Partial differential equations {excess function} {E-function} can represent energy function.
Energy or force equations can minimize quantities {least constraint principle} {principle of least constraint}. For example, sum of kinetic-energy-to-potential-energy changes over time {action} can be minimum: integral of (kinetic energy - potential energy) * dt.
Partial differential equations {Hamilton-Jacobi equation} can represent potential energy plus kinetic energy equals total energy. Sum of second partial derivatives of potential with respect to each coordinate and partial derivative of potential with respect to time equals zero: (D^2)V / Dx + (D^2)V / Dy + (D^2)V / Dz - DV / Dt = 0, where V is potential, (D^2) is second partial derivative, D is partial derivative, and x, y, z, and t are coordinates.
Operators {Laplace operator} {Laplace's operator}, on vector fields or potentials {del squared of f}, can be second derivatives, describe field-variation smoothness, be vectors, and be non-linear.
potential
Partial differential equations {potential equation} {Laplace's equation} can represent potentials. Potential V depends on distance r from mass or charge center: r = (x^2 + y^2 + z^2)^0.5.
Second partial derivative of potential V with respect to distance along x-axis plus second partial derivative of potential V with respect to distance along y-axis plus second partial derivative of potential V with respect to distance along z-axis equals zero: (D^2)V / Dx + (D^2)V / Dy + (D^2)V / Dz = 0, where (D^2) is second partial derivative, D is partial derivative, and V is constant times distance from center, because dx^2 / dx = 2 * x and d(2*x) / dx = 0.
solution
Spherical functions or Legendre polynomials can solve potential equation.
(1 - x^2) * y'' - 2 * x * y' + n * (n + 1) * y = 0, where n is parameter {Legendre differential equation}. Solutions are polynomials {Legendre polynomial}, potential equation spherical coordinates derived by variable separation, or spherical harmonics of second kind.
For boundaries with potential change zero, calculations can find potential change normal to region {Neumann problem} {second fundamental problem}.
If potential-equation right side equals -4 * pi * (energy density), rather than zero, equation describes gravitation and electrostatics {Poisson's equation} {Poisson equation}. Energy density is pressure.
3-Calculus-Differential Equation-Kinds-Partial
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Date Modified: 2022.0225