Laplace's Equation
Laplace's Equation is a second-order partial differential equation named after the French mathematician Pierre-Simon Laplace. It plays a central role in various fields of physics, particularly in the study of gravitational fields, electric potentials, and fluid dynamics.
Equation Formulation
The equation in three-dimensional Cartesian coordinates is expressed as:
∇²φ = 0
where:
- ∇² is the Laplacian operator.
- φ (phi) is a scalar function of position, typically representing potential.
Historical Context
Pierre-Simon Laplace introduced this equation in the late 18th century while working on problems in celestial mechanics and potential theory. His work on the Celestial Mechanics led him to consider the distribution of gravitational forces in a system, where he encountered this equation as a fundamental descriptor of potential fields.
Applications
- Electrostatics: In electrostatics, Laplace's Equation describes the electric potential in a region where there are no charges (or when considering the field outside the charges).
- Gravitation: It is used to determine gravitational potentials outside of masses.
- Fluid Dynamics: In irrotational and inviscid flow, it describes the velocity potential of the fluid.
- Heat Conduction: For steady-state heat transfer in a medium with no heat sources or sinks, the temperature distribution can be described by this equation.
Mathematical Properties and Solutions
The solutions to Laplace's Equation are called harmonic functions. These functions have several important properties:
- They are infinitely differentiable.
- They obey the Maximum Principle, stating that the maximum (or minimum) value of a harmonic function in a domain must occur on the boundary of that domain.
- They satisfy the Mean-Value Property, where the value of the function at any point equals the average value over any sphere centered at that point.
Solutions can be found through various methods:
External Links
Related Concepts