Expression frequently encountered in mathematical physics, generalization of Laplace's 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 the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.[1][2]
Statement of the equation
Poisson's equation is
where is the Laplace operator, and and are real or complex-valued functions on a manifold. Usually, is given, and is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as ∇2, and so Poisson's equation is frequently written as
In the case of a gravitational field g due to an attracting massive object of density ρ, Gauss's law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity:
Since the gravitational field is conservative (and
irrotational), it can be expressed in terms of a scalar potential
ϕ:
Substituting this into Gauss's law,
yields Poisson's equation for gravity:
If the mass density is zero, Poisson's equation reduces to Laplace's equation. The corresponding Green's function can be used to calculate the potential at distance r from a central point mass m (i.e., the fundamental solution). In three dimensions the potential is
One of the cornerstones of electrostatics is setting up and solving problems described by the Poisson equation. Solving the Poisson equation amounts to finding the electric potentialφ for a given charge distribution .
The mathematical details behind Poisson's equation in electrostatics are as follows (
Substituting this into Gauss's law and assuming that ε is spatially constant in the region of interest yields
In electrostatics, we assume that there is no magnetic field (the argument that follows also holds in the presence of a constant magnetic field). Then, we have that
where ∇× is the curl operator. This equation means that we can write the electric field as the gradient of a scalar function φ (called the electric potential), since the curl of any gradient is zero. Thus we can write
where the minus sign is introduced so that φ is identified as the electric potential energy per unit charge.
The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field,
directly produces Poisson's equation for electrostatics, which is
Using Green's function, the potential at distance r from a central point charge Q (i.e., the fundamental solution) is
which is Coulomb's law of electrostatics. (For historic reasons, and unlike gravity's model above, the factor appears here and not in Gauss's law.)
The above discussion assumes that the magnetic field is not varying in time. The same Poisson equation arises even if it does vary in time, as long as the
This solution can be checked explicitly by evaluating ∇2φ.
Note that for r much greater than σ, the erf function approaches unity, and the potential φ(r) approaches the
point-charge
potential,
as one would expect. Furthermore, the error function approaches 1 extremely quickly as its argument increases; in practice, for r > 3σ the relative error is smaller than one part in a thousand.
Surface reconstruction
Surface reconstruction is an
surface normal ni.[3] Poisson's equation can be utilized to solve this problem with a technique called Poisson surface reconstruction.[4]
The goal of this technique is to reconstruct an
least-squares
fit to minimize the difference between V and the gradient of f.
In order to effectively apply Poisson's equation to the problem of surface reconstruction, it is necessary to find a good discretization of the vector field V. The basic approach is to bound the data with a
finite-difference grid. For a function valued at the nodes of such a grid, its gradient can be represented as valued on staggered grids, i.e. on grids whose nodes lie in between the nodes of the original grid. It is convenient to define three staggered grids, each shifted in one and only one direction corresponding to the components of the normal data. On each staggered grid we perform trilinear interpolation on the set of points. The interpolation weights are then used to distribute the magnitude of the associated component of ni onto the nodes of the particular staggered grid cell containing pi. Kazhdan and coauthors give a more accurate method of discretization using an adaptive finite-difference grid, i.e. the cells of the grid are smaller (the grid is more finely divided) where there are more data points.[4] They suggest implementing this technique with an adaptive octree
^Poisson (1823). "Mémoire sur la théorie du magnétisme en mouvement" [Memoir on the theory of magnetism in motion]. Mémoires de l'Académie Royale des Sciences de l'Institut de France (in French). 6: 441–570. From p. 463: "Donc, d'après ce qui précède, nous aurons enfin:
selon que le point M sera situé en dehors, à la surface ou en dedans du volume que l'on considère." (Thus, according to what preceded, we will finally have:
depending on whether the point M is located outside, on the surface of, or inside the volume that one is considering.) V is defined (p. 462) as
where, in the case of electrostatics, the integral is performed over the volume of the charged body, the coordinates of points that are inside or on the volume of the charged body are denoted by , is a given function of and in electrostatics, would be a measure of charge density, and is defined as the length of a radius extending from the point M to a point that lies inside or on the charged body. The coordinates of the point M are denoted by and denotes the value of (the charge density) at M.