In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed.
[8] 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] Many physics textbooks restrict the Helmholtz decomposition to the three-dimensional space and limit its application to vector fields that decay sufficiently fast at infinity or to bump functions that are defined on a bounded domain.
According to the definition of the Helmholtz decomposition, the condition is equivalent to Taking the divergence of each member of this equation yields
This theorem is of great importance in electrostatics, since Maxwell's equations for the electric and magnetic fields in the static case are of exactly this type.
The Helmholtz decomposition can be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives).
where φ is in the Sobolev space H1(Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and A ∈ H(curl, Ω), the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.
For a slightly smoother vector field u ∈ H(curl, Ω), a similar decomposition holds:
In the three-dimensional case, the matrix elements just correspond to the components of the vector potential
The rotational field, on the other hand, is defined in the general case as the row divergence of the matrix:
In place of the definition of the vector Laplacian used above, we now make use of an identity for the Levi-Civita symbol
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.
[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.
is the only harmonic function with this property, which follows from Liouville's theorem, this guarantees the uniqueness of the gradient and rotation fields.
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.
[16] In fluid dynamics, the Helmholtz projection plays an important role, especially for the solvability theory of the Navier-Stokes equations.
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.
[32] 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:
The quadratic scalar potential provides motion in the direction of the coordinate origin, which is responsible for the stable fix point for some parameter range.
For other parameters, the rotation field ensures that a strange attractor is created, causing the model to exhibit a butterfly effect.
[8][37] In magnetic resonance elastography, a variant of MR imaging where mechanical waves are used to probe the viscoelasticity of organs, the Helmholtz decomposition is sometimes used to separate the measured displacement fields into its shear component (divergence-free) and its compression component (curl-free).
[38] In this way, the complex shear modulus can be calculated without contributions from compression waves.
This includes robotics, image reconstruction but also computer animation, where the decomposition is used for realistic visualization of fluids or vector fields.