Newtonian potential
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 f is defined as the convolution
Here ωd is the volume of the unit d-ball (sometimes sign conventions may vary; compare (Evans 1998) and (Gilbarg & Trudinger 1983)). For example, for we have
The Newtonian potential w of f is a solution of the Poisson equation
The Newtonian potential is defined more broadly as the convolution
If f is a compactly supported continuous function (or, more generally, a finite measure) that is rotationally invariant, then the convolution of f with Γ satisfies for x outside the support of f
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 electrostatics in the context of the electrostatic potential associated to a charge distribution on a closed surface. If dμ = f dH is the product of a continuous function on S with the (d − 1)-dimensional Hausdorff measure, then at a point y of S, the normal derivative undergoes a jump discontinuity f(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 potential
- Green's function
- Riesz potential
- Green's function for the three-variable Laplace equation
References
- Evans, L.C. (1998), Partial Differential Equations, Providence: American Mathematical Society, ISBN 0-8218-0772-2.
- Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7.
- Solomentsev, E.D. (2001) [1994], "Newton potential", Encyclopedia of Mathematics, EMS Press
- Solomentsev, E.D. (2001) [1994], "Simple-layer potential", Encyclopedia of Mathematics, EMS Press
- Solomentsev, E.D. (2001) [1994], "Surface potential", Encyclopedia of Mathematics, EMS Press