Local Maxima of the Potential Energy on Spheres
Let S d be a unit sphere in ℝ d+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N ∈ S d , we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ i≠j ||x i − x j ||−α . The following theorem is proved: for α ≥ d − 2, the functional E α (x 1,x 2, ....
Gespeichert in:
| Datum: | 2013 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2013
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2520 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508430017495040 |
|---|---|
| author | Radchenko, D. V. Радченко, Д. В. |
| author_facet | Radchenko, D. V. Радченко, Д. В. |
| author_sort | Radchenko, D. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:17:19Z |
| description | Let S d be a unit sphere in ℝ d+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N ∈ S d , we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ i≠j ||x i − x j ||−α . The following theorem is proved: for α ≥ d − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima. |
| first_indexed | 2026-03-24T02:25:05Z |
| format | Article |
| fulltext |
UDC 517.5
D. V. Radchenko (Kyiv Nat. Taras Shevchenko Univ.)
LOCAL MAXIMA OF THE POTENTIAL ENERGY ON SPHERES
ЛОКАЛЬНI МАКСИМУМИ ПОТЕНЦIАЛЬНОЇ ЕНЕРГIЇ НА СФЕРАХ
Let Sd be a unit sphere in Rd+1, and let α be a positive real number. For pairwise different points x1, x2, . . . , xN ∈ Sd,
we consider a functional Eα(x1, x2, . . . , xN ) =
∑
i 6=j
‖xi − xj‖−α. The following theorem is proved: for α ≥ d − 2,
the functional Eα(x1, x2, . . . , xN ) does not have local maxima.
Нехай Sd — одинична сфера в Rd+1, а α — додатне число. Для попарно рiзних точок x1, x2, . . . , xN ∈ Sd
розглядається функцiонал Eα(x1, x2, . . . , xN ) =
∑
i 6=j
‖xi − xj‖−α. Доведено, що при α ≥ d − 2 функцiонал
Eα(x1, x2, . . . , xN ) не має локальних максимумiв.
1. Introduction. In this short note we will prove that certain potential energy functionals on sphere
attain no local maxima. This partially answers the question that Professor Edward Saff asked on the
conference “Optimal Configurations on the Sphere and Other Manifolds” at Vanderbilt University in
2010.
For d ∈ N, denote by Sd a unit sphere in Rd+1. For α > 0 and any configuration of N ≥ 2
distinct points x1, x2, . . . , xN ∈ Sd consider the following energy functional:
Eα(x1, x2, . . . , xN ) =
∑
i 6=j
1
‖xi − xj‖α
,
where ‖ · ‖ is the Euclidean norm on Rd+1.
For d = 2, α = 1 this functional has a physical interpretation as the electrostatic potential energy
of a system containing N equally charged particles on the sphere.
The problem of finding configurations which minimize these functionals is closely related to the
problem of finding uniformly distributed collections of points on sphere, in particular, of spherical
designs (see [3, 1]), as well as to the problem of finding optimal spherical codes (see [2]).
It is clear that for each d ∈ N, N ≥ 2, and α > 0 there exists a configuration of N points on
sphere Sd at which Eα has a local (and even global) minimum. In his closing speech at the conference
“Optimal Configurations on the Sphere and Other Manifolds” Professor Saff asked whether Eα can
have local maxima. We prove the following theorem, which says that for sufficiently large α this is
impossible.
Theorem 1. For positive α ≥ d− 2 the functional Eα(x1, x2, . . . , xN ) has no local maxima.
2. Proof of Theorem 1. For convenience, we rescale the energy by a factor of 2α/2. Let r = α/2
and denote gr(t) = (1− t)−r. Then
2r
‖xi − xj‖α
= gr(〈xi, xj〉),
where 〈·, ·〉 is the usual inner product on Rd+1. Therefore, the energy functional can be written as
c© D. V. RADCHENKO, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10 1427
1428 D. V. RADCHENKO
Eα(x1, . . . , xN ) =
∑
i 6=j
gr(〈xi, xj〉).
Introduce arbitrary vectors h1, h2, . . . , hN orthogonal to corresponding xi, (i.e. 〈xi, hi〉 = 0) and
consider the function f : R→ R defined by
f(t) = Eα
(
x1 + th1
‖x1 + th1‖
, . . . ,
xN + thN
‖xN + thN‖
)
.
If Eα attains a local maximum at x1, . . . , xN , then we must have f ′(0) = 0 and f ′′(0) ≤ 0. The
expression for second derivative is
f ′′(0) =
∑
i 6=j
[
g′′r (〈xi, xj〉)(〈xi, hj〉+ 〈xj , hi〉)2+
+ g′r(〈xi, xj〉)
(
2〈hi, hj〉 − (‖hi‖2 + ‖hj‖2)〈xi, xj〉
)]
. (1)
Therefore, in order to prove that our energy has no local maxima it is sufficient to find hi such that
(1) is strictly positive. To do so, take h2 = h3 = . . . = hN = 0 and h1 = h, where ‖h‖ = 1. Then
f ′′(0)/2 is equal to
N∑
j=2
[
g′′r (〈x1, xj〉)〈xj , h〉2 − g′r(〈x1, xj〉)〈x1, xj〉
]
. (2)
Suppose that (2) is nonpositive for all h orthogonal to x1. Then the average value of (2) over
all such h is also nonpositive. More specifically, let H = {h ∈ Sd : 〈x1, h〉 = 0}, then H is a
(d − 1)-dimensional sphere, and we take µd−1 to be the normalized Lebesgue measure on H. We
have ∫
H
〈xj , h〉2dµd−1(h) =
∫
H
〈xj − x1〈x1, xj〉, h〉2dµd−1(h) =
1− 〈xj , x1〉2
d
,
because x′j = xj − x1〈x1, xj〉 is orthogonal to x1 and ‖x′j‖2 = 1− 〈xj , x1〉2. Therefore, integrating
(2) over H with respect to µd−1(h) gives us
N∑
j=2
(
g′′r (〈x1, xj〉)
1− 〈x1, xj〉2
d
− g′r(〈x1, xj〉)〈x1, xj〉
)
≤ 0. (3)
After substituting gr(t) = (1− t)−r into (3) we get
N∑
j=2
(
r(r + 1)(1 + 〈x1, xj〉)
d(1− 〈x1, xj〉)r+1
− r〈x1, xj〉
(1− 〈x1, xj〉)r+1
)
≤ 0,
or, equivalently,
N∑
j=2
r
(r + 1) + (r + 1− d)〈x1, xj〉
d(1− 〈x1, xj〉)r+1
≤ 0. (4)
Since α ≥ d− 2, we have |r+1− d| ≤ r+1 and hence every term on the left of (4) is nonnegative.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
LOCAL MAXIMA OF THE POTENTIAL ENERGY ON SPHERES 1429
In fact, we have 〈x1, xj〉 < 1, so every term is strictly positive, and therefore the sum on the left of
(4) must be strictly positive. This contradiction concludes the proof.
1. Bondarenko A., Viazovska M. Spherical designs via Brouwer fixed point theorem // SIAM J. Discrete Math. – 2010. –
24. – P. 207 – 217.
2. Cohn H., Kumar A. Universally optimal distribution of points on spheres // J. Amer. Math. Soc. – 2006. – 20, № 1. –
P. 99 – 148.
3. Saff E. B., Kuijlaars A. B. J. Distributing many points on a sphere // Math. Intelligencer. – 1997. – 19, № 1. – P. 5 – 11.
Received 16.10.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
|
| id | umjimathkievua-article-2520 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:05Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/4d/764c3d9b862f60e23541e95c978d484d.pdf |
| spelling | umjimathkievua-article-25202020-03-18T19:17:19Z Local Maxima of the Potential Energy on Spheres Локальні максимуми потенціальної енергії на сферах Radchenko, D. V. Радченко, Д. В. Let S d be a unit sphere in ℝ d+1, and let α be a positive real number. For pairwise different points x 1,x 2, . . . ,x N ∈ S d , we consider a functional E α (x 1,x 2, . . . ,x N ) = Σ i≠j ||x i − x j ||−α . The following theorem is proved: for α ≥ d − 2, the functional E α (x 1,x 2, . . . ,x N ) does not have local maxima. Нехай $S^d$ — одинична сфера в $ℝ^{d+1}$, а a — додатне число. Для попарно різних точок $x_1, x_2,...,x_N \in S^d$ розглядається функціонал $E_{α} (x_1, x_2, ... ,x_N) = Σ_{i≠j} ||x_i − x_j ||^{−α}$. Доведено, що при $α ≥ d − 2$ функціонал $E_{α} (x_1, x_2, ... ,x_N)$ не має локальних максимумів. Institute of Mathematics, NAS of Ukraine 2013-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2520 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 10 (2013); 1427–1429 Український математичний журнал; Том 65 № 10 (2013); 1427–1429 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2520/1804 https://umj.imath.kiev.ua/index.php/umj/article/view/2520/1805 Copyright (c) 2013 Radchenko D. V. |
| spellingShingle | Radchenko, D. V. Радченко, Д. В. Local Maxima of the Potential Energy on Spheres |
| title | Local Maxima of the Potential Energy on Spheres |
| title_alt | Локальні максимуми потенціальної енергії на сферах |
| title_full | Local Maxima of the Potential Energy on Spheres |
| title_fullStr | Local Maxima of the Potential Energy on Spheres |
| title_full_unstemmed | Local Maxima of the Potential Energy on Spheres |
| title_short | Local Maxima of the Potential Energy on Spheres |
| title_sort | local maxima of the potential energy on spheres |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2520 |
| work_keys_str_mv | AT radchenkodv localmaximaofthepotentialenergyonspheres AT radčenkodv localmaximaofthepotentialenergyonspheres AT radchenkodv lokalʹnímaksimumipotencíalʹnoíenergíínasferah AT radčenkodv lokalʹnímaksimumipotencíalʹnoíenergíínasferah |