Laplace's equation

In mathematics and physics, Laplace's equation is a special math rule named after Pierre-Simon Laplace. It helps describe balanced or unchanging situations, 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. Both Laplace's and Poisson's equations are types of elliptic partial differential equations.

Solutions to Laplace's equation are called harmonic functions. They 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 work with 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 an answer inside a space where the numbers on the edge are known. Imagine fixing the temperature around the edge of a metal plate and waiting until the temperature stops changing. The temperature inside the plate will match the answer to the Dirichlet problem.

The Neumann boundary conditions deal with how the answer changes normal to the edge, instead of the answer itself. This can show, for example, how heat moves through the edge of an object.

Answers to Laplace's equation are called harmonic functions. If two answers exist, adding them together is also an answer. This makes solving hard problems easier by adding simpler answers 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 work 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 that uses Laplace's equation. It uses a process called inversion in a sphere to switch between the inside and outside of that sphere.

When we use this transform on a function that follows Laplace's equation in one place, we get a new function that also follows Laplace's equation. This new function is in a different, flipped place.

This tool helps us change problems about the inside of a space into problems about the outside. It also helps us study special points and understand how these 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 an important idea in math and physics. It helps us understand how things balance 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 useful in many areas, like studying electric and magnetic fields. When there are no sources (like charges) in a region, the potential there follows Laplace's equation. This means the potential changes smoothly, 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 follows a 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 be understood using something called Brownian motion. Brownian motion is when a tiny particle moves randomly in space.

When we watch how such a particle behaves inside a bounded area, we can learn about harmonic functions. Harmonic functions are solutions to Laplace's equation.

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 edge 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.