Laplace's equation

In mathematics and physics, Laplace's equation is a special math rule named after Pierre-Simon Laplace, who first studied it in 1786. It helps describe situations that are balanced or unchanging over time, like steady heat flow or electric fields. The equation is written as ∇²f = 0, where ∇² is called the Laplace operator. This operator uses tools like the divergence and gradient to show how a function changes in space.

When the equation includes a specific function on the right side, Δf = h, it becomes Poisson's equation, a more general version. Both Laplace's and Poisson's equations are types of elliptic partial differential equations, which are important in many areas of science.

Solutions to Laplace's equation are called harmonic functions and appear in many physical situations, such as electrostatics, gravitation, and fluid dynamics. In heat conduction, Laplace's equation describes the steady-state heat equation. Overall, Laplace's equation helps us understand how things balance out and stay stable in the natural world.

Forms in different coordinate systems

Laplace's equation can look different depending on the coordinate system used. In rectangular coordinates, it is written as ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z² = 0. In cylindrical coordinates, the equation becomes ∇²f = (1/r)∂/∂r(r∂f/∂r) + (1/r²)∂²f/∂ϕ² + ∂²f/∂z² = 0. For spherical coordinates, it is expressed as ∇²f = (1/r²)∂/∂r(r²∂f/∂r) + (1/(r²sinθ))∂/∂θ(sinθ∂f/∂θ) + (1/(r²sin²θ))∂²f/∂φ² = 0. These different forms help scientists and mathematicians solve problems in various shapes and spaces.

In more complex curvilinear coordinates, the equation uses the metric tensor and Christoffel symbols to adapt to any coordinate system. This makes it useful for studying shapes and patterns in many areas of science and math.

Boundary conditions

See also: Boundary value problem

The Dirichlet problem for Laplace's equation is about finding a solution inside a space where the values on the edge are known. Imagine fixing the temperature around the edge of a metal plate and waiting until the temperature stops changing. The final temperature inside the plate will match the solution to the Dirichlet problem.

The Neumann boundary conditions deal with the changes of the solution normal to the edge, rather than the solution itself. This can represent, for example, how heat flows through the edge of an object.

Solutions to Laplace's equation are called harmonic functions. If two solutions exist, their sum is also a solution. This makes solving complex problems easier by adding simpler solutions together.

Weak solutions and Dirichlet principle

Laplace's equation can also be understood in a less strict way, called the weak sense. This helps us study functions that might not be perfectly smooth but still behave well with Laplace's equation.

There is a special idea called Dirichlet's principle. It tells us that, among all functions with fixed values on the edges of a shape, the solutions to Laplace's equation are the ones that make a certain energy amount as small as possible. This helps us find solutions in a different, useful way.

Main article: Dirichlet's principle

Kelvin transform

The Kelvin transform is a special way to change a problem involving Laplace's equation. It uses a process called inversion in a sphere to switch between inside and outside regions. When we apply this transform to a function that follows Laplace's equation in one area, we get a new function that also follows Laplace's equation, but in a different, inverted area.

This tool is helpful for turning problems about the inside of a space into problems about the outside, studying points where solutions might act strangely, and understanding how these special functions behave when we look very far away.

In two dimensions

Laplace's equation describes special patterns in two dimensions. It looks like this: ∂²ψ/∂x² + ∂²ψ/∂y² = 0. This equation helps us understand how things flow or how electric fields behave in flat spaces.

The equation connects closely to analytic functions, where the real and imaginary parts both follow Laplace's rule. This link shows up in fluid flow and electrostatics, making the equation useful in many areas of science.

In three dimensions

Laplace's equation is a key idea in math and physics. It helps us understand how things balance out in space. The equation looks like this: ∇²f = 0. Here, ∇² is called the Laplace operator. It takes a function f and shows how it changes in different directions.

This equation is important because it appears in many areas, like studying electric and magnetic fields. When there are no sources (like charges) in a region, the potential there satisfies Laplace's equation. This means the potential doesn't suddenly change—it smooths out evenly, which makes sense for fields in empty space.

Gravitation

Laplace's equation helps us understand gravity in empty space. When there is no mass around, the gravitational potential satisfies this special math rule. This rule shows how gravity behaves when there are no objects creating it.

Brownian motion and harmonic measure

Laplace's equation can also be understood using something called Brownian motion. Imagine a tiny particle moving randomly in space—this is Brownian motion. When we study how such a particle behaves inside a bounded area, we can learn about harmonic functions, which are solutions to Laplace's equation.

The idea is that the value of a harmonic function inside an area can be found by looking at its values on the edge of that area. This is linked to where the particle first leaves the area, which is described by something called harmonic measure. It helps us understand how values on the boundary affect points inside.

In the Schwarzschild metric

S. Persides solved the Laplace equation in Schwarzschild spacetime on special surfaces called hypersurfaces of constant time. The solution uses special math functions called spherical harmonics and Legendre functions. These functions help describe patterns and shapes in space, especially around objects like black holes. The parameter l used in these functions is any whole number that is zero or positive.