Newtonian potential
Encyclopedia
In mathematics
, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory
. In its general nature, it is a singular integral operator, defined by convolution
with a function having a mathematical singularity
at the origin, the Newtonian kernel Γ which is the fundamental solution
of the Laplace equation. It is named for Isaac Newton
, who first discovered it and proved that it was a harmonic function
in the special case of three variables
, where it served as the fundamental gravitational potential in Newton's law of universal gravitation
. In modern potential theory, the Newtonian potential is instead thought of as an electrostatic potential.
The Newtonian potential of a compactly supported integrable function ƒ is defined as the convolution
where the Newtonian kernel Γ in dimension d is defined by
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
. In its general nature, it is a singular integral operator, defined by convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
with a function having a mathematical singularity
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
at the origin, the Newtonian kernel Γ which is the fundamental solution
Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function...
of the Laplace equation. It is named for Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, who first discovered it and proved that it was a harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
in the special case of three variables
Green's function for the three-variable Laplace equation
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...
, where it served as the fundamental gravitational potential in Newton's law of universal gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...
. In modern potential theory, the Newtonian potential is instead thought of as an electrostatic potential.
The Newtonian potential of a compactly supported integrable function ƒ is defined as the convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
where the Newtonian kernel Γ in dimension d is defined by
-
Here ωd is the volume of the unit d-ball, and sometimes sign conventions may vary; compare and .
The Newtonian potential w of ƒ is a solution of the Poisson equation
which is to say that the operation of taking the Newtonian potential of a function is a partial inverse to the Laplace operator. The solution is not unique, since addition of any harmonic function to w will not affect the equation. This fact can be used to prove existence and uniqueness of solutions to the Dirichlet problemDirichlet problemIn mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
for the Poisson equation in suitably regular domains, and for suitably well-behaved functions ƒ: one first applies a Newtonian potential to obtain a solution, and then adjusts by adding a harmonic function to get the correct boundary data.
The Newtonian potential is defined more broadly as the convolution
when μ is a compactly supported Radon measureRadon measureIn mathematics , a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.-Motivation:...
. It satisfies the Poisson equation
in the sense of distributionsDistribution (mathematics)In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
. Moreover, when the measure is positive, the Newtonian potential is subharmonicSubharmonic functionIn mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory....
on Rd.
If ƒ is a compactly supported continuous functionContinuous functionIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
(or, more generally, a finite measure) that is rotationally invariantRotationA rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
, then the convolutionConvolutionIn mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
of ƒ with Γ satisfies for x outside the support of ƒ
In dimension d = 3, this reduces to Newton's theorem that the potential energy of a small mass outside a much larger spherically symmetric mass distribution is the same as if all of the mass of the larger object were concentrated at its center.
When the measure μ is associated to a mass distribution on a sufficiently smooth hypersurface S (a Lyapunov surface of Hölder class C1,α) that divides Rd into two regions D+ and D−, then the Newtonian potential of μ is referred to as a simple layer potential. Simple layer potentials are continuous and solve the Laplace equation except on S. They appear naturally in the study of electrostaticsElectrostaticsElectrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....
in the context of the electrostatic potential associated to a charge distribution on a closed surface. If dμ = ƒ dH is the product of a continuous function on S with the (d − 1)-dimensional Hausdorff measureHausdorff measureIn mathematics a Hausdorff measure is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in Rn or, more generally, in any metric space. The zero dimensional Hausdorff measure is the number of points in the set or ∞ if the set is infinite...
, then at a point y of S, the normal derivative undergoes a jump discontinuity ƒ(y) when crossing the layer. Furthermore, the normal derivative is of w a well-defined continuous function on S. This makes simple layers particularly suited to the study of the Neumann problem for the Laplace equation.
See also
- Double layer potentialDouble layer potentialIn potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface S in three-dimensions...
- Green's functionGreen's functionIn mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions...
- Riesz potentialRiesz potentialIn mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space...
- Green's function for the three-variable Laplace equationGreen's function for the three-variable Laplace equationIn 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...
- Double layer potential