Editing Helmholtz decomposition (section)

== 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.<ref name="petrascheck2015" />

=== 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 flow|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 <math>P\Delta</math> is called the [[Stokes operator]].<ref name="chorin1990" />

=== Dynamical systems theory ===

In the theory of [[dynamical system]]s, Helmholtz decomposition can be used to determine "quasipotentials" as well as to compute [[Lyapunov function]]s in some cases.<ref name="suda2019" /><ref name="suda2020" /><ref name="zhou2012" />

For some dynamical systems such as the [[Lorenz system]] ([[Edward N. Lorenz]], 1963<ref name="lorenz1963" />), a simplified model for [[atmosphere|atmospheric]] [[convection]], a [[closed-form expression]] of the Helmholtz decomposition can be obtained:
<math display="block">\dot \mathbf{r} = \mathbf{F}(\mathbf{r}) = \big[a (r_2-r_1), r_1 (b-r_3)-r_2, r_1 r_2-c r_3 \big].</math>
The Helmholtz decomposition of <math>\mathbf{F}(\mathbf{r})</math>, with the scalar potential <math>\Phi(\mathbf{r}) = \tfrac{a}{2} r_1^2 + \tfrac{1}{2} r_2^2 + \tfrac{c}{2} r_3^2</math> is given as:

<math display="block">\mathbf{G}(\mathbf{r}) = \big[-a r_1, -r_2, -c r_3 \big],</math>
<math display="block">\mathbf{R}(\mathbf{r}) = \big[+ a r_2, b r_1 - r_1 r_3, r_1 r_2 \big].</math>

The quadratic scalar potential provides motion in the direction of the coordinate origin, which is responsible for the stable [[fixed point (mathematics)|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]].<ref name="glotzl2023" /><ref name="peitgen1992" />

=== 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).<ref name="manduca2021" /> 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.<ref name="bhatia2013" /><ref name="bhatia2014" />