Laplace's equation

Last updated

Pierre-Simon Laplace Laplace, Pierre-Simon, marquis de.jpg
Pierre-Simon Laplace

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

Contents

where is the Laplace operator, [note 1] 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 solutions of Laplace's equation are the harmonic functions, [1] 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. [2] 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, [3]

In cylindrical coordinates , [3]

In spherical coordinates , using the convention, [3]

More generally, in curvilinear coordinates ,

or

Boundary conditions

Laplace's equation on an annulus (inner radius r = 2 and outer radius R = 4) with Dirichlet boundary conditions u(r=2) = 0 and u(R=4) = 4 sin(5 th) Laplace's equation on an annulus.svg
Laplace's equation on an annulus (inner radius r = 2 and outer radius R = 4) with Dirichlet boundary conditions u(r=2) = 0 and u(R=4) = 4 sin(5 θ)

The Dirichlet problem for Laplace's equation consists of finding a solution φ on some domain D such that φ on the boundary of D 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 conditions for Laplace's equation specify not the function φ itself on the boundary of D, but its normal derivative. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of D alone.

Solutions of Laplace's equation are called harmonic functions; 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 both satisfy the Laplace equation. That is, if z = x + iy, and if

then the necessary condition that f(z) be analytic is that u and v be differentiable and that the Cauchy–Riemann equations be satisfied:

where ux is the first partial derivative of u with respect to x. It follows that

Therefore u satisfies the Laplace equation. A similar calculation shows that v also satisfies the Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function, f(z) (at least locally). If a trial form is

then the Cauchy–Riemann equations will be satisfied if we set

This relation does not determine ψ, but only its increments:

The Laplace equation for φ implies that the integrability condition for ψ is satisfied:

and thus ψ 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 r and θ are polar coordinates and

then a corresponding analytic function is

However, the angle θ 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, which generally have less regularity.

There is an intimate connection between power series and Fourier series. If we expand a function f in a power series inside a circle of radius R, this means that

with suitably defined coefficients whose real and imaginary parts are given by

Therefore

which is a Fourier series for f. These trigonometric functions can themselves be expanded, using multiple angle formulae.

Fluid flow

Let the quantities u and v 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 ψ by

then the continuity condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines. The first derivatives of ψ are given by

and the irrotationality condition implies that ψ satisfies the Laplace equation. The harmonic function φ that is conjugate to ψ 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, an electric field (u, v) in two space dimensions that is independent of time satisfies

and

where ρ is the charge density. The first Maxwell equation is the integrability condition for the differential

so the electric potential φ may be constructed to satisfy

The second of Maxwell's equations then implies that

which is the Poisson equation. 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 of Laplace's equation satisfies

where the Dirac delta function δ denotes a unit source concentrated at the point (x′, y′, z′). No function has this property: in fact it is a distribution 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). 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. The definition of the fundamental solution thus implies that, if the Laplacian of u 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 r from the source point. If we choose the volume to be a ball of radius a around the source point, then Gauss' divergence theorem implies that

It follows that

on a sphere of radius r that is centered on the source point, and hence

Note that, with the opposite sign convention (used in physics), this is the potential generated by a point particle, for an inverse-square law force, arising in the solution of Poisson equation. A similar argument shows that in two dimensions

where log(r) denotes the natural logarithm. Note that, with the opposite sign convention, this is the potential generated by a pointlike sink (see point particle), which is the solution of the Euler equations in two-dimensional incompressible flow.

Green's function

A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary S of a volume V. For instance,

may satisfy

Now if u is any solution of the Poisson equation in V:

and u assumes the boundary values g on S, then we may apply Green's identity, (a consequence of the divergence theorem) which states that

The notations un and Gn denote normal derivatives on S. In view of the conditions satisfied by u and G, this result simplifies to

Thus the Green's function describes the influence at (x′, y′, z′) of the data f and g. For the case of the interior of a sphere of radius a, the Green's function may be obtained by means of a reflection ( Sommerfeld 1949 ): the source point P at distance ρ from the center of the sphere is reflected along its radial line to a point P' that is at a distance

Note that if P is inside the sphere, then P' will be outside the sphere. The Green's function is then given by

where R denotes the distance to the source point P and R′ denotes the distance to the reflected point P′. A consequence of this expression for the Green's function is the Poisson integral formula . Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ 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 g inside the sphere is given by

( Zachmanoglou 1986 , p. 228)

where

is the cosine of the angle between (θ, φ) and (θ′, φ′). A simple consequence of this formula is that if u is a harmonic function, then the value of u 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

Real (Laplace) spherical harmonics Yl for l = 0, ..., 4 (top to bottom) and m = 0, ..., l (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics
Y
l
-
m
{\displaystyle Y_{\ell }^{-m}}
would be shown rotated about the z axis by
90
[?]
/
m
{\displaystyle 90^{\circ }/m}
with respect to the positive order ones.) Rotating spherical harmonics.gif
Real (Laplace) spherical harmonics Y for = 0, …, 4 (top to bottom) and m = 0, …, (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics would be shown rotated about the z axis by with respect to the positive order ones.)

Laplace's equation in spherical coordinates is: [4]

Consider the problem of finding solutions of the form f(r, θ, φ) = R(r) Y(θ, φ). By separation of variables, two differential equations result by imposing Laplace's equation:

The second equation can be simplified under the assumption that Y has the form Y(θ, φ) = Θ(θ) Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations

for some number m. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e± imφ. The solution function Y(θ, φ) is regular at the poles of the sphere, where θ = 0, π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ( + 1) for some non-negative integer with ≥ |m|; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pm(cos θ) . Finally, the equation for R has solutions of the form R(r) = A r + B r − 1; requiring the solution to be regular throughout R3 forces B = 0. [5]

Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m. These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

which fulfill

Here Ym is called a spherical harmonic function of degree and order m, Pm is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively. In particular, the colatitude θ, or polar angle, ranges from 0 at the North Pole, to π/2 at the Equator, to π at the South Pole, and the longitude φ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer , every solution Y(θ, φ) of the eigenvalue problem

is a linear combination of Ym. In fact, for any such solution, r Y(θ, φ) is the expression in spherical coordinates of a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are 2 + 1 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 r,

where the fm are constants and the factors r Ym are known as solid harmonics. 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 (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 for electricity (Maxwell's first equation) in differential form states [6]

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. [6]

Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity, [6]

In the particular case of a source-free region, and Poisson's equation reduces to Laplace's equation for the electric potential. [6]

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. [7]

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 [8] solved the Laplace equation in Schwarzschild spacetime on hypersurfaces of constant t. Using the canonical variables r, θ, φ the solution is

where Yl(θ, φ) is a spherical harmonic function, and

Here Pl and Ql are Legendre functions of the first and second kind, respectively, while rs is the Schwarzschild radius. The parameter l is an arbitrary non-negative integer.

See also

Notes

  1. 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.

Related Research Articles

Divergence Vector operator producing the scalar quantity of a flow at a point

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

Spherical coordinate system 3-dimensional coordinate system

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 measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

Navier–Stokes equations Equations describing the motion of viscous fluid substances

In physics, the Navier–Stokes equations are a set of differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.

Potential flow velocity field as the gradient of a scalar function

In fluid dynamics, potential 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 approximation for several applications. The irrotationality of a potential flow is due to the curl of the gradient of a scalar always being equal to zero.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, 2 or Δ. The Laplacian ∇·∇f(p) of a function f at a point p is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere shrinks towards 0. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

Cylindrical coordinate system 3-dimensional coordinate system

A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

Spherical harmonics special function defined on the surface of a sphere; eigenfunction of the Laplace–Beltrami operator on the sphere

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.

In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

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 is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Helmholtz equation

In mathematics, the eigenvalue problem for the laplace operator is called Helmholtz equation. It corresponds to the linear partial differential equation:

In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces.

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

In physics, the Green's function for Laplace's equation in three variables is used to describe the response of a particular type of physical system to a point source. In particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

Mild-slope equation Combined effects of diffraction and refraction for water waves propagating over variable depth and with lateral boundaries

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

Potential flow around a circular cylinder

In mathematics, potential flow around a circular cylinder is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and can be modeled as a potential flow. Unlike a real fluid, this solution indicates a net zero drag on the body, a result known as d'Alembert's paradox.

In general relativity, the Weyl metrics are a class of static and axisymmetric solutions to Einstein's field equation. Three members in the renowned Kerr–Newman family solutions, namely the Schwarzschild, nonextremal Reissner–Nordström and extremal Reissner–Nordström metrics, can be identified as Weyl-type metrics.

Conformastatic spacetimes refer to a special class of static solutions to Einstein's equation in general relativity.

References

  1. Stewart, James. Calculus : Early Transcendentals . 7th ed., Brooks/Cole, Cengage Learning, 2012. Chapter 14: Partial Derivatives. p. 908. ISBN   978-0-538-49790-9.
  2. Zill, Dennis G, and Michael R Cullen. Differential Equations with Boundary-Value Problems. 8th edition / ed., Brooks/Cole, Cengage Learning, 2013. Chapter 12: Boundary-value Problems in Rectangular Coordinates. p. 462. ISBN   978-1-111-82706-9.
  3. 1 2 3 Griffiths, David J. Introduction to Electrodynamics . 4th ed., Pearson, 2013. Inner front cover. ISBN   978-1-108-42041-9.
  4. The approach to spherical harmonics taken here is found in ( Courant & Hilbert 1966 , §V.8, §VII.5).
  5. Physical applications often take the solution that vanishes at infinity, making A = 0. This does not affect the angular portion of the spherical harmonics.
  6. 1 2 3 4 Griffiths, David J. Introduction to Electrodynamics. Fourth ed., Pearson, 2013. Chapter 2: Electrostatics. p. 83-4. ISBN   978-1-108-42041-9.
  7. Griffiths, David J. Introduction to Electrodynamics. Fourth ed., Pearson, 2013. Chapter 3: Potentials. p. 119-121. ISBN   978-1-108-42041-9.
  8. Persides, S. (1973). "The Laplace and poisson equations in Schwarzschild's space-time". Journal of Mathematical Analysis and Applications. 43 (3): 571–578. doi: 10.1016/0022-247X(73)90277-1 .

Further reading