History[edit]
The Helmholtz decomposition in three dimensions was first described in 1849[9] by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858,[10][11] which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines.[11] Their derivation required the vector fields to decay sufficiently fast at infinity. Later, this condition could be relaxed, and the Helmholtz decomposition could be extended to higher dimensions.[8][12][13] For riemannian manifolds, the Helmholtz-Hodge decomposition using differential geometry and tensor calculus was derived.[8][11][14][15]
The decomposition has become an important tool for many problems in theoretical physics,[11][14] but has also found applications in animation, computer vision as well as robotics.[15]
Generalization to higher dimensions[edit]
Matrix approach[edit]
The generalization to dimensions cannot be done with a vector potential, since the rotation operator and the cross product are defined (as vectors) only in three dimensions.
Let be a vector field on a bounded domain which decays faster than for and .
The scalar potential is defined similar to the three dimensional case as:
Differential forms[edit]
The Hodge decomposition is closely related to the Helmholtz decomposition,[25] generalizing from vector fields on R3 to differential forms on a Riemannian manifold M. Most formulations of the Hodge decomposition require M to be compact.[26] Since this is not true of R3, the Hodge decomposition theorem is not strictly a generalization of the Helmholtz theorem. However, the compactness restriction in the usual formulation of the Hodge decomposition can be replaced by suitable decay assumptions at infinity on the differential forms involved, giving a proper generalization of the Helmholtz theorem.
Extensions to fields not decaying at infinity[edit]
Most textbooks only deal with vector fields decaying faster than with at infinity.[16][13][27] However, Otto Blumenthal showed in 1905 that an adapted integration kernel can be used to integrate fields decaying faster than with , which is substantially less strict.
To achieve this, the kernel in the convolution integrals has to be replaced by .[28]
With even more complex integration kernels, solutions can be found even for divergent functions that need not grow faster than polynomial.[12][13][24][29]
For all analytic vector fields that need not go to zero even at infinity, methods based on partial integration and the Cauchy formula for repeated integration[30] can be used to compute closed-form solutions of the rotation and scalar potentials, as in the case of multivariate polynomial, sine, cosine, and exponential functions.[8]
Applications[edit]
Electrodynamics[edit]
The Helmholtz theorem is of particular interest in electrodynamics, since it can be used to write Maxwell's equations in the potential image and solve them more easily. The Helmholtz decomposition can be used to prove that, given electric current density and charge density, the electric field and the magnetic flux density can be determined. They are unique if the densities vanish at infinity and one assumes the same for the potentials.[16]
Fluid dynamics[edit]
In fluid dynamics, the Helmholtz projection plays an important role, especially for the solvability theory of the Navier-Stokes equations. If the Helmholtz projection is applied to the linearized incompressible Navier-Stokes equations, the Stokes equation is obtained. This depends only on the velocity of the particles in the flow, but no longer on the static pressure, allowing the equation to be reduced to one unknown. However, both equations, the Stokes and linearized equations, are equivalent. The operator is called the Stokes operator.[32]
Dynamical systems theory[edit]
In the theory of dynamical systems, Helmholtz decomposition can be used to determine "quasipotentials" as well as to compute Lyapunov functions in some cases.[33][34][35]
For some dynamical systems such as the Lorenz system (Edward N. Lorenz, 1963[36]), a simplified model for atmospheric convection, a closed-form expression of the Helmholtz decomposition can be obtained:
for a further decomposition of the divergence-free component .
