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Analytical Helmholtz Decomposition and Potential Functions for Many N-Dimensional Unbounded Vector fields

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Analytical Helmholtz Decomposition and Potential Functions for Many N-Dimensional Unbounded Vector fields Analytical Helmholtz Decomposition and Potential Functions for many n-dimensional unbounded vector fields Erhard Glötzl1 , Oliver Richters2;3 1: Institute of Physical Chemistry, Johannes Kepler University Linz, Austria. 2: ZOE. Institute for Future-Fit Economies, Bonn, Germany. 3: Department of Business Administration, Economics and Law, Carl von Ossietzky University, Oldenburg, Germany. Version 2 – April 2021 Abstract: We present a Helmholtz Decomposition for many n-dimensional, continuously differentiable vector fields on unbounded domains that do not decay at infinity. Existing methods are restricted to fields not growing faster than polynomially and require solving n-dimensional volume integrals over unbounded domains. With our method only one- dimensional integrals have to be solved to derive gradient and rotation potentials. Analytical solutions are obtained for smooth vector fields f (x) whose components are separable into a product of two functions: fk(x) = uk(xk) · vk(x,k), where uk(xk) depends only on xk and vk(x,k) depends not on xk. Additionally, an integer λk must exist such that the 2λk-th integral of one of the functions times the λk-th power of the Laplacian applied to the other function yields a product that is a multiple of the original product. A similar condition is well-known from repeated partial integration, where the calculation can be finalized if the shifting of derivatives yields a multiple of the original integrand. Also linear combinations of such vector fields can be decomposed. These conditions include periodic and exponential functions, combinations of polynomials with arbitrary integrable functions, their products and linear combinations, and examples such as Lorenz or Rössler attractor. Keywords: Helmholtz Decomposition, Fundamental Theorem of Vector Calculus, Gradi- ent and Rotation Potentials, Unbounded Domains, Poisson Equation. Licence: Creative-Commons CC-BY-NC-ND 4.0. arXiv:2102.09556v2 [math-ph] 22 Apr 2021 1. Introduction The Helmholtz Decomposition splits a sufficiently smooth vector field f into an irrotational (curl- free) ‘gradient field’ g and a solenoidal (divergence-free) ‘rotation field’ r. This ‘Fundamental Theorem of Vector Calculus’ is indispensable for many problems in mathematical physics (Dassios and Lindell, 2002; Kustepeli, 2016; Sprössig, 2009; Tran-Cong, 1993), but is also used in animation, computer vision or robotics (Bhatia et al., 2013), or for describing ‘quasi-potential’ landscapes and Lyapunov functions for high-dimensional non-gradient systems (Suda, 2019; Zhou et al., 2012). 2 Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . The challenge is to derive the potential G and the rotation field r such that: f (x) = grad G(x) + r(x); (1) div r(x) = 0: (2) Only if f is curl-free, path integration yields the potential G, and r = 0. In other cases, the Poisson equation has to be solved: ∆G(x) = div f (x): (3) On bounded domains, a unique solution is guaranteed by appropriate boundary conditions (Chorin and Marsden, 1990; Schwarz, 1995). Numerical methods based on finite elements or difference methods, Fourier or wavelet domains have been developed to derive L2-orthogonal decompositions. This concept has been extended as Hodge or Helmholtz–Hodge Decomposition to compact Rieman- nian manifolds (Bhatia et al., 2013). For fields decaying sufficiently fast at infinity, the Helmholtz Decomposition can be derived by numerically computing infinite integrals over Rn for each point x. For many unboundedly growing vector fields, these integrals diverge. To overcome this problem, we provide an alternative method that only requires finding analytical solutions for one-dimensional integrals. Sec. 2 describes our notation. Sec. 3 introduces methods to calculate a ‘potential matrix’ from which gradient and rotation potentials and corresponding fields can be derived. Theorem 2 derives the potential matrix numerically for decaying fields using convolution integrals over the entire space, reformulating well-known results of the classical Helmholtz Decomposition. In preparation for the analytical Helmholtz Decomposition for unbounded fields, an example with exponentially diverging fields presents the intuition that we try to compensate terms created by the gradient of the gradient potential by the choice of appropriate rotation potentials, and inversely. Theorem 3 formalizes this method for unbounded fields with only one non-zero component, and Theorem 4 extends it to linear combinations, which levies the restriction to one-component fields. Five corollaries provide simplified versions for special cases, including linear functions. Sec. 4 discusses our results. In the appendix, the theorems are applied to seven examples. 2. Notation Square brackets [ fk; 1≤k≤n] indicate a n-dimensional vector. If a vector component fk depends not on its ‘own’ coordinate xk, but only on ‘foreign’ coordinates xi with i , k, we denote it as fk(x,k). k We denote the partial derivative as @x j , the k-th partial derivative as @x j , the Laplace operator as ∆ p 0 0 and the Laplacian to the power of p as ∆ , with the convention that @x j f = ∆ f = f . We denote the antiderivative of a scalar field fi with respect to x j as: Z x j Ax j fi(x) B fi(ξ)dξ j: (4) 0 The p-th antiderivative of a scalar fi with respect to x j is given by the Cauchy formula for repeated integration or the Riemann–Liouville integral (Cauchy, 1823; Riesz, 1949): Z x j p p−1 1 p−1 Ax j fi(x) B Ax j Ax j fi(x) = (x j − t j) fi(t)dt j; (5) (p − 1)! 0 A p Ap A0 such that @x j x j fi = @x j x j fi = x j fi = fi. 3 Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . Figure 1: The figure depicts the ways to obtain a Helmholtz Decomposition, starting from a vector field f . The variables are defined in Definitions 1 and 2. For sufficiently fast decaying fields on unbounded domains, the convolution of the Jacobian Ji j with a solution of Laplace’s equation yields the potential matrix Fi j (Theorem 2, green arrows). The Laplacian applied to the potential matrix yields again the Jacobian matrix (light blue arrow). The analytical approach (Theorems 3, 4) allows to drop the condition that the vector field f decays sufficiently fast at infinity by directly calculating the potential matrix (red arrow). The Helmholtz Decomposition of f is obtained by calculating the gradient field and the rotation field from the potential matrix (dark blue arrow). 3. Helmholtz Decomposition Theorems using potential matrices First we define the terms of the paper, whose relations are illustrated in Figure 1. Definition 1 (Helmholtz Decomposition, gradient field and rotation field). For a vector field f 2 C1(Rn; Rn), a ‘gradient field’ g 2 C1(Rn; Rn) and a ‘rotation field’ r 2 C1(Rn; Rn) are called a ‘Helmholtz Decomposition of f ’, if they sum to f , if g is gradient of some ‘gradient potential’ G, and if r is divergence-free: f (x) = g(x) + r(x); (6) g(x) = grad G(x); (7) 0 = div r(x): (8) 2 Definition 2 (Potential matrix, gradient potential and rotation potential). For a matrix F 2 C2(Rn; Rn ), 2 define a scalar function G 2 C2(Rn; R) as the trace of F, and a matrix R 2 C2(Rn; Rn ) as two times the antisymmetric part of F: X G(x) = Fkk(x) = Tr Fi j(x); (9) k Rik(x) = Fik(x) − Fki(x); (10) 4 Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . Two vector fields g; r 2 C1(Rn; Rn) are defined as gradient of G, and as divergence of the antisym- metric second-rank tensor Ri j: X g(x) = grad G(x) = @x G(x); 1≤i≤n = @x Fkk(x); 1≤i≤n ; (11) i i k X X r(x) = @x Rik(x); 1≤i≤n = @x Fik(x) − Fki(x) ; 1≤i≤n : (12) k k k k Let f 2 C1(Rn; Rn) be a vector field. If the ‘gradient field’ g and the ‘rotation field’ r are a ‘Helmholtz Decomposition of f ’ as per Definition 1, we call F a ‘potential matrix of f ’, G a ‘gradient potential of f ’, and R a ‘rotation potential of f ’. Lemma 1. For every matrix F, the gradient field g as defined by Eq. (11) is rotation-free, and the rotation field r as defined by Eq. (12) is divergence-free. If f = g + r, the matrix F is a ‘potential matrix of f ’. Proof. The ‘gradient field’ g is gradient of a potential G, and therefore rotation-free by definition. The divergence of the rotation field r is zero, as X X X div r(x) = @x ri(x) = @x @x Rik(x) i i i i k k X (13) = @xi @xk (Fik(x) − Fki(x)) = 0; i;k because the partial derivatives can be exchanged. If f = g + r, then g and r are a ‘Helmholtz Decomposition of f ’ as per Definition 1. Theorem 2 shows how such a potential matrix can be derived for decaying fields. Theorem 2 (Helmholtz Theorem for decaying fields). For a vector field f 2 C1(Rn; Rn) that decays faster than jxj−δ for jxj ! 1 with δ > 0, define the matrix F as the convolution of the Jacobian matrix of f with a solution of Laplace’s equation: Z @ fi n Fi j(x) = K(x; ξ)dξ ; (14) Rn @x j with the kernel K(x), 8 > 1 log jx − ξj − log jξj n = 2; K(x; ξ) = < 2π :> 1 jx − ξj2−n − jξj2−n otherwise; n(2−n)Vn n 2 n with Vn = π =Γ 2 + 1 the volume of a unit n-ball and Γ(x) the gamma function. With Definition 2 and Eqs.
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