Electrostatic Equilibria on the Unit Circle via Jacobi Polynomials
EELECTROSTATIC EQUILIBRIA ON THE UNIT CIRCLE VIA JACOBIPOLYNOMIALS
Kev Johnson & Brian Simanek
Abstract.
We use classical Jacobi polynomials to identify the equilibrium configurationsof charged particles confined to the unit circle. Our main result unifies two theorems froma 1986 paper of Forrester and Rogers. Introduction
The use of Jacobi polynomials to describe configurations of charged particles that are inelectrostatic equilibrium goes back at least to the work of Heine and Stieltjes in the 19thcentury (see [4, 9, 10, 11]). Their work considered particles of identical charge confined to aninterval in the real line. The key to the calculations is to relate the condition of being a criticalpoint of the appropriate Hamiltonian to the second order differential equation satisfied bythe polynomial whose zeros mark the equilibrium points (see [12]). In the case of n particlesconfined to an interval with charged particles fixed at the endpoints, the relevant secondorder differential equation is precisely the ODE satisfied by the degree n Jacobi polynomial P ( α,β ) n ( x ), namely(1) (1 − x ) y (cid:48)(cid:48) + ( β − α − ( α + β + 2) x ) y (cid:48) + n ( n + α + β + 1) y = 0 , where the real numbers α and β are related to the magnitude of the fixed charges at theendpoints of the interval. Many variations and generalizations of Stieltjes’ work have beenrealized since his original papers (see for example [5, 6]).It was approximately 100 years before the work of Heine and Stieltjes was adapted tothe setting of the unit circle by Forrester and Rogers in [1]. In that paper, the authorsstudied highly symmetric configurations of charged particles that are on the unit circle andin electrostatic equilibrium, meaning the total force on each particle is normal to the circleat its location. They described the equilibrium configurations in terms of the zeros of theappropriate Jacobi polynomials. Our main result (Theorem 1 below) will generalize theresults from that paper by allowing for a broader collection of configurations and charges.Since we will be working with the two-dimensional electrostatic interaction, we will con-sider Hamiltonians of the form(2) H ( { t j } Mj =1 ) = (cid:88) ≤ j The Hamiltonian ˜ H attains its maximum on S precisely when the points { e iφ j } mj =1 mark the m th roots of , the points { e iψ j } mj =1 mark the m th roots of − , and thepoints { e iθ j } mnj =1 mark the zeros of the polynomial (3) z mn P ( p − / ,q − / n (cid:18) z m z m (cid:19) An example of a configuration described in Theorem 1 can be seen in Figure 1. Thespecial case of Theorem 1 in which m = 1 is precisely [1, Theorem 2.1]. The special case ofTheorem 1 in which p = q and m is a power of 2 is precisely [1, Theorem 4.1]. The proof ofTheorem 1 will require some intermediate steps where we consider related Hamiltonians, allof the form (2) for an appropriate choice of parameters. More precisely, we will proceed byusing the ODE (1) to find a critical point of the Hamiltonian ˜ H , which we will deduce is themaximizer from a uniqueness result that we prove in Section 2. To find the critical point, wewill first consider the case in which the particles with charge p and q are fixed (see Section3) and then use a symmetry argument to handle the general case in Section 4. ●● ●●●● ●●●●●● ●●●● ●●●● ●● ●●●● ●● ●●●● ●●●●●● ●● ●●●● ●● ●●●● ■■■ ■ ■◆◆ ◆ ◆◆ - - - - Figure 1. The equilibrium configuration described by Theorem 1 when n =5, m = 5, p = 2, and q = 2 . 5. The squares mark the 5 th roots of unity,the diamonds mark the 5 th roots of − 1, and the circles mark the roots of thepolynomial (3). 2. Critical Points of H Our main result of this section is a uniqueness result that applies to all Hamiltonians H of the form (2). We will apply it in several special cases in later sections. Theorem 2. The Hamiltonian H defined in (2) has a unique critical point on each connectedcomponent of the domain on which H is finite.Proof. We follow the method used to prove similar statements in [5, 6]. Define the Hessian H of H to be H jk = ∂ H∂t j ∂t k . We will first show that −H is strictly positive definite, which will then imply that H isstrictly concave on each connected component of its domain (using the fact that every suchconnected component is a convex set; see [8, Theorem 1.5]). To this end, we calculate thepartial derivatives ∂H∂t k = M (cid:88) j =1 j (cid:54) = k σ ( e it j ) σ ( e it k )2 cot (cid:18) t k − t j (cid:19) + K (cid:88) b =1 σ ( e iη b ) σ ( e it k )2 cot (cid:18) t k − η b (cid:19) ∂ H∂t j ∂t k = σ ( e it j ) σ ( e it k )4 csc (cid:18) t k − t j (cid:19) , j (cid:54) = k∂ H∂t k = − M (cid:88) j =1 j (cid:54) = k σ ( e it j ) σ ( e it k )4 csc (cid:18) t k − t j (cid:19) − K (cid:88) b =1 σ ( e iη b ) σ ( e it k )4 csc (cid:18) t k − η b (cid:19) . Observe that the negative of each diagonal entry is precisely the sum of the off diagonalentries of the same row plus a positive term. It follows that −H is diagonally dominant andhas only positive eigenvalues and is therefore strictly positive definite. It follows that H isstrictly concave on each connected component of its domain.Notice that H ( x ) approaches −∞ as x approaches the boundary of a connected componentof the domain of H . It follows from the upper semicontinuity of H that H attains a maximumon every such connected component and therefore must have a critical point on every suchconnected component. Uniqueness of this critical point follows from the strict concavity justproven. (cid:3) Recall the convention that if a particle of charge q is located at a point a ∈ C and aparticle of charge p is located at a point b ∈ C , then the force on the particle at b due tothe particle at a is 2 pq/ (¯ b − ¯ a ) (as in [2, 3, 7]). With this convention, we have the followinglemma relating critical points of general Hamiltonians of the form (2) to the condition ofelectrostatic equilibrium (see also [2]). Lemma 3. For the Hamiltonian H from (2) , it holds that ∂H∂t k ( { t ∗ j } Mj =1 ) = 0 if and only if (4) M (cid:88) j =1 j (cid:54) = k σ ( e it ∗ j ) σ ( e it ∗ k ) e it ∗ k − e it ∗ j + K (cid:88) j =1 σ ( e iη j ) σ ( e it ∗ k ) e it ∗ k − e iη j = e − it ∗ k σ ( e it ∗ k ) M (cid:88) j =1 j (cid:54) = k σ ( e it ∗ j )2 + K (cid:88) b =1 σ ( e iη b )2 . Proof. We have already seen that ∂H∂t k = M (cid:88) j =1 j (cid:54) = k σ ( e it j ) σ ( e it k )2 cot (cid:18) t k − t j (cid:19) + K (cid:88) b =1 σ ( e iη b ) σ ( e it k )2 cot (cid:18) t k − η b (cid:19) We can rewrite this as ∂H∂t k = M (cid:88) j =1 j (cid:54) = k σ ( e it j ) σ ( e it k ) e it k e it k − e it j − M (cid:88) j =1 j (cid:54) = k σ ( e it j ) σ ( e it k )2 + K (cid:88) b =1 σ ( e iη b ) σ ( e it k ) e it k e it k − e iη b − K (cid:88) b =1 σ ( e iη b ) σ ( e it k )2and the desired result follows. (cid:3) It follows from Lemma 3 that { t ∗ j } Mj =1 is a critical point of H if and only if we have equalityin (4) for all k = 1 , , . . . , M . For future reference, notice that the expression(5) M (cid:88) j =1 j (cid:54) = k σ ( e it ∗ j )2 + K (cid:88) b =1 σ ( e iη b )2on the right-hand side of (4) is one half of the sum of the charges on all of the particles inthe system except the one at e it ∗ k .3. p and q Charges Fixed In this section, we will take a preliminary step towards the proof of Theorem 1 and considerthe Hamiltonian ˆ H given byˆ H (cid:0) { θ j } mnj =1 (cid:1) = p m (cid:88) j =1 2 mn (cid:88) k =1 log | e πij/m − e iθ k | + q m (cid:88) j =1 2 mn (cid:88) k =1 log | e (2 j +1) iπ/m − e iθ k | + (cid:88) ≤ k The unique configuration that maximizes ˆ H on ˆ S occurs when the points { e iθ j } mnj =1 mark the zeros of the polynomial in (3) .Proof. Notice that the Hamiltonian ˆ H is of the form H from (2) with M = 2 mn ; K = 2 m ; { e iη j } mj =1 equal to the roots of z m − σ ( e iθ k ) = 1 for all k = 1 , . . . , mn ; σ ( e πij/m ) = p ;and σ ( e (2 j +1) iπ/m ) = q for all j = 1 , . . . , m . By Theorem 2, it will suffice to show that thezeros of the polynomial in (3) form a critical point of ˆ H on ˆ S because the maximum mustoccur at a critical point. Let us define the polynomial Q nm ( z ) to be the polynomial given in (3). It follows that(where we abbreviate P ( α,β ) n by P n ) Q (cid:48) nm ( z ) = nmz nm − P n (cid:18) z m z m (cid:19) + mz nm − (cid:18) z m − z m (cid:19) P (cid:48) n (cid:18) z m z m (cid:19) Q (cid:48)(cid:48) nm ( z ) = nm ( nm − z nm − P n (cid:18) z m z m (cid:19) + (2 nm − m ) z nm − (cid:18) z m − z m (cid:19) P (cid:48) n (cid:18) z m z m (cid:19) + m z mn − (cid:18) z m z m (cid:19) P (cid:48) n (cid:18) z m z m (cid:19) + m z mn − (cid:18) z m − z m (cid:19) P (cid:48)(cid:48) n (cid:18) z m z m (cid:19) Using these calculations and the differential equation (1), one can verify that Q nm satisfiesthe ODE y (cid:48)(cid:48) + (cid:20) mz m − (cid:18) α + 1 z m − − α + β + 2 nz m + 2 β + 1 z m + 1 (cid:19) − m − z (cid:21) y (cid:48) ( z ) + ( mz m − ) T ( z m ) y ( z ) = 0 , where T ( z ) = − n (cid:20) α + β + 1 z − β − α ) z − ( α + β + 1)( z + 1) z ( z − (cid:21) We see that the only poles of T are at 0, 1 and − Q nm and the polesof T ( z m ) are disjoint sets. We also notice that m (2 α + 1) z m − z m − m (cid:88) j =1 α + 1 z − e πij/m , m (2 β + 1) z m − z m + 1 = m (cid:88) j =1 β + 1 z − e πi (2 j +1) /m Thus, if Q nm ( e iθ ∗ j ) = 0 for j = 1 , , . . . , mn , then it holds that(6) mn (cid:88) k =1 k (cid:54) = j e iθ ∗ j − e iθ ∗ k + m (cid:88) j =1 α + 1 e iθ ∗ j − e iφ j − m ( α + β + 2 n ) + m − e iθ ∗ j + m (cid:88) j =1 β + 1 e iθ ∗ j − e iψ j = 0 , where we used the identity Q (cid:48)(cid:48) ( e iθ ∗ j ) Q (cid:48) ( e iθ ∗ j ) = mn (cid:88) k =1 k (cid:54) = j e iθ ∗ j − e iθ ∗ k . Now set p = α + 1 / q = β + 1 / mn (cid:88) k =1 k (cid:54) = j σ ( e iθ ∗ k )2 + m (cid:88) k =1 σ ( e kiπ/m )2 + m (cid:88) k =1 σ ( e (2 k +1) iπ/m )2 = 2 mn − m ( α + β + 1)2Since this is true for every j = 1 , . . . , mn , Lemma 3 and (6) show that the zeros of Q nm form a critical point of ˆ H . By Theorem 2, this is the only critical point on ˆ S and hence isthe maximizing configuration. (cid:3) p and q Charges Mobile Now we will consider the full Hamiltonian ˜ H . This Hamiltonian is of the form H with K = 1, η = 0 and M = 2 mn + 2 m − 1, and with the t (cid:48) j s denoting the arguments of all ofthe particles in the system other than the particle at 1. Proof of Theorem 1. By Theorem 2, it suffices to show that the suggested configuration isa critical point of ˜ H . Let { e iθ ∗ j } mnj =0 denote the zeros of Q nm from the previous section. Weknow from Proposition 1 that ∂∂θ k ˜ H (cid:32)(cid:26) jπm (cid:27) m − j =0 , (cid:26) (2 j + 1) πm (cid:27) m − j =0 , { θ ∗ j } mnj =1 (cid:33) = 0for all k = 1 , , . . . , mn . It remains to check that the partial derivatives with respect toeach φ k ( k = 2 , . . . , m ) and ψ k ( k = 1 , . . . , m ) vanish at this configuration and for this wewill use Lemma 3.From symmetry, we know that in this configuration, the sum of the forces on each particleof charge p or q is radial at that point. Also, the magnitude of the force on all particles ofcharge p is the same and the magnitude of the force on all particles of charge q is the same.This means that if e iφ j = e iπ ( j − /m and e iψ j = e iπ (2 j − /m , then for each k = 1 , , . . . , m there are real constants C and C (cid:48) so that mn (cid:88) j =1 pe iφ k − e iθ ∗ j + m (cid:88) j =1 pqe iφ k − e iψ j + m (cid:88) j =1 j (cid:54) = k p e iφ k − e iφ j = Ce iφ k mn (cid:88) j =1 qe iψ k − e iθ ∗ j + m (cid:88) j =1 pqe iψ k − e iφ j + m (cid:88) j =1 j (cid:54) = k q e iψ k − e iψ j = C (cid:48) e iψ k We can rewrite these expressions as2 p Q (cid:48) nm ( e iφ k ) Q nm ( e iφ k ) + 2 pq D (cid:48) m ( e iφ k ) D m ( e iφ k ) + p B (cid:48)(cid:48) m ( e iφ k ) B (cid:48) m ( e iφ k ) = Ce iφ k q Q (cid:48) nm ( e iψ k ) Q nm ( e iψ k ) + 2 pq B (cid:48) m ( e iψ k ) B m ( e iψ k ) + q D (cid:48)(cid:48) m ( e iψ k ) D (cid:48) m ( e iψ k ) = C (cid:48) e iψ k , where B m ( z ) = z m − D m ( z ) = z m + 1. The above expressions simplify to2 pnm + pqm + p ( m − 1) = C qnm + pqm + q ( m − 1) = C (cid:48) This shows that we can write mn (cid:88) j =1 pe iφ k − e iθ ∗ j + m (cid:88) j =1 pqe iφ k − e iψ j + m (cid:88) j =1 j (cid:54) = k p e iφ k − e iφ j = e − iφ k pnm + pqm + p ( m − mn (cid:88) j =1 qe iψ k − e iθ ∗ j + m (cid:88) j =1 pqe iψ k − e iφ j + m (cid:88) j =1 j (cid:54) = k q e iψ k − e iψ j = e − iψ k qnm + pqm + q ( m − k = 1 , . . . , m . At e iφ k , the sum of the charges on all of the other particles is 2 mn + mq +( m − p . At e iψ k , the sum of the charges on all of the other particles is 2 mn + mp + ( m − q .We can now apply Lemma 3 to conclude that the suggested configuration is a criticalpoint of ˜ H on S (note that to reach this conclusion, we do not need to apply (7) when k = 1because we assume φ = 0 always). By Theorem 2, this is the only critical point in S andhence must be the maximizing configuration. (cid:3) References [1] P. Forrester and J. B. Rogers, Electrostatics and the zeros of the classical polynomials , SIAM J. Math.Anal. 17 (1986), 461–468.[2] A. Grinshpan, A minimum energy problem and Dirichlet spaces , Proc. Amer. Math. Soc. 130 (2002),no. 2, 453–460.[3] A. Grinshpan, Electrostatics, hyperbolic geometry and wandering vectors , J. London Math. Soc. (2) 69(2004), no. 1, 169–182.[4] E. Heine, Handbruch der Kugelfunktionen , vol. II, second ed. G. Reimer, Berlin, 1878.[5] M. E. H. Ismail, An electrostatics model for zeros of general orthogonal polynomials , Pacific J. Math.193 (2000), no. 2, 355–369.[6] F. Marcell´an, A. Mart´ınez-Finkelshtein, and P. Mart´ınez-Gonz´alez, Electrostatic models for zeros ofpolynomials: old, new, and some open problems , J. Comput. Appl. Math. 207 (2007), no. 2, 258–272.[7] B. Simanek, An electrostatic interpretation of the zeros of paraorthogonal polynomials on the unit circle ,SIAM J. Math. Anal. 48 (2016), no. 3, 2250–2268.[8] B. Simon, Convexity: An analytic viewpoint , Cambridge Tracts in Mathematics 187, Cambridge Uni-versity Press, Cambridge, 2011.[9] T. Stieltjes, Sur certains polynˆomes qui v´erifient une ´equation diff´erentielle lin´eaire du second ordre etsur la theorie des fonctions de lam´e , Acta Math. 6 (1885), no. 1, 321–326.[10] T. Stieltjes, Sur quelques th´eor`ems d’alg`ebre , Comptes Rendus de l’Academie des Sciences, Paris 100(1885), 439–440; Oeuvres Compl`etes, Vol. 1, 440–441.[11] T. Stieltjes, Sur les polynˆomes de Jacobi , Comptes Rendus de l’Academie des Sciences, Paris, 100 (1885),620–622; Oevres Compl`etes, Vol. 1, 442–444.[12] G. Szeg˝o,