What is Potential theory: Definition and 13 Discussions

In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravity and the electrostatic force, could be modeled using functions called the gravitational potential and electrostatic potential, both of which satisfy Poisson's equation—or in the vacuum, Laplace's equation.
There is considerable overlap between potential theory and the theory of Poisson's equation to the extent that it is impossible to draw a distinction between these two fields. The difference is more one of emphasis than subject matter and rests on the following distinction: potential theory focuses on the properties of the functions as opposed to the properties of the equation. For example, a result about the singularities of harmonic functions would be said to belong to potential theory whilst a result on how the solution depends on the boundary data would be said to belong to the theory of the Laplace equation. This is not a hard and fast distinction, and in practice there is considerable overlap between the two fields, with methods and results from one being used in the other.
Modern potential theory is also intimately connected with probability and the theory of Markov chains. In the continuous case, this is closely related to analytic theory. In the finite state space case, this connection can be introduced by introducing an electrical network on the state space, with resistance between points inversely proportional to transition probabilities and densities proportional to potentials. Even in the finite case, the analogue I-K of the Laplacian in potential theory has its own maximum principle, uniqueness principle, balance principle, and others.

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  1. Ashphy

    Given the potential find the eigenfunction

    Hi, this was one of the oral exam questions my teacher asked so i tried to solve it. Consider y>0 the energy spectrum here is continuous and non degenerate while for y<0 the spectrum is discrete and non degenerate because E<0. for y>0 i thought of 2 cases case 1 there is no wave function for...
  2. M

    Potential functions for separation and isochronic gauges

    Most potentials in physics are expressed as a radius or another geometric norm/gauge. I am looking to understand the significance of the choice of potential functions for force/pressure separation in harmonic analysis before this creates a topology. To my understanding this is the decision of...
  3. Beelzedad

    I Does This Limit Exist?

    This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary. PRELIMINARY: Let potential due to a small volume ##\delta## at a point ##(1,2,3)## inside it be...
  4. M

    I Why ignoring the contribution from point r=0 in eq (1) and (2)?

    The potential of a dipole distribution at a point ##P## is: ##\psi=-k \int_{V'} \dfrac{\vec{\nabla'}.\vec{M'}}{r}dV' +k \oint_{S'}\dfrac{\vec{M'}.\hat{n}}{r}dS'## If ##P\in V'##, the integrand is discontinuous (infinite) at the point ##r=0##. So we need to use improper integrals by removing...
  5. N

    Potential theory for water waves

    hi :) http://ocw.mit.edu/courses/mechanical-engineering/2-017j-design-of-electromechanical-robotic-systems-fall-2009/course-text/MIT2_017JF09_ch06.pdf In page 37 they use Newton's 2nd law for a fluid element (while ignoring viscous forces) to derive the bernoulli equation for unsteady flow...
  6. N

    Geometries of potential theory (fluid mechanics)

    Hello! My book (fluid mechanics by White) doesn't explain the formulas it uses for finding geometric information about a potential field. For instance, sometimes if a stream-function is kept constant it, will form a figure like the one in this picture...
  7. F

    Induced charge on a conductive shell, potential theory

    There's no electric field inside a conductor, a classic observation of electrostatics. Any field that "should" exist is compensated for by charge redistribution on the surface of the conductor. This produces classic results like shielding since in a hollow conductive shell, the field is still...
  8. M

    How Is Voltage Across a Resistor Equivalent to That Across a Capacitor?

    Alright, so there is a very basic theory involving capacitors and electric potential that is throwing me off. I have a very basic problem here: http://img444.imageshack.us/img444/2251/73619554.png Assume the switch is closed and the capacitor is fully charged. From here I'm prompted to find...
  9. L

    Newton gravitational potential theory

    Homework Statement we all know to measure g.p,e(gravitational potential enery) we would likely to use g.p.e = mgh, however from the theory of Newton gravitational potential energy theory, g.p.e = G\frac{m1m2}{r}, my lecturer told me that mgh is not equal to G\frac{m1m2}{r} Homework...
  10. B

    Vanishing Electrostatic Force in a Plane with Multiple Point Charges

    Given n (finite) point charges in the xy-plane, is it possible to have a curve (in the plane) along which the electrostatic force vanishes (F=0)? I know that it's possible to have a curve through space along which the force vanishes when all of the charges are in the plane. For instance...
  11. T

    A couple of electric potential theory problems

    A couple of electric potential "theory" problems OK, so let's say Person A is on Planet A, where the ground potential is 1,000,000 V. Person A touches an object insulated from ground at a potential of 1,000,001 V. Person B is on Planet B, where the ground potential is 0 V. Person B touches...
  12. R

    Solving Potential Theory Problem: Step-by-Step Guide

    Hello, I have a problem with an exercice of potential theory, and don't know how to continue. The ecuations are this: \Phi {\left ( \nabla \frac{{\partial }}{{\partial t}} + \nabla (c· \nabla) \right ) = {\left ( - \frac{{\partial (c· \nabla)}}{{\partial t}} - \frac{{\partial^2...
  13. R

    Potential Theory: Study Caracteristics of Formula

    Dear friends, I need to know, in the theory of the potential, what must I do to study the caracteristics of this formulae: 0=\vec{v} [\nabla \vec{j}+ \frac{ \partial \vec{D}}{\partial t} ]+ [\nabla \vec{D} \frac {\partial \vec{v}}{\partial t} ]+ \nabla \vec{D} (\vec{v} \nabla ) \vec{v}...