Analytical Helmholtz Decomposition and Potential Functions for many n-dimensional unbounded vector fields
AAnalytical Helmholtz Decomposition and Potential Functionsfor many n-dimensional unbounded vector fields
Erhard Glötzl , Oliver Richters ,
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.February 2021
Abstract:
We present a Helmholtz Decomposition for many n-dimensional, continuouslydi ff erentiable vector fields on unbounded domains that do not decay at infinity. Existingmethods are restricted to fields not growing faster than polynomially and require solvingn-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 areseparable into a product of two functions: f k ( x ) = u k ( x k ) · v k ( x (cid:44) k ), where u k ( x k ) dependsonly on x k and v k ( x (cid:44) k ) depends not on x k . Additionally, an integer λ k must exist such thatthe 2 λ k -th integral of one of the functions times the λ k -th power of the Laplacian appliedto the other function yields a product that is a multiple of the original product. A similarcondition is well-known from repeated partial integration: If the shifting of derivativesyields a multiple of the original integrand, the calculation can be terminated.Also linear combinations of such vector fields can be decomposed. These conditionsinclude periodic and exponential functions, combinations of polynomials with arbitraryintegrable functions, their products and linear combinations, and examples such as Lorenzor 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.
1. Introduction
The Helmholtz Decomposition splits a su ffi ciently smooth vector field f ( x ) into an irrotational(curl-free) and a solenoidal (divergence-free) vector field. This ‘Fundamental Theorem of VectorCalculus’ 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 orrobotics (Bhatia et al., 2013), or for describing ‘quasi-potential’ landscapes and Lyapunov functionsfor high-dimensional non-gradient systems (Suda, 2019; Zhou et al., 2012). a r X i v : . [ m a t h - ph ] F e b Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
The challenge is to derive the potential G ( x ) and the rotation field r ( x ) such that: f ( x ) = grad G ( x ) + r ( x ) , (1)div r ( x ) = . (2)Only if f ( x ) is curl-free, path integration yields the potential G ( x ). In other cases, the Poissonequation has to be solved: ∆ G ( x ) = div f ( x ) . (3)On bounded domains, a unique solution is guaranteed by appropriate boundary conditions (Chorinand Marsden, 1990; Schwarz, 1995). Numerical methods based on finite elements or di ff erence meth-ods, Fourier or wavelet domains have been developed to derive L -orthogonal decompositions. Thisconcept has been extended as Hodge or Helmholtz–Hodge Decomposition to compact Riemannianmanifolds (Bhatia et al., 2013).To solve the Poisson equation on unbounded domains, the standard approach as summarizedin Sec. 2 is limited to fields decaying su ffi ciently fast and requires to calculate numerically infi-nite integrals over R n for each point x . This paper provides an alternative, applicable to manyunboundedly growing vector fields. Only one-dimensional integrals have to be solved to derivegradient and rotation potentials. Sec. 3 introduces our notation. Sec. 4 illustrates our method usingan exponentially growing field. Sec. 5 states two theorems, a special case for fields with only onenon-zero component and a more general case for linear combinations of the former which includesfields with more than one non-zero component. Three corollaries provide simplified versions forspecial cases, including linear functions. Sec. 6 concludes.
2. Literature Review
On unbounded domains as in the classical Helmholtz decomposition in R , solving Eq. (3) leads tovolume integrals over the entire space. The Helmholtz Decomposition in R n proceeds analogouslyto R (Glötzl and Richters, 2021), see Figure 1. Starting from a twice continuously di ff erentiablevector field f , the scalar source density γ ( x ) and the rotation densities ρ i j ( x ) are calculated. Todetermine the ‘gradient potential’ G ( x ) and the antisymmetric ‘rotation potentials’ R i j ( x ) = − R ji ( x ),the Newton Integral operator N (cid:82) convolves these densities with the fundamental solutions of theLaplace equation, performing a multiple integral over R n : G ( x ) = N (cid:90) γ ( x ) = (cid:90) R n K ( x − ξ ) div f ( ξ ) d ξ n , (4) R i j ( x ) = N (cid:90) ρ i j ( x ) = (cid:90) R n K ( x − ξ )ROT i j f ( ξ ) d ξ n , (5)with the Newtonian kernel K ( x , ξ ) = π log | x − ξ | n = n ) ! n (2 − n ) π n | x − ξ | − n n (cid:44) , (6)and the rotation operator ROT i j f ( x ) = ∂ f i ∂ x j − ∂ f j ∂ x i . (7) Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
Figure 1: For unbounded domains, the usual approach derives scalar and rotation densities fromthe vector field f . These densities are convolved with the fundamental solutions of theLaplace equation to derive scalar and rotation potentials (‘Newton Integrals’ N (cid:82) ). Thisintegration over the entire space requires the vector field f to decay su ffi ciently fast atinfinity. Our approach allows to drop this condition by directly calculating the potentials.Figure adapted from (Glötzl and Richters, 2021).To ensure that the ‘Newton Integrals’ (Eqs. 4–6) used to calculate the ‘Newton Potentials’ G ( x )and R i j ( x ) converge, f ( x ) must decay faster than 1 / | x | for | x | → ∞ . Replacing K ( x , ξ ) by K ( x , ξ ) − K (0 , ξ ) makes these integrals converge if f ( x ) decays faster than x − δ for some δ > | x | → ∞ (Blumenthal, 1905; Gregory, 1996; Gurtin, 1962). By choosing more complicated kernel functions,this method can also be applied for functions that grow slower than a polynomial x q with q > g ( x ) as gradient of G and a ‘rotation field’ r ( x ) as divergenceof the antisymmetric second-rank tensor R = (cid:126) R i j (cid:127) can be calculated (Glötzl and Richters, 2021): g ( x ) (cid:66) grad G ( x ) = (cid:104) ∂ x k G ( x ); ≤ k ≤ n (cid:105) , (8) r ( x ) (cid:66) ROT R ( x ) (cid:66) (cid:20)(cid:88) nj = ∂ x j R k j ( x ); ≤ k ≤ n (cid:21) . (9)In sum, they yield the original vector field and provide a Helmholtz Decomposition of f ( x ), as r satisfies div r ( x ) =
0, and f = g + r (see proof in Glötzl and Richters, 2021). Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
3. Notation and Definitions
Square brackets [ f k ; ≤ k ≤ n ] indicate a n-dimensional vector. We denote the partial derivative ∂ x j , thek-th partial derivative ∂ kx j , the Laplace operator ∆ and the Laplace to the power of p with ∆ p , withthe convention that ∂ x j f = ∆ f = f .We denote the antiderivative of a scalar field f i with respect to x j as: A x j f i ( x ) (cid:66) (cid:90) x j f i ( ξ ) d ξ j . (10)The p -th antiderivative of a scalar f i with respect to x j , using the convention that A x j f = f , isgiven by the Cauchy formula for repeated integration or the Riemann–Liouville integral (Cauchy,1823; Riesz, 1949): A px j f i ( x ) (cid:66) A p − x j A x j f i ( x ) = p − (cid:90) x j ( x j − t j ) p − f i ( t ) dt j , (11)The p-th partial derivative of the p-th antiderivative yields the original function: ∂ x j A x j f i ( x ) = ∂ px j A px j f i ( x ) = f i ( x ) . (12)
4. Examples with exponentially diverging vector field In R , for the exponentially growing vector field f ( x , x , x ) = (cid:104) f ( x ) , , (cid:105) = (cid:104) e ax e bx , , (cid:105) , (13)with a (cid:44) b , the Newton Integrals according to Eqs. (4–6) diverge and a Helmholtz Decompositionbased on Newton Potentials does not exist. As an alternative, we present two approaches (a) and(b) to derive the gradient potential G ( x ) and the rotation potentials with an analytical calculation.Denote u ( x ) = e ax and v ( x ) = e bx , such that f ( x ) = u ( x ) · v ( x ).For approach (a), to get ∂ x G ( x ) = f ( x ), choose G ( x ) = (cid:16) A x u ( x ) (cid:17)(cid:16) v ( x ) (cid:17) = a e ax e bx (14)by integrating the first component. This results in g ( x ) = (cid:104) ∂ x k G ( x ) , ≤ k ≤ (cid:105) = (cid:104) u ( x ) v ( x ) , (cid:16) A x u ( x ) (cid:17)(cid:16) ∂ x v ( x ) (cid:17) , (cid:105) = (cid:104) e ax e bx , ba e ax e bx , (cid:105) , (15)with an unwanted term in the component g ( x ). To get rid of this part, choose R ( x ) = − R ( x ) = A x g ( x ) = (cid:16) A x u ( x ) (cid:17)(cid:16) ∂ x v ( x ) (cid:17) = ba e ax e bx , (16) R ( x ) = R ( x ) = R ( x ) = R ( x ) = , (17)such that r ( x ) = (cid:104) b a e ax e bx , − ba e ax e bx , (cid:105) . (18) Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
This results in g ( x ) + r ( x ) = (cid:104)(cid:16) + b a (cid:17) e ax e bx , , (cid:105) = (cid:16) + b a (cid:17) f ( x ) . (19)Integrating u ( x ) twice and taking the derivative ∂ x v ( x ) yielded the function C u ( x ) v ( x ) with b / a (cid:67) C ∈ R \ { } , a multiple C of f ( x ). This allows to multiply all potentials by 1 / (1 − C ) toget fields that add to f and provide a Helmholtz Decomposition of f .Approach (b) works inversely by taking derivatives of u ( x ) and integrating v ( x ): Start bychoosing R ( x ) = − R ( x ) = u ( x ) (cid:0) A x v ( x ) (cid:1) = b e ax e bx , (20) R ( x ) = R ( x ) = R ( x ) = R ( x ) = , (21) r ( x ) = (cid:104) e ax e bx , − ab e ax e bx , (cid:105) . (22)To correct for the unwanted term in the second component r ( x ), choose G ( x ) = A x r ( x ) = (cid:16) ∂ x u ( x ) (cid:17)(cid:16) A x v ( x ) (cid:17) = − ab e ax e bx , (23) g ( x ) = (cid:104) − a b e ax e bx , ab e ax e bx , (cid:105) . (24)This results in g ( x ) + r ( x ) = (cid:104)(cid:16) − a b (cid:17) e ax e bx , , (cid:105) = (cid:16) − a b (cid:17) f ( x ) . (25)Integrating v ( x ) twice and taking the derivative ∂ x u ( x ) yielded a function C (cid:48) u ( x ) v ( x ) with a / b (cid:67) C (cid:48) ∈ R \ { } . This allows to multiply all potentials by 1 / (1 − C (cid:48) ) to get fields that add to f and provide a Helmholtz Decomposition of f .Another example is the vector field f ( x ) = (cid:2) e x + e x , e x − e x (cid:3) . (26)The first part [ e x , e x ] is rotation-free and the second part [ e x , − e x ] is divergence free, the corre-sponding gradient and rotation potentials can be guessed by integration for each part. Because oflinearity of all the operators, the total potentials, gradient and rotation fields are: G ( x ) = e x + e x , (27) R ( x ) = − R ( x ) = e x + e x , (28) R kk ( x ) = , (29) g ( x ) = (cid:2) + e x , + e x (cid:3) , (30) r ( x ) = (cid:2) + e x , − e x (cid:3) . (31)
5. Helmholtz Decomposition Theorems for many unbounded vector fields
Theorem 1 formalizes the process described for the above examples for fields with only one non-zero component. Theorem 2 extends it to linear combinations, which levies the restriction toone-component fields. The intuition for the theorem is that we try to compensate terms created bythe gradient of the scalar potential by the choice of appropriate rotation potentials, and inversely.
Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
Theorem 1 (Special case for fields with one non-zero component) . Let f ∈ C ( R n , R n ) be a vector field with only one non-zero component f k ( x ) that satisfiesconditions 1 (separability) and either 2a or 2b (partial integrability). Then, a potential matrix F , agradient potential G and rotation potentials R = (cid:126) R i j (cid:127) can be specified such that the rotation-freegradient field g = grad G and the divergence-free rotation field r = ROT R yield a HelmholtzDecomposition of f = g + r . Condition 1: Separability:
The component f k ( x ) = u k ( x k ) · v k ( x (cid:44) k ) is separable into a product of u k ( x k ) that depends only on the ‘own’ coordinate x k and v k ( x (cid:44) k ) that depends only on the ‘foreign’coordinates x j with j (cid:44) k . Condition 2: Partial integrability:
It has to be possible to integrate one of the two functions u k and v k and di ff erentiate the other until a multiple C k of the original function is obtained. Thistechnique is known from repeated integration by parts (Polyanin and Chernoutsan, 2010, p. 173).Two cases can be distinguished: Condition 2a: Integrate ‘own’ and di ff erentiate ‘foreign’ coordinates: It exists λ k ∈ N and C k ∈ R \ { } such that 2 λ k antiderivatives of u k ( x k ) can be determined and( − λ k (cid:16) A λ k x k u k ( x k ) (cid:17)(cid:16) ∆ λ k v k ( x (cid:44) k ) (cid:17) = C k u k ( x k ) v k ( x (cid:44) k ) . (32)Define the n × n potential matrix F ∈ C ( R n , R n ) as: F k j ( x ) (cid:66) λ k − (cid:88) p k = ( − p k − C k ∂ x j (cid:16) A p k + x k u k ( x k ) (cid:17)(cid:16) ∆ p k v k ( x (cid:44) k ) (cid:17) , F i j ( x ) (cid:66) i (cid:44) k . (33) Condition 2b: Di ff erentiate ‘own’ and integrate one ‘foreign’ coordinate: v k ( x m ) dependsonly on one coordinate x m with m (cid:44) k , and it exists λ k ∈ N and C k ∈ R \ { } such that 2 λ k antiderivatives of v k ( x m ) can be determined, and( − λ k (cid:16) ∂ λ k x k u k ( x k ) (cid:17)(cid:16) A λ k x m v k ( x m ) (cid:17) = C k u k ( x k ) v k ( x m ) . (34)Define the n × n potential matrix F ∈ C ( R n , R n ) as: F k j ( x ) (cid:66) λ k − (cid:88) p k = ( − p k − C k ∂ x j (cid:16) ∂ p k x k u k ( x k ) (cid:17)(cid:16) A p k + x m v k ( x m ) (cid:17) for j = k , m , and F i j ( x ) (cid:66) . (35) Definition of potentials and vector fields:
Define the ‘gradient potential’ G ∈ C ( R n , R ) and the‘rotation potential’ matrix R ∈ C ( R n , R n ) as: G ( x ) (cid:66) (cid:88) i F ii ( x ) = Tr F i j ( x ) , (36) R i j ( x ) (cid:66) F i j ( x ) − F ji ( x ) = F − F (cid:124) . (37)Note that R i j ( x ) = − R ji ( x ) is antisymmetric, and R kk ( x ) = g ( x ) = grad G ( x ) and the divergence-free rotation field r ( x ) = ROT R ( x ) according to Eqs. (8–9) yields a Helmholtz Decomposition of f with f = g + r : g ( x ) = grad Tr F i j ( x ) = (cid:20) ∂ x k (cid:88) i F ii ( x ); ≤ k ≤ n (cid:21) , (38) r ( x ) = (cid:20)(cid:88) nj = ∂ x j (cid:0) F k j ( x ) − F jk ( x ) (cid:1) ; ≤ k ≤ n (cid:21) . (39) Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
Theorem 2 (General case for linear combinations of the special case) . This theorem removes the restriction of Theorem 1 to fields that are non-zero in only onecomponent and extends it further to field components that contains sums. A vector field f (cid:48) ∈ C ( R n , R n ) that is a linear combination of vector fields satisfying Theorem 1 can be decomposedinto two vector fields, one rotation-free gradient field g (cid:48) and one divergence-free rotation field r (cid:48) .The decomposition can be obtained by deriving the potentials resp. the vectors g and r for each fieldseperately and adding them.The proof of Theorem 2 follows directly from Theorem 1 by linearity of all the operators. Proof of Theorem 1:
Since g ( x ) is defined as gradient of some potential Tr F i j ( x ), it is knownto be rotation-free, thus ROT g ( x ) = curl g ( x ) =
0, and div r ( x ) = R ( x ) forsome antisymmetric potential R ( x ) (see Prop. 2–3 in Glötzl and Richters, 2021). Note that in theconventional Helmholtz Decomposition, the fact that R i j is antisymmetric is guaranteed before theNewton Integration on the level of the rotation densities.It remains to be shown that f = g + r for both Conditions 2a and 2b.For Condition 2a that integrates the ‘own’ and di ff erentiates the ‘foreign’ coordinates, gradientand rotation potentials yield: G ( x ) = λ k − (cid:88) p k = ( − p k − C k (cid:16) A p k + x k u k ( x k ) (cid:17)(cid:16) ∆ p k v k ( x (cid:44) k ) (cid:17) , (40) R k j ( x ) = λ k − (cid:88) p k = ( − p k − C k (cid:16) A p k + x k u k ( x k ) (cid:17)(cid:16) ∂ x j ∆ p k v k ( x (cid:44) k ) (cid:17) , R jk ( x ) = − R k j ( x ) for j (cid:44) k , and R i j ( x ) = g ( x ) and the rotation field r ( x ) are, using Eqs. (38–39): g k = λ k − (cid:88) p k = ( − p k − C k (cid:16) A p k x k u k ( x k ) (cid:17)(cid:16) ∆ p k v k ( x (cid:44) k ) (cid:17) , (42) g j (cid:44) k = λ k − (cid:88) p k = ( − p k − C k (cid:16) A p k + x k u k ( x k ) (cid:17)(cid:16) ∂ x j ∆ p k v k ( x (cid:44) k ) (cid:17) , (43) r k = λ k − (cid:88) p k = ( − p k − C k (cid:16) A p k + x k u k ( x k ) (cid:17)(cid:16) ∆∆ p k v k ( x (cid:44) k ) (cid:17) = λ k (cid:88) p k = − ( − p k − C k (cid:16) A p k x k u k ( x k ) (cid:17)(cid:16) ∆ p k v k ( x (cid:44) k ) (cid:17) , (44) r j (cid:44) k = λ k − (cid:88) p k = − ( − p k − C k (cid:16) A p k + x k u k ( x k ) (cid:17)(cid:16) ∂ x j ∆ p k v k ( x (cid:44) k ) (cid:17) . (45)In the k -component of r , the additional ∆ arises out of the sum over ∂ x i ∂ x i . This sum was simplifiedby a shift of the sum index. g j (cid:44) k + r j (cid:44) k = = f j (cid:44) k (46) Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . . as the sums cancel out. For g k + r k , only the terms with equal index p k cancel out, and it remains theterm with p k = g k and the term with p k = λ k of r k . With the definitions for A and ∆ in Sec. 3and Eq. (32), it holds: g k + r k = − C k u k ( x k ) v k ( x (cid:44) k ) − ( − λ k − C k (cid:16) A λ k x k u k ( x k ) (cid:17)(cid:16) ∆ λ k v k ( x (cid:44) k ) (cid:17) = − C k (cid:16) u k ( x k ) v k ( x (cid:44) k ) − C k u k ( x k ) v k ( x (cid:44) k ) (cid:17) = u k ( x k ) v k ( x (cid:44) k ) = f k ( x ) , (47)which proves that g ( x ) + r ( x ) = f ( x ). (cid:3) For
Condition 2b that di ff erentiates the ‘own’ and integrates one ‘foreign’ coordinate, gradientand rotation potentials yield: G ( x ) = λ k − (cid:88) p k = ( − p k − C k (cid:16) ∂ p k + x k u k ( x k ) (cid:17)(cid:16) A p k + x m v k ( x m ) (cid:17) , (48) R km ( x ) = λ k − (cid:88) p k = ( − p k − C k (cid:16) ∂ p k x k u k ( x k ) (cid:17)(cid:16) A p k + x m v k ( x m ) (cid:17) , R mk ( x ) = − R km , and R i j ( x ) = g k = λ k − (cid:88) p k = ( − p k − C k (cid:16) ∂ p k + x k u k ( x k ) (cid:17)(cid:16) A p k + x m v k ( x m ) (cid:17) = λ k (cid:88) p k = − ( − p k − C k (cid:16) ∂ p k x k u k ( x k ) (cid:17)(cid:16) A p k x m v k ( x m ) (cid:17) , (50) g m = λ k − (cid:88) p k = ( − p k − C k (cid:16) ∂ p k + x k u k ( x k ) (cid:17)(cid:16) A p k + x m v k ( x m ) (cid:17) , (51) r k = λ k − (cid:88) p k = ( − p k − C k (cid:16) ∂ p k x k u k ( x k ) (cid:17)(cid:16) A p k x m v k ( x m ) (cid:17) , (52) r m = λ k − (cid:88) p k = − ( − p k − C k (cid:16) ∂ p k + x k u k ( x k ) (cid:17)(cid:16) A p k + x m v k ( x m ) (cid:17) , (53) g i = r i = i (cid:44) k , m . (54)Summing f = r + g , most of the terms cancel. Note that the term − C k f k is provided by the rotationfield, not the gradient field as with Condition 2a. f i = g i + r i = i (cid:44) k , (55) g k + r k = − − C k ( − λ k (cid:16) ∂ λ k x k u k ( x k ) (cid:17)(cid:16) A λ k x m v k ( x m ) (cid:17) + − C k (cid:16) u k ( x k ) (cid:17)(cid:16) v k ( x m ) (cid:17) = − C k (cid:16) u k ( x k ) v k ( x (cid:44) k ) − C k u k ( x k ) v k ( x (cid:44) k ) (cid:17) = u k ( x k ) v k ( x (cid:44) k ) = f k ( x ) , (56)which proves that g ( x ) + r ( x ) = f ( x ). (cid:3) Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
Corollary 3 (Fields linear in ‘foreign’ coordinates) . Let f ( x ) have only one non-zero component f k ( x ) = u k ( x k ) v k ( x (cid:44) k ) (57)and let v k ( x (cid:44) k ) be linear in all ‘foreign’ coordinates x i with i (cid:44) k . This implies ∆ v k ( x (cid:44) k ) =
0, andCondition 2a is satisfied with λ k = C k =
0. The equations simplify to: F k j ( x ) = ∂ x j (cid:16) A x k u k ( x k ) (cid:17) v k ( x (cid:44) k ) , F i j ( x ) = i (cid:44) k . (58) G ( x ) = (cid:16) A x k u k ( x k ) (cid:17) v k ( x (cid:44) k ) . (59) R k j ( x ) = (cid:16) A x k u k ( x k ) (cid:17)(cid:16) ∂ x i v k ( x (cid:44) k ) (cid:17) = − R jk for j (cid:44) k , (60) g k ( x ) = f k ( x ) , g j (cid:44) k ( x ) = + (cid:0) A x k u k ( x k ) (cid:1)(cid:0) ∂ x i v k ( x (cid:44) k ) (cid:1) , (61) r k ( x ) = , r j (cid:44) k ( x ) = − (cid:0) A x k u k ( x k ) (cid:1)(cid:0) ∂ x i v k ( x (cid:44) k ) (cid:1) . (62) Corollary 4 (Fields linear in ‘own’ coordinate) . Let f ( x ) have only one non-zero component f k ( x ) = u k ( x k ) v k ( x m ) , (63)with u k ( x k ) a linear function of its ‘own’ coordinate x k and v k ( x m ) dependent only on one ‘foreign’coordinate x m . This implies ∆ u k ( x ) =
0, and Condition 2b is satisfied with λ k = C k =
0. Theequations simplify to: F k j ( x ) = ∂ x j u k ( x k ) (cid:16) A x m v k ( x m ) (cid:17) for j = k , m , F i j ( x ) = , (64) G ( x ) = (cid:16) ∂ x k u k ( x k ) (cid:17)(cid:16) A x m v k ( x m ) (cid:17) , (65) R km ( x ) = u k ( x k ) (cid:16) A x m v k ( x m ) (cid:17) , R mk ( x ) = − R km , and R i j ( x ) = g k ( x ) = , (67) g m ( x ) = + (cid:16) ∂ x k u k ( x k ) (cid:17)(cid:16) A x m v k ( x m ) (cid:17) , (68) r k ( x ) = u k ( x k ) v k ( x m ) , (69) r m ( x ) = − (cid:16) ∂ x k u k ( x k ) (cid:17)(cid:16) A x m v k ( x m ) (cid:17) , (70) g i ( x ) = r i ( x ) = i (cid:44) k , m . (71) Corollary 5 (Linear vector fields) . If f ( x ) is a linear vector field f ( x ) = Mx , (72) f k ( x ) = n (cid:88) j = M k j x j , (73) Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . . with a n × n matrix M = (cid:126) M i j (cid:127) , the easiest way to derive a Helmholtz Decomposition is to split thevector f into two parts using Theorem 2, one containing the ‘own’ coordinate and one the ‘foreign’coordinates: f (1) k ( x ) = M kk x k , (74) f (2) k ( x ) = (cid:88) j (cid:44) k M k j x j . (75)The components of the first part f (1) satisfy Condition 2a with u k ( x k ) = M kk x k , v k ( x (cid:44) k ) = λ k = C k =
0, which yields using Corollary 3: F (1) kk ( x ) = M kk x k , F (1) i j ( x ) = i (cid:44) j , (76) G (1) ( x ) = n (cid:88) k = M kk x k , (77) R (1) i j ( x ) = ∀ i , j . (78)All the summands M k j x j of the components of the second part f (2) satisfy Condition 2b with u k ( x k ) = v k ( x j ) = M k j x j , λ k = C k =
0, which yields using Theorem 2 and Corollary 4: F (2) i j ( x ) = M i j x i for i (cid:44) j , F (2) kk ( x ) = , (79) G (2) ( x ) = , (80) R (2) i j ( x ) = M i j x i − M ji x j . (81)These results show that g = f (1) and r = f (2) was already a Helmholtz Decomposition of f . The totalpotentials are: F i j ( x ) = M i j x j , (82) G ( x ) = n (cid:88) k = M kk x k , (83) R i j ( x ) = M i j x i − M ji x j . (84)
6. Discussion
The theorems presented in this paper allows to analytically derive a Helmholtz Decompositionand scalar and rotation potentials for many unbounded vector fields on unbounded domains. Thisdecomposition is not unique, because adding a harmonic function H ( x ) with ∆ H ( x ) = r to maintain r + g = f yields g (cid:48) ( x ) = grad( G ( x ) + H ( x )) and r (cid:48) ( x ) = r ( x ) − grad H ( x ) (85)that are also a Helmholtz Decomposition of f ( x ). By the choice of this harmonic function H ,additional boundary conditions can be satisfied. In some cases, harmonic functions can be chosensuch that the decomposition is orthogonal with (cid:104) g ( x ) , r ( x ) (cid:105) = ∀ x (Suda, 2020).The method is applicable to all vector fields described by multivariate polynomial functions. Eachof the terms of a polynomial satisfies both Conditions 2a and 2b with C k = λ k not higher thanthe total degree (sum of exponents) of the polynomial – but the choice of the appropriate method Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . . matters: Applying Condition 2a is practical for vector fields such as f ( x ) = [ x x , , f ( x ) = [ x x , , λ k = ffi cient, while the worse approach requires λ k =
51. This is similar to the choice forthe integration by parts which function to integrate and which to di ff erentiate.Further applications includes exponential functions, sine or cosine functions, and their product withpolynomials. For example: f ( x ) = [cos( wx ) exp( ax ) , ,
0] with u ( x ) = cos( wx ) and v ( x (cid:44) ) = exp( ax ). The appendix explains the process step-by-step for some polynomials, exponentialfunctions, trigonometric functions, and two examples from complex system theory that require theapplication of the general theorem for linear combinations, the Rössler and Lorenz attractors.Thanks to its versatility and the possibility of obtaining analytical solutions for Helmholtz decom-position, this method may prove helpful for problems of vector analysis, theoretical physics, andcomplex systems theory. Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
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EG thanks Walter Zulehner from Johannes Kepler University Linz. OR thanks Ulrike Feudel andJan Freund from Carl von Ossietzky University of Oldenburg. Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
A. ExamplesA.1. Example 1 with Multipolynomial using Condition 2a f ( x ) = (cid:104) x a x x , , (cid:105) , with a ∈ R + . (86)This vector field satisfies Conditions 1 and 2a with u ( x ) = x a , A x u ( x ) = x a + a + , A px u ( x ) = x a + p a !( a + p )! , (87)and v ( x (cid:44) ) = x x , ∆ v ( x (cid:44) ) = x + x , ∆ v ( x (cid:44) ) = , ∆ v ( x (cid:44) ) = . (88)As ∆ v ( x (cid:44) ) =
0, chose λ = C = F ( x ) = G ( x ) = x a + a + · x x − x a + a !( a + · (2 x + x ) + x a + a !( a + · , (89) F i ( x ) = R i ( x ) = − R i ( x ) = − x a + ( a + a + · ∂ x i ( x x ) + x a + a !( a + · x i , (90) F i j ( x ) = R i j ( x ) = . (91)The Helmholtz Decomposition f ( x ) = g ( x ) + r ( x ) is given by: g ( x ) = x a x x − x a + ( a + a + · (2 x + x ) + x a + a !( a + · x a + a + · x x − x a + a !( a + · x x a + a + · x x − x a + a !( a + · x , (92) r ( x ) = + x a + ( a + a + · (2 x ) − x a + a !( a + · + x a + ( a + a + · (2 x ) − x a + a !( a + · − x a + a + · (2 x x ) + x a + a !( a + · x − x a + a + · (2 x x ) + x a + a !( a + · x . (93) A.2. Example 2 with Multipolynomial using Condition 2b f ( x ) = (cid:104) x x a , , (cid:105) with u ( x ) = x and v ( x ) = x a . (94)As ∂ x u ( x ) = ∂ x u ( x ) =
0, Conditions 1 and 2b are satisfied with λ = C =
0. Thisimplies: F ( x ) = G ( x ) = x x a + ( a + a + , (95) F ( x ) = R ( x ) = − R ( x ) = x a + a !( a + − x x a + a + , (96) F i j ( x ) = R i j ( x ) = f ( x ) = g ( x ) + r ( x ) is given by: g ( x ) = + x a + ( a + a + , + x x a + a + , , (98) r ( x ) = x x a − x a + ( a + a + , − x x a + a + , . (99) Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
A.3. Example 3 with cosine and exponential function using Condition 2a f ( x ) = (cid:104) cos( wx ) exp( ax ) , , (cid:105) . (100)This implies u ( x ) = cos( wx ) and v ( x (cid:44) ) = exp( ax ). For λ = C = a w , Condition 2a issatisfied:1 w cos( wx ) a exp( az ) = ( − λ (cid:16) A λ x u ( x ) (cid:17)(cid:16) ∆ λ v ( x (cid:44) ) (cid:17) = C u ( x ) v ( x (cid:44) ) . (101)Therefore, for a (cid:44) w , set: F ( x ) = G ( x ) = / w − a / w sin( wx ) exp( ax ) , (102) F ( x ) = R ( x ) = − R ( x ) = − a / w − a / w cos( wx ) exp( ax ) , (103) F i j ( x ) = R i j ( x ) = f ( x ) = g ( x ) + r ( x ) is given by: g ( x ) = (cid:34) − a / w cos( wx ) exp( ax ) , , + a / w − a / w sin( wx ) exp( ax ) (cid:35) , (105) r ( x ) = (cid:34) − a / w − a / w cos( wx ) exp( ax ) , , − a / w − a / w sin( wx ) exp( ax ) (cid:35) . (106) A.4. Example 4: Rössler attractor
The Rössler attractor, a classic example from complex system theory (Peitgen et al., 1992; Rössler,1976), is given by: f ( x ) = (cid:2) − ( x + x ) , x + ax , b + ( x − c ) x (cid:3) . (107)Splitting the sums, deriving the potentials, and adding them yields: G ( x ) = a x + bx − c x + x x , (108) R ( x ) = − x − x , (109) R ( x ) = − x − x , (110) R ( x ) = . (111)The Helmholtz Decomposition f ( x ) = g ( x ) + r ( x ) is given by: g ( x ) = grad G ( x ) = (cid:2) + x , + ax , b − cx + x x (cid:3) , (112) r ( x ) = ROT R ( x ) = (cid:2) − x − x − x , + x , (cid:3) . (113) Glötzl, Richters: Analytical Helmholtz Decomposition and Potential Functions . . .
A.5. Example 5: Lorenz attractor
The Lorenz attractor, another classic example from complex system theory (Lorenz, 1963; Peitgenet al., 1992), is given by: f ( x ) = (cid:2) a ( x − x ) , x ( b − x ) − x , x x − cx (cid:3) . (114)Splitting the sums, deriving the potentials, and adding them yields: G ( x ) = − a x − x − c x , (115) R ( x ) = − a x + b x − x x − x , (116) R ( x ) = x x + x , (117) R ( x ) = x x + x x + x . (118)The Helmholtz Decomposition f ( x ) = g ( x ) + r ( x ) is given by: g ( x ) = grad G ( x ) = (cid:2) − ax , − x , − cx (cid:3) , (119) r ( x ) = ROT R ( x ) = (cid:2) ax , bx − x x , x x (cid:3) . (120)The Lorenz system contains a square gradient potential, pushing the dynamics into the directionof the origin. This is responsible for the stable fixed point at the origin for some parameters. Ifthe rotation field rr