Helmholtz decomposition explained

In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field. It is named after Hermann von Helmholtz.

Definition

For a vector field

F \in C ^{1(V,} R ^{n)} defined on a domain V \subseteq R ^{n} , a Helmholtz decomposition is a pair of vector fields G \in C ^{1(V,} R ^{n)} and R \in C ^{1(V,} R ^{n)} such that:Here, \Phi \in C ^{2(V,} R ) is a scalar potential, \nabla \Phi is its gradient, and \nabla ⋅ R is the divergence of the vector field R . The irrotational vector field G is called a gradient field and R is called a solenoidal field or rotation field. This decomposition does not exist for all vector fields and is not unique.

History

The Helmholtz decomposition in three dimensions was first described in 1849 by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. 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. For Riemannian manifolds, the Helmholtz-Hodge decomposition using differential geometry and tensor calculus was derived.

The decomposition has become an important tool for many problems in theoretical physics, but has also found applications in animation, computer vision as well as robotics.

Three-dimensional space

A can be defined, such that the rotation field is given by R = \nabla x A , using the curl of a vector field.

Let

F be a vector field on a bounded domain V\subseteq R ^{3} , which is twice continuously differentiable inside V , and let S be the surface that encloses the domain V . Then F can be decomposed into a curl-free component and a divergence-free component as follows:

$$\backslash mathbf=-\backslash nabla\; \backslash Phi+\backslash nabla\backslash times\backslash mathbf,$$where

\, \mathrmV' -\frac 1 \oint_S \mathbf' \cdot \frac \, \mathrmS' \\[8pt]\mathbf(\mathbf) & =\frac 1 \int_V \frac \, \mathrmV' -\frac 1 \oint_S \mathbf'\times\frac \, \mathrmS'\end

and

\nabla' is the nabla operator with respect to r' , not r .

If

V = \R ^{3} and is therefore unbounded, and F vanishes faster than 1 /r as r \to inf ty , then one has

\, \mathrmV' \\[8pt]\mathbf (\mathbf) & =\frac\int_ \frac \, \mathrmV'\endThis holds in particular if F is twice continuously differentiable in R ^{3} and of bounded support.

Solution space

If

(\Phi _{1,} { A _{1})} is a Helmholtz decomposition of F , then (\Phi _{2,} { A _{2})} is another decomposition if, and only if,\Phi _{1-\Phi} _{2} = λ and A _{1} - A _{2} = { A} _{λ} + \nabla \varphi,

where

λ is a harmonic scalar field,{ A} _{λ} is a vector field which fulfills \nabla x { A} _{λ} = \nabla λ , \varphi is a scalar field.

Proof: Set

λ = \Phi _{2} - \Phi _{1} and { B = A _{2} - A _{1}} . According to the definitionof the Helmholtz decomposition, the condition is equivalent to - \nabla λ + \nabla x B = 0 .

Taking the divergence of each member of this equation yields

\nabla ^{2} λ = 0 , hence λ is harmonic.

Conversely, given any harmonic function

λ ,\nabla λ is solenoidal since\nabla ⋅ (\nabla λ ) = \nabla ^{2} λ = 0 .

Thus, according to the above section, there exists a vector field

{ A} _{λ} such that\nabla λ = \nabla x { A} _{λ} .

If

{ A'} _{λ} is another such vector field, then C = { A} _{λ} - { A'} _{λ} fulfills \nabla x { C} = 0 , hence C = \nabla \varphi for some scalar field \varphi .

Fields with prescribed divergence and curl

The term "Helmholtz theorem" can also refer to the following. Let be a solenoidal vector field and d a scalar field on which are sufficiently smooth and which vanish faster than at infinity. Then there exists a vector field such that

$$\backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; d\; \backslash quad\; \backslash text\; \backslash quad\; \backslash nabla\; \backslash times\; \backslash mathbf\; =\; \backslash mathbf;$$

if additionally the vector field vanishes as, then is unique.

In other words, a vector field can be constructed with both a specified divergence and a specified curl, and if it also vanishes at infinity, it is uniquely specified by its divergence and curl. 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 proof is by a construction generalizing the one given above: we set

$$\backslash mathbf\; =\; \backslash nabla(\backslash mathcal\; (d))\; -\; \backslash nabla\; \backslash times\; (\backslash mathcal(\backslash mathbf)),$$

where

l{G} represents the Newtonian potential operator. (When acting on a vector field, such as, it is defined to act on each component.)

Weak formulation

The Helmholtz decomposition can be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field has an orthogonal decomposition:

$$\backslash mathbf=\backslash nabla\backslash varphi+\backslash nabla\; \backslash times\; \backslash mathbf$$

where is in the Sobolev space of square-integrable functions on whose partial derivatives defined in the distribution sense are square integrable, and, the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl.

For a slightly smoother vector field, a similar decomposition holds:

$$\backslash mathbf=\backslash nabla\backslash varphi+\backslash mathbf$$

where .

Derivation from the Fourier transform

Note that in the theorem stated here, we have imposed the condition that if

F is not defined on a bounded domain, then F shall decay faster than 1 /r . Thus, the Fourier transform of F , denoted as G , is guaranteed to exist. We apply the convention$$\backslash mathbf(\backslash mathbf)\; =\; \backslash iiint\; \backslash mathbf(\backslash mathbf)\; e^\; dV\_k$$

The Fourier transform of a scalar field is a scalar field, and the Fourier transform of a vector field is a vector field of same dimension.

Now consider the following scalar and vector fields:

Hence

Longitudinal and transverse fields

\hat F of the vector field F . Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k. So far, we have

$$\backslash hat\backslash mathbf\; (\backslash mathbf)\; =\; \backslash hat\backslash mathbf\_t\; (\backslash mathbf)\; +\; \backslash hat\backslash mathbf\_l\; (\backslash mathbf)$$$$\backslash mathbf\; \backslash cdot\; \backslash hat\backslash mathbf\_t(\backslash mathbf)\; =\; 0.$$$$\backslash mathbf\; \backslash times\; \backslash hat\backslash mathbf\_l(\backslash mathbf)\; =\; \backslash mathbf.$$

Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive:

$$\backslash mathbf(\backslash mathbf)\; =\; \backslash mathbf\_t(\backslash mathbf)+\backslash mathbf\_l(\backslash mathbf)$$$$\backslash nabla\; \backslash cdot\; \backslash mathbf\_t\; (\backslash mathbf)\; =\; 0$$$$\backslash nabla\; \backslash times\; \backslash mathbf\_l\; (\backslash mathbf)\; =\; \backslash mathbf$$

Since

\nabla x (\nabla\Phi) =0 and \nabla ⋅ (\nabla x A ) =0 ,

we can get

$$\backslash mathbf\_t=\backslash nabla\backslash times\backslash mathbf=\backslash frac\backslash nabla\backslash times\backslash int\_V\backslash frac\backslash mathrmV\text{'}$$$$\backslash mathbf\_l=-\backslash nabla\backslash Phi=-\backslash frac\backslash nabla\backslash int\_V\backslash frac\backslash mathrmV\text{'}$$

so this is indeed the Helmholtz decomposition.

Generalization to higher dimensions

Matrix approach

The generalization to

d 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

F be a vector field on a bounded domain V\subseteq R ^{d} which decays faster than | r | ^{-\delta} for | r | \to inf ty and \delta > 2 .

The scalar potential is defined similar to the three dimensional case as:$$\backslash Phi(\backslash mathbf)\; =\; -\; \backslash int\_\; \backslash operatorname(\backslash mathbf(\backslash mathbf\text{'}))\; K(\backslash mathbf,\; \backslash mathbf\text{'})\; \backslash mathrmV\text{'}\; =\; -\; \backslash int\_\; \backslash sum\_i\; \backslash frac(\backslash mathbf\text{'})\; K(\backslash mathbf,\; \backslash mathbf\text{'})\; \backslash mathrmV\text{'},$$where as the integration kernel

K( r , r ') is again the fundamental solution of Laplace's equation, but in d-dimensional space:with V _{d} = \pi ^{(d)/(2)} / \Gamma ( \tfrac{d}{ 2 } +1 ) the volume of the d-dimensional unit balls and \Gamma( r ) the gamma function.

For

d = 3 , V _{d} is just equal to (4 \pi)/(3) , yielding the same prefactor as above.The rotational potential is an antisymmetric matrix with the elements:$$A\_(\backslash mathbf)\; =\; \backslash int\_\; \backslash left(\backslash frac(\backslash mathbf\text{'})\; -\; \backslash frac(\backslash mathbf\text{'})\; \backslash right)\; K(\backslash mathbf,\; \backslash mathbf\text{'})\; \backslash mathrmV\text{'}.$$Above the diagonal are style\binom{d}{2} entries which occur again mirrored at the diagonal, but with a negative sign.In the three-dimensional case, the matrix elements just correspond to the components of the vector potential A = [A _{1,} A _{2,} A _{3]} = [A _{23} , A _{31} , A _{12} ] .However, such a matrix potential can be written as a vector only in the three-dimensional case, because style\binom{d}{2} = d is valid only for d = 3 .

As in the three-dimensional case, the gradient field is defined as$$\backslash mathbf(\backslash mathbf)\; =\; -\; \backslash nabla\; \backslash Phi(\backslash mathbf).$$The rotational field, on the other hand, is defined in the general case as the row divergence of the matrix:$$\backslash mathbf(\backslash mathbf)\; =\; \backslash left[\backslash sum\backslash nolimits\_k\; \backslash partial\_\{r\_k\}\; A\_\{ik\}(\backslash mathbf\{r\});\; \{1\; \backslash leq\; i\; \backslash leq\; d\}\; \backslash right].$$In three-dimensional space, this is equivalent to the rotation of the vector potential.

Tensor approach

In a

d -dimensional vector space with d ≠ 3 , $-\backslash frac$ can be replaced by the appropriate Green's function for the Laplacian, defined by$$\backslash nabla^2\; G(\backslash mathbf,\backslash mathbf\text{'})\; =\; \backslash frac\backslash fracG(\backslash mathbf,\backslash mathbf\text{'})\; =\; \backslash delta^d(\backslash mathbf-\backslash mathbf\text{'})$$where Einstein summation convention is used for the index \mu . For example, $G(\backslash mathbf,\backslash mathbf\text{'})=\backslash frac\backslash ln\backslash left|\backslash mathbf-\backslash mathbf\text{'}\backslash right|$ in 2D.

Following the same steps as above, we can write$$F\_\backslash mu(\backslash mathbf)\; =\; \backslash int\_V\; F\_\backslash mu(\backslash mathbf\text{'})\; \backslash frac\backslash fracG(\backslash mathbf,\backslash mathbf\text{'})\; \backslash ,\backslash mathrm^d\; \backslash mathbf\text{'}\; =\; \backslash delta\_\backslash delta\_\backslash int\_V\; F\_\backslash nu(\backslash mathbf\text{'})\; \backslash frac\backslash fracG(\backslash mathbf,\backslash mathbf\text{'})\; \backslash ,\backslash mathrm^d\; \backslash mathbf\text{'}$$where

\delta _{\mu\nu} is the Kronecker delta (and the summation convention is again used). In place of the definition of the vector Laplacian used above, we now make use of an identity for the Levi-Civita symbol \varepsilon ,$$\backslash varepsilon\_\backslash varepsilon\_\; =\; (d-2)!(\backslash delta\_\backslash delta\_\; -\; \backslash delta\_\backslash delta\_)$$which is valid in d\ge 2 dimensions, where \alpha is a (d -2 ) -component multi-index. This gives$$F\_\backslash mu(\backslash mathbf)\; =\; \backslash delta\_\backslash delta\_\backslash int\_V\; F\_\backslash nu(\backslash mathbf\text{'})\; \backslash frac\backslash fracG(\backslash mathbf,\backslash mathbf\text{'})\; \backslash ,\backslash mathrm^d\; \backslash mathbf\text{'}+\; \backslash frac\backslash varepsilon\_\backslash varepsilon\_\; \backslash int\_V\; F\_\backslash nu(\backslash mathbf\text{'})\; \backslash frac\backslash fracG(\backslash mathbf,\backslash mathbf\text{'})\; \backslash ,\backslash mathrm^d\; \backslash mathbf\text{'}$$

We can therefore write$$F\_\backslash mu(\backslash mathbf)\; =\; -\backslash frac\; \backslash Phi(\backslash mathbf)\; +\; \backslash varepsilon\_\backslash frac\; A\_(\backslash mathbf)$$whereNote that the vector potential is replaced by a rank-

(d -2 ) tensor in d dimensions.

Because

G( r , r ') is a function of only r - r ' , one can replace (\partial)/(\partial r _{\mu}) → - (\partial)/(\partial r' _{\mu}) , givingIntegration by parts can then be used to givewhere S = \partial V is the boundary of V . These expressions are analogous to those given above for three-dimensional space.

For a further generalization to manifolds, see the discussion of Hodge decomposition below.

Differential forms

The Hodge decomposition is closely related to the Helmholtz decomposition, 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. 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

Most textbooks only deal with vector fields decaying faster than

| r | ^{-\delta} with \delta > 1 at infinity. However, Otto Blumenthal showed in 1905 that an adapted integration kernel can be used to integrate fields decaying faster than | r | ^{-\delta} with \delta > 0 , which is substantially less strict.To achieve this, the kernel K( r , r ') in the convolution integrals has to be replaced by K'( r , r ') = K( r , r ') - K( 0 , r ') .With even more complex integration kernels, solutions can be found even for divergent functions that need not grow faster than polynomial.

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 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.

Uniqueness of the solution

H( r ) is a function that satisfies \Delta H( r ) = 0 .By adding H( r ) to the scalar potential \Phi( r ) , a different Helmholtz decomposition can be obtained:

For vector fields

F , decaying at infinity, it is a plausible choice that scalar and rotation potentials also decay at infinity. Because H( r ) = 0 is the only harmonic function with this property, which follows from Liouville's theorem, this guarantees the uniqueness of the gradient and rotation fields.

This uniqueness does not apply to the potentials: In the three-dimensional case, the scalar and vector potential jointly have four components, whereas the vector field has only three. The vector field is invariant to gauge transformations and the choice of appropriate potentials known as gauge fixing is the subject of gauge theory. Important examples from physics are the Lorenz gauge condition and the Coulomb gauge. An alternative is to use the poloidal–toroidal decomposition.

Applications

Electrodynamics

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.

Fluid dynamics

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

P\Delta is called the Stokes operator.

Dynamical systems theory

In the theory of dynamical systems, Helmholtz decomposition can be used to determine "quasipotentials" as well as to compute Lyapunov functions in some cases.

For some dynamical systems such as the Lorenz system (Edward N. Lorenz, 1963), a simplified model for atmospheric convection, a closed-form expression of the Helmholtz decomposition can be obtained:$$\backslash dot\; \backslash mathbf\; =\; \backslash mathbf(\backslash mathbf)\; =\; \backslash big[a\; (r\_2-r\_1),\; r\_1\; (b-r\_3)-r\_2,\; r\_1\; r\_2-c\; r\_3\; \backslash big].$$The Helmholtz decomposition of

F ( r ) , with the scalar potential \Phi( r ) = \tfrac{a}{ 2 } r ^{2} _{1} + \tfrac{ 1 }{ 2 } r ^{2} _{2} + \tfrac{c}{ 2 } r ^{2} _{3} is given as:

$$\backslash mathbf(\backslash mathbf)\; =\; \backslash big[-a\; r\_1,\; -r\_2,\; -c\; r\_3\; \backslash big],$$$$\backslash mathbf(\backslash mathbf)\; =\; \backslash big[+\; a\; r\_2,\; b\; r\_1\; -\; r\_1\; r\_3,\; r\_1\; r\_2\; \backslash big].$$

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.

Medical Imaging

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). In this way, the complex shear modulus can be calculated without contributions from compression waves.

Computer animation and robotics

The Helmholtz decomposition is also used in the field of computer engineering. This includes robotics, image reconstruction but also computer animation, where the decomposition is used for realistic visualization of fluids or vector fields.

See also

• Clebsch representation for a related decomposition of vector fields
• Darwin Lagrangian for an application
• Poloidal–toroidal decomposition for a further decomposition of the divergence-free component

\nabla x A .

• Scalar–vector–tensor decomposition
• Hodge theory generalizing Helmholtz decomposition
• Polar factorization theorem
• Helmholtz–Leray decomposition used for defining the Leray projection

References

• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 4th edition, Academic Press: San Diego (1995) pp. 92–93
• George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists – International Edition, 6th edition, Academic Press: San Diego (2005) pp. 95–101
• Rutherford Aris, Vectors, tensors, and the basic equations of fluid mechanics, Prentice-Hall (1962),, pp. 70–72