across the unit 2-sphere x^2+y^2+z^2=1. They are so named because the loci where one coordinate is const ...
, a coordinate system under which Laplace's equation becomes
''R''-separable
*
Helmholtz equation, a general case of Laplace's equation.
*
Spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form ...
*
Quadrature domains
*
Potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
*
Potential flow
In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid app ...
*
Bateman transform
In the mathematical study of partial differential equations, the Bateman transform is a method for solving the Laplace equation in four dimensions and wave equation in three by using a line integral of a holomorphic function in three complex v ...
*
Earnshaw's theorem uses the Laplace equation to show that stable static ferromagnetic suspension is impossible
*
Vector Laplacian
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 ...
*
Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
Notes
References
Further reading
*
*
*
*
* *
External links
*
Laplace Equation (particular solutions and boundary value problems)at EqWorld: The World of Mathematical Equations.
Example initial-boundary value problemsusing Laplace's equation from exampleproblems.com.
* {{MathWorld , urlname= LaplacesEquation , title= Laplace's Equation
Elliptic partial differential equations
Harmonic functions
Equations
Fourier analysis
Equation