Newton Algorithm on Constraint Manifolds and the 5-electron Thomson problem

0.46 MB·

by  Petre Birtea

Newton Algorithm on Constraint Manifolds and the 5-electron Thomson
problem
Petre Birtea, Dan Coma˘nescu
Department of Mathematics, West University of Timi¸soara
Bd. V. Paˆrvan, No 4, 300223 Timi¸soara, Romaˆnia
[e-mail], [e-mail]
6
1
0 Abstract
2
We give a description of numerical Newton algorithm on a constraint manifold using only the

n ambient coordinates (usually Euclidean coordinates) and the geometry of the constraint manifold.
a We apply the numerical Newton algorithm on a sphere in order to find the critical configurations
J
of the 5-electron Thomson problem. As a result, we find a new critical configuration of a regular

9 pentagonal type. We also make an analytical study of the critical configurations found previously
1
and determine their nature using Morse-Bott theory. Last section contains an analytical study of

critical configurations for Riesz s-energy of 5-electron on a sphere and their bifurcation behavior is
]
h pointed out.
p
MSC: 49M15, 53-XX
-
h Keywords: Newton algorithm, constraint manifold, Hessian operator, Morse theory, 5-electron
at Thomson problem, Riesz s-energy
m
[ 1 Introduction

1
v Newton algorithm is a efficient numerical iterative method that can be applied for finding the critical
8 pointsofthecostfunction. Thistypeofalgorithmusessecond-orderinformationaboutthecostfunction,
2
fact that guarantees super-linear convergence (in cases it converges). Initially it was constructed on
8
vector spaces, but later was adapted to manifolds for practical reasons, see [1] for a careful and detailed
4
0 discussion of the geometry involved in such a context.
1. Let (S,g ) be a smooth Riemannian manifold and G(cid:101) : S → R be a smooth cost function. The
S
0 iterative scheme is given by, see [1]:
16 xn+1 =R(cid:101)xn(v(cid:101)xn), (1.1)
v: where R(cid:101) : TS → S is a smooth retraction and the tangent vector v(cid:101)xn ∈ TxnS is the solution of the
(contravariant) Newton equation
i
X
ar HG(cid:101)(xn)·v(cid:101)xn =−∇gSG(cid:101)(xn). (1.2)
In the above equation we have the Hessian operator HG(cid:101) : TS → TS. The link between the Hessian
operator HG(cid:101) and the associated symmetric bilinear form HessG(cid:101) :TS×TS →R is given by:
(cid:16) (cid:17)
HG(cid:101)(x)·v(cid:101)x =g(cid:93)(x) HessG(cid:101)(x)·v(cid:101)x ,
S
where g(cid:93)(x):T∗S →T S is the sharp operator in Riemannian geometry. The (contravariant) Newton
S x x
equation (1.2) can be rewritten as HessG(cid:101)(xn) · v(cid:101)xn = −gS(cid:91)(xn) · ∇gSG(cid:101)(xn), or equivalently as the
(covariant) Newton equation:
HessG(cid:101)(xn)·v(cid:101)xn =−dG(cid:101)(xn). (1.3)
1
To solve the above equation on the manifold S one needs to put local charts on the manifold in order
to make the computations and to have a good knowledge on the Riemannian geometry of the manifold.
Another way is to embed the manifold S in a larger manifold (usually an Euclidean space) and work
with local charts (Euclidean coordinates) on this ambient space. The details of these constructions
in the case when S is described by constraint functions are given in the Section 2. We end this sec-
tion presenting a new form of the Newton algorithm on constraint manifolds that we call Embedded
Newton algorithm. The construction relies on the formula, given in [6], for the Hessian of a cost
function restricted to a manifold defined by constraint functions, formula which involves only ambient
coordinates.
In Section 3 we apply the Embedded Newton algorithm to the 5-electron Thomson problem. We
find numerically the following critical configurations: bi-pyramidal configuration, square right pyramid
configuration and a new family of saddle critical points in the configuration of a regular pentagon. We
also carry out the analytical study of the three critical configurations discovered numerically. Each
type of critical configuration comes as differential curves of critical points for the embedded gradient
vector field introduced in [5] and [6]. We prove that all this three critical configurations are non-
degenerate in the sense of Morse-Bott theory.
InSection4weproceedwithananalyticalstudyoftheRieszs-energyof5-pointsonasphereproblem
which is a generalization of 5-electron Thomson problem. First we prove that the three types of critical
configurations of the 5-electron Thomson problem discussed in Section 3 remain critical configurations
for the Riesz s-energy problem. Various bifurcation phenomena do appear when s varies and we find
a new bifurcating value for s = 13.5204990011... when the square right pyramid goes from a saddle
critical point into a local minima. Some aspects of this bifurcation behavior have been previously
pointed out in [21] and [22]. Analyzing the nature of the three critical configurations for s in the
interval (13.5204990011..., 21.1471229401...), Mountain Pass type theorem suggest that there should
be another critical configuration with Morse index one. Indeed, we find the critical configuration of a
double-tetrahedronandprovethatithasMorseindexone. Thisconfigurationwaspreviouslydiscovered
in [22] for various values of s in this interval. We also prove that this configuration persists also for
s>21.1471229401....
2 Embedded Newton algorithm
Wewanttorewritethe(covariant)Newtonequation(1.3)onthemanifoldS intermsoftheRiemannian
geometry of an ambient space M. Assume that there exists an ambient smooth Riemannian manifold
(M,g) and a smooth map F:M →Rk such that the manifold S can be written as S =F−1(c), where
c ∈ Rk is a regular value of the map F. Furthermore, suppose that the induced metric on S by the
ambient metric g is the metric g , i.e. g =g . Let G:M →R be a smooth prolongation of the cost
S S |S
function G(cid:101).
We fix an adapted frame {b ,...,b ,∇F ,...,∇F } on the regular foliation induced by F. More
1 s 1 k
precisely, fori=1,swehavethatthevectorfieldb restrictedtothesubmanifoldS isatangentvector
i
field to this submanifold, i.e. b ∈ X(S) and ∇F ,...,∇F generate the normal subspace to the
i|S 1|S k|S
submanifold S.
Startingwiththeaboveadaptedframeweconsideraco-base{Θ ,...Θ ,dF ,...,dF }forthemodule
1 s 1 k
of 1-forms Λ1(M), where for i,j =1,s we have Θ (b )=δ and for α=1,k we have Θ (∇F )=0.
i j ij i α
On the manifold S the equation (1.3) is equivalent with
HessG(cid:101)(xn)(v(cid:101)xn,w(cid:101)xn)=−dG(cid:101)(xn)·w(cid:101)xn, ∀w(cid:101)xn ∈TxnS. (2.1)
Regarding the tangent vectors v ,w ∈ T S as vectors v ,w ∈ T S ⊂ T M in the ambient
(cid:101)xn (cid:101)xn xn xn xn xn xn
space and using the formula for the Hessian of a constrained function, see Theorem 2.1, formula 2.5
from [6] and the Appendix 1., the above equation can be rewritten in the equivalent form:
(cid:32) k (cid:33)
(cid:88)
HessG(x )− σ (x )HessF (x ) (v ,w )=−dG(x )·w , ∀w ∈T S ⊂T M, (2.2)
n α n α n xn xn n xn xn xn xn
α=1
2
where σ :M →R are the Lagrange multiplier functions, see Appendix 2. eq. (5.1).
α
In the adapted co-frame, we have
s k
(cid:88) (cid:88)
dG(x )= g (x )Θ (x )+ f (x )dF (x ),
n i n i n α n α n
i=1 α=1
and using the orthogonality between the vector fields ∇F and b , we obtain the formula for the
α j
coordinate functions
g (x )=dG(x )·b (x ).
i n n i n
Taking a tangent vector w ∈T S to the submanifold S written as a vector in the ambient space
M, we have w = (cid:80)s wj(cid:101)xbn (x )x.nConsequently, using again the orthogonality between the vector
xn j=1 xn j n
fields ∇F and b , for the right hand side of (2.2), we have
α j
s k s
(cid:88) (cid:88)(cid:88)
dG(x )·w = g (x )wj Θ (b )(x )+ f (x )wj dF (b )(x )
n xn i n xn i j n α n xn α j n
i,j=1 α=1j=1
s k s
(cid:88) (cid:88)(cid:88)
= g (x )wj δ + f (x )wj g(∇F (x ),b (x ))
i n xn ij α n xn α n j n
i,j=1 α=1j=1
s
(cid:88)
= g (x )wj .
i n xn
i=1
In the adapted co-frame, we have
k s s k k
(cid:88) (cid:88) (cid:88)(cid:88) (cid:88)
HessG− σ HessF = h Θ ⊗Θ + (h Θ ⊗dF +h dF ⊗Θ )+ h dF ⊗dF .
α α ij i j iα i α αi α i αβ α β
α=1 i,j=1 i=1α=1 α,β=1
Consequently,
(cid:32) k (cid:33) s
(cid:88) (cid:88)
HessG(x )− σ (x )HessF (x ) (v ,w )= h (x )vi wj ,
n α n α n xn xn ij n xn xn
α=1 i,j=1
where v =(cid:80)s vi b (x ) and
xn i=1 xn i n
(cid:32) k (cid:33)
(cid:88)
h (x )= HessG(x )− σ (x )HessF (x ) (b (x ),b (x )).
ij n n α n α n i n j n
α=1
Erasing the vector w , equation (2.2) is equivalent with the system of s equations with s unknown
xn
variables (v1 ,...,vs ):
xn xn  (cid:80)s h (x )vi =−g (x )
 i=1 i1 n xn 1 n
... (2.3)
 (cid:80)s h (x )vi =−g (x )
i=1 is n xn s n
Theconsiderationsabovearemeanttoconstructanumericalalgorithminordertosolvethefollowing
problem.
Problem: FindcriticalpointsforthesmoothfunctionG(cid:101) :S →R,whereS =F−1(c)isthepreimage
of a regular value for the smooth map F:M →Rk.
The following is the Newton algorithm on constraint manifolds written in the ambient coordinates
(usually Euclidean coordinates) on the manifold M (usually an Euclidean space).
3
Embedded Newton algorithm:
1. Consider a smooth prolongation G:M →R of the cost function G(cid:101) :S →R.
2. Construct an adapted frame {b ,...,b ,∇F ,...,∇F }, where the vector fields b ∈ TM are
1 s 1 k i
tangent to the submanifold S.
3. Compute the coordinate functions
g =dG·b , i∈1,s. (2.4)
i i
4. ComputetheLagrangemultiplierfunctions,seeAppendixeq. (5.1),σ :M →R,forα=1,k.
α
5. Compute the components of the Hessian matrix HessG(cid:101) of the cost function G(cid:101)
(cid:32) k (cid:33)
(cid:88)
h = HessG− σ HessF (b ,b ), i,j ∈1,s. (2.5)
ij α α i j
α=1
6. Choose a retraction R:T M →M such that for any v∈T S ⊂T M we have R (v)∈S.
x x x x
7. Input x ∈S and n=0.
0
8. repeat
• Solve the linear system with the unknowns (v1 ,...,vs ),
xn xn
(cid:80)s h (x )vi =−g (x )
 i=1 i1 n xn 1 n
... (2.6)
(cid:80)s h (x )vi =−g (x ).
i=1 is n xn s n
• Construct the line search vector
s
(cid:88)
v = vj b (x ).
xn xn j n
j=1
• Set x =R (v ).
n+1 xn xn
until xn+1 sufficiently minimizes G(cid:101).
3 5-electron Thomson problem
We will apply the above numerical algorithm to the problem of finding critical configurations for the
5-electron Thomson problem. The 5-electron Thomson problem is the following: consider 5 electrons
constrained on a unit sphere interacting through Coulomb force. Which are the configurations that
renders the Coulomb potential its minimum value? The mathematical 5-electron Thomson problem is:
(cid:88) 1
argmin , (3.1)
||pi||=1 ||p −p ||
i j
1≤i<j≤5
where p is the position vector of the electron P on the unit sphere.
i i
4
Without minimizing the generality of the problem we can suppose that the electron P is fixed in
5
the North pole, i.e. p :=(0,0,1). The phase space of the problem is the manifold
5
S :=S2×S2×S2×S2\{p:=(p ,p ,p ,p )|∃(i,j),1≤i<j ≤5, s.t.p =p }.
1 2 3 4 i j
The cost function G(cid:101) :S →R is a Coulomb potential and it is given by
(cid:88) 1
G(cid:101)(p):= .
||p −p ||
i j
1≤i<j≤5
The aim is to find all critical configurations of the above Coulomb potential. First we will pro-
ceed by constructing Embedded Newton algorithm for the constraint configuration space S. This will
give us numerically three types of critical configurations. We also make an analytical study of this
configurations.
3.1 Numerical study: Embedded Newton algorithm
We embed this optimization problem into R12 using the constraint functions F := (F ,F ,F ,F ) :
1 2 3 4
M →R4, where the components are F (p):= 1||p ||2, i=1,4 and the ambient space M is the open set
i 2 i
R12\{p|∃(i,j),1≤i<j ≤5, s.t.p =p }. The extended cost function G:M →R is given by
i j
(cid:88) 1
G(p):= .
||p −p ||
i j
1≤i<j≤5
On the unit sphere S2 ⊂ R3 we can construct a local frame using the North pole stereographic
projection: e (x,y,z) := (1 − z − x2,−xy,x(1 − z)), e (x,y,z) := (−xy,1 − z − y2,y(1 − z)), see
1 2
Appendix 3..
A local frame on the constraint manifold S is given by:
b (p):=(e (p ),0,0,0), b (p):=(e (p ),0,0,0),
1 1 1 2 2 1
b (p):=(0,e (p ),0,0), b (p):=(0,e (p ),0,0),
3 1 2 4 2 2
b (p):=(0,0,e (p ),0), b (p):=(0,0,e (p ),0),
5 1 3 6 2 3
b (p):=(0,0,0,e (p )), b (p):=(0,0,0,e (p )).
7 1 4 8 2 4
The eight coordinate functions are given by:
∂G ∂G
g (p)=< (p),e (p )>, g (p)=< (p),e (p )>,
1 ∂p 1 1 2 ∂p 2 1
1 1
∂G ∂G
g (p)=< (p),e (p )>, g (p)=< (p),e (p )>,
3 ∂p 1 2 4 ∂p 2 2
2 2
∂G ∂G
g (p)=< (p),e (p )>, g (p)=< (p),e (p )>,
5 ∂p 1 3 6 ∂p 2 3
3 3
∂G ∂G
g (p)=< (p),e (p )>, g (p)=< (p),e (p )>.
7 ∂p 1 4 8 ∂p 2 4
4 4
By a straightforward computation we obtain the Lagrange multipliers functions:
∂G ∂G
σ (p):=< (p),p )>, σ (p):=< (p),p )>,
1 ∂p 1 2 ∂p 2
1 2
∂G ∂G
σ (p):=< (p),p )>, σ (p):=< (p),p )>.
3 ∂p 3 4 ∂p 4
3 4
5
We have all the necessary elements to compute the Hessian matrix of the constraint function G(cid:101)
applying formula (2.5). The expressions are long and irrelevant for what follows and we choose not to
write them down.
We need to construct a retract on M. A natural choice is R(cid:101)p :TpM →M given by
R(cid:101)p(vp):=(Rp1(vp1),...,Rp4(vp4)),
where vpi ∈TpiS2, vp =(vp1,...,vp4), and Rpi(vpi)= ||ppii++vvppii|| is the usual retraction on the sphere.
For a detailed discussion of retractions and their use see [1].
After running many numerical experiments using the Newton algorithm described above we find
three types of configurations that are critical points for the cost function G(cid:101).
1. Bi-pyramidal configuration. In this configuration two points are opposed one to another and
theotherthreelieonanequilateraltriangleinaplaneperpendicularonthediameterformedbythefirst
twopoints. Forourproblemwefindtwotypesofbi-pyramidalconfigurations, oneinwhichP isoneof
5
the vertex of the equilateral triangle and other one is when P is not a vertex of the equilateral triangle
5
butoneofdiametricallyopposedpoints. Thevalueofthecostfunction√G(cid:101) on√bothtypesofbi-pyramidal
configurations is the same and it is equal with the known value 1 +3 2+ 3=6.47461494....
2
2. Square right pyramid. This configuration has a right square at the base located at a distance
of 1+0.2432010309... from the apex. The value of the cost function G(cid:101) on this configuration is equal
with 6.483660519....
3. Regular pentagon. In this configurations the five electrons form a regular pentagon which is
inscribedinagreatcirclethatgoestoNorthpole. ThevalueofthecostfunctionG(cid:101) onthisconfiguration
is equal with 6.881909602....
The bi-pyramidal configurations are the well known local minima and indeed, we find that the
HessianmatrixofthecostfunctionG(cid:101) insuchaconfigurationhassevenstrictlypositiveeigenvaluesand
one eigenvalue is equal with zero. In [21] it has been given a rigorous computer-assisted proof that this
configurations are the only global minima. See also [16] for a computer-assisted proof for the solution
of a similar problem.
For the square right pyramid configuration we find that the Hessian matrix of the cost function G(cid:101)
in such a configuration has six strictly positive eigenvalues, one strictly negative eigenvalue, and one
eigenvalue is equal with zero. This configuration has been extensively studied in [8], [17], [22].
For the regular pentagon configuration we find that the Hessian matrix of the cost function G(cid:101) in
such a configuration has five strictly positive eigenvalues, two strictly negative eigenvalues, and one
eigenvalue is equal with zero.
We will show in the next section that the zero eigenvalue is in fact a degeneracy that can be
eliminated using Morse-Bott Theorem and that all three configurations are nondegenerate in the sense
of Morse-Bott theory.
3.2 Analytical study
In this section we will show that for all three types of configurations, each belongs to a differential
curve of critical configurations. Next we will compute the Hessian matrix of the cost function G(cid:101) in
this critical configurations and study their nature. For all this three types of critical configurations,
the Hessian matrix has one eigenvalue equal to zero. We will prove that this is due to the fact that
these configurations belong to one dimensional submanifolds of critical configurations. We will apply
the Morse-Bott theory to study them.
To prove that a configuration is critical we have to verify the equation dG(cid:101)(p) = 0. In order to do
this we have to introduce local coordinates on the manifold S, see [18]. An alternative method is to
embed S in the ambient space M and solve an equivalent equation written in Cartesian coordinates. In
[5] and [6] we have introduced and studied the vector field on the ambient space M
4
(cid:88)
∂G(p)=∇G(p)− σ (p)∇F (p), (3.2)
i i
i=1
6
which has the property that when restricted to the submanifold S it is equal with the gradient of the
cost function G(cid:101) with respect to the induced metric, i.e. (∂G)|S =∇gS G(cid:101). Due to this property we will
ind
call the vector field ∂G the embedded gradient vector field. To verify that a configuration is
critical for G(cid:101) is equivalent with verifying the equation ∂G(p)=0, which is an equation written
in Cartesian coordinates. This technique has also been used in [7].
For the case of the constraint functions F ,...,F , a direct computation shows that the equation
1 4
∂G(p)=0 is equivalent with the following system:
∂G ∂G
∂G(p)=0 ⇔ −< ,p >p =0, i=1,4. (3.3)
∂p ∂p i i
i i
FortheparticularcaseoftheCoulombpotentialwehavethefollowingsystemoftwelvescalarequations:
5
∂G(p)=0 ⇔ (cid:88) pj−<pj,pi >pi =0, i=1,4. (3.4)
||p −p ||3
i j
j=1,j(cid:54)=i
Next we will give analytical expressions for the three types of critical configurations previously
discovered numerically.
1. Bi-pyramidal configuration.
Let c:[−1,1]→M be the differential curve c(λ):=(P (λ),P (λ),P (λ),P (λ),P ), where
1 2 3 4 5
√ √ √
P (λ)=( 3λ,− 3 1−λ2,−1)
P1(λ)=(−2√3λ,√23√1−λ2,−21)
2 √ 2 2 2 . (3.5)
PP3((λλ))==((−√1−1−λ2λ,2λ,,−0)λ,0)
4
By direct computation we obtain that ∂G(c(λ)) = 0, for all λ ∈ [−1,1]. Consequently, c is a curve
of critical points for the Coulomb potential G(cid:101). Every point of this curve generates a bi-pyramidal
configuration with P being one of the vertex of the equilateral triangle.
5
Using formula (2.5) we obtain the eight eigenvalues of the Hessian matrix HessG(cid:101)(c(λ)) of the
Coulomb potential as follows:
λ =0
1
9 √ 1 15√ 1 (cid:113) √ √ √ √
λ = 2+ + 3+ 1717+72 2−270 2 3−120 3=2.297...
2 32 8 32 32
9 √ 1 15√ 1 (cid:113) √ √ √ √
λ = 2+ + 3− 1717+72 2−270 2 3−120 3=0.371...
3 32 8 32 32
1 9 √ 5 √ 1 (cid:113) √ √ √ √
λ = + 2+ 3+ 541+72 2−40 3−90 2 3=1.380...
4 8 32 32 32
1 9 √ 5 √ 1 (cid:113) √ √ √ √
λ = + 2+ 3− 541+72 2−40 3−90 2 3=0.206...
5 8 32 32 32
1√ 9√ 1(cid:113) √ √
λ = 3+ 2+ 93+18 2 3=3.487...
6 4 8 8
1√ 9√ 1(cid:113) √ √
λ = 3+ 2− 93+18 2 3=0.560...
7 4 8 8
√
9 2
λ = =3.181...
8 4
We prove that the eigenvalue λ = 0 is a manifestation of the Morse-Bott Theory, see [9] and [20].
1
The connected and compact set C := c([−1,1]) ⊂ S of critical points of Coulomb potential G(cid:101) is a
1-dimensional submanifold of S. We prove that Tc(λ)C = Ker[HessG(cid:101)(c(λ))], for all λ ∈ [−1,1]. For
7
this we write the tangent vector c(cid:48)(λ) ∈ R12 as a vector in T S (cid:39) R8, i.e. solving the equation
c(λ)
c(cid:48)(λ)=(cid:80)8 w (λ)b (c(λ)) for unknowns w . The solution is given by the tangent vector
i=1 i i i
√ √ √ √
3 3λ 3 3λ λ λ
w(λ)=( , √ ,− ,− √ ,−√ ,1,√ ,−1)∈T S.
3 3 1−λ2 3 3 1−λ2 1−λ2 1−λ2 c(λ)
AdirectcomputationshowsthatHessG(cid:101)(c(λ))·w(λ)=0, ∀λ∈(−1,1),whichistheconditionofMorse-
Bott Theorem, that is the eigenspace corresponding to the eigenvalue λ =0 is equal with the tangent
1
space of the submanifold of critical points C that generate the bi-pyramidal configuration. Because all
other eigenvalues are strictly positive we have that the bi-pyramidal configuration are non-degenerate
in the sense of Morse-Bott local minima.
Any relabeling in (3.5) will give connected, compact 1-dimensional submanifolds of critical points
that will generates the same bi-pyramidal configuration with P is one of the vertex of the equilateral
5
triangle. Their study is analogous, all of them being non-degenerate in the sense of Morse-Bott local
minima. Also the bi-pyramidal configurations where P is not one of the vertex of the equilateral
5
triangle come from 1-dimensional submanifolds of critical points and are non-degenerate in the sense of
Morse-Bott local minima.
2. Square right pyramid. As the numerical study shows we have a critical configuration of
squarerightpyramidwhereP istheapexandthebaseisatthedistance1+0.2432010309.... Weprove
5
analytically this is indeed true.
Let c :[0,2π]→M be the differential curve c (λ):=(Pα(λ),Pα(λ),Pα(λ),Pα(λ),P ), where
α α 1 2 3 4 5
√ √
Pα(λ)=( 1−α2cos(λ), 1−α2sin(λ),α)
Pα1(λ)=(√1−α2cos(λ+ π),√1−α2sin(λ+ π),α)
2 √ 2 √ 2 , (3.6)
PPα3α((λλ))==((√11−−αα22ccooss((λλ++π3π),),√11−−αα22sisnin(λ(λ++π)3,πα),)α)
4 2 2
and |α| ∈ [0,1) is the distance from the center of the sphere to the center of the square base of the
pyramid. By direct computation we obtain that
(cid:32) √ √
α 1−α2 1−α2
∂G(c (λ))=− √ Q(α) cos(λ),sin(λ),− ,−sin(λ),cos(λ),− ,
α 2 1−α2 α α
√ √ (cid:33)
1−α2 1−α2
−cos(λ),−sin(λ),− ,sin(λ),−cos(λ),− ,
α α
for all λ∈[0,2π], α∈(−1,1), and
Q(α)=(2−2α)−12 +2 (cid:0)2−2α2(cid:1)−21 α+α (cid:0)4−4α2(cid:1)−21 +(2−2α)−12 α.
Consequently,∂G(c (λ))=0iffQ(α)=0. Theonlysolutionintheinterval(−1,1)isα∗ =−0.24320103..
α
Usingformula(2.5)weobtaintheeighteigenvaluesoftheHessianmatrixHessG(cid:101)(cα∗(λ))oftheCoulomb
potential as follows:
λ =0; λ =−0.205..; λ =0.565..; λ =0.565..;λ =2.232..; λ =2.232..; λ =2.260..; λ =3.592..
1 2 3 4 5 6 7 8
As before, we show that we have a Morse-Bott non-degeneracy. A straightforward computation shows
(cid:16) (cid:17)
that HessG(c (λ))−(cid:80)4 σ (c (λ))HessF (c (λ))+ · c(cid:48) (λ) = 0, which implies that the tan-
α∗ i=1 i α∗ i α∗ α∗
gent vector c(cid:48) (λ) to the critical submanifold c ([0,2π]) is also an eigenvector corresponding to the
α∗ α∗
eigenvalue λ = 0. Thus the square right pyramid configurations are non-degenerate in the sense of
1
Morse-Bott saddle points for the Coulomb potential G(cid:101).
Anyrelabelingin(4.4)willgiveconnected,compact1-dimensionalsubmanifoldsofcriticalpointsthat
will generate the same square right pyramid configuration with P the apex. Their study is analogous,
5
8
all of them being non-degenerate in the sense of Morse-Bott saddle points. We also find square right
pyramid critical configurations where P is one of the vertex for the base square of the pyramid. These
5
configurations are also nondegenerate Morse-Bott saddle points with same value 6.483660519.... of the
Coulomb potential.
3. Regular pentagon. Thenumericalstudyshowsthatweobtainaconfigurationofcriticalpoints
in the shape of a regular pentagon sitting on the big circles that go through P that is fixed in the
5
North pole. This configuration seems to be new, at least the authors could not find it in the literature.
Let c:[0,2π]→M be the differential curve c(λ):=(P (λ),P (λ),P (λ),P (λ),P ), where
1 2 3 4 5
P (λ)=(−sin(λ)cos(π),−cos(λ)cos(π),sin(π))
P1(λ)=(−sin(λ)cos(31π0),−cos(λ)cos(130π),−si1n0(3π))
2 10 10 10 . (3.7)
PP3((λλ))==((ssiinn((λλ))ccooss((31ππ0)),,ccooss((λλ))ccooss((31ππ0)),,s−ins(inπ()31)π0))
4 10 10 10
By direct computation we obtain that ∂G(c(λ)) = 0, for all λ ∈ [0,2π]. Consequently, c is a
curve of critical points for the Coulomb potential G(cid:101). Every point of this curve generates a pentagonal
configuration.
Using formula (2.5) we obtain the eight eigenvalues of the Hessian matrix HessG(cid:101)(c(λ)) of the
Coulomb potential as follows:
λ =0;λ =−2.628..;λ =−0.453..;λ =0.490..;λ =1.084..;λ =1.992..;λ =4.932..;λ =11.156..
1 2 3 4 5 6 7 8
(cid:16) (cid:17)
A straightforward computation shows that HessG−(cid:80)4 σ HessF ·c(cid:48)(λ) = 0 which implies that
i=1 i i
thetangentvectorc(cid:48)(λ)tothecriticalsubmanifoldc([0,2π])isalsoaneigenvectorcorrespondingtothe
eigenvalue λ = 0. Thus the pentagonal configurations are non-degenerate in the sense of Morse-Bott
1
saddle points for the Coulomb potential G(cid:101).
4 Riesz s-energy of 5-points on a sphere
Ageneralizationof5-electronThomsonproblemisgivenbytheminimizationproblemofRieszs-energy
of 5-points on a sphere:
(cid:88) 1
argmin , (4.1)
||pi||=1 ||p −p ||s
i j
1≤i<j≤5
where we use the notations of Section 3 and s>0 represents for example the soft or strong repulsion in
VSEPR(ValenceShellElectronPairRepulsion)model, see[13], [14], and[15]. Amoregeneralproblem
offindingoptimalconfigurationsusinggeneralpotentialsond-dimensionalspheresisstudiedin[4],[10],
and [11].
InthissectionwestudythecriticalpointsoftheRieszs-energyof5pointsonthesphereG(cid:101)s :S →R
given by
(cid:88) 1
G(cid:101)s(p):= ||p −p ||s,
i j
1≤i<j≤5
where the constraint manifold is S := S2 ×S2 ×S2 ×S2\{p := (p ,p ,p ,p )|∃(i,j),1 ≤ i < j ≤
1 2 3 4
5, s.t.p =p }.
i j
Using the same embedding mechanism as in Section 3 we obtain that the critical points of the Riesz
s-energy G(cid:101)s are the solutions of the system of twelve scalar equations:
5
∂G (p)=0 ⇔ (cid:88) pj−<pj,pi >pi =0, i=1,4, (4.2)
s ||p −p ||s+2
i j
j=1,j(cid:54)=i
where Gs is the natural extension of G(cid:101)s on M =R12\{p|∃(i,j),1≤i<j ≤5, s.t.pi =pj}.
9
1. Bi-pyramidconfiguration. Bydirectcomputationweobtainthatthebi-pyramidconfiguration
(3.5) verifies equation (4.2) and consequently, it remains a critical configuration of G(cid:101)s for any s > 0
with
3 6 1
G(cid:101)s(bi-pyramid)= √3s + √2s + 2s.
Computing the Hessian of G(cid:101)s in this bi-pyramid configuration we obtain that for s<21.1471229401...
we have non-degenerate local minima in the sense of Morse-Bott. This has been previously noticed and
discussedin[21]and[22]. Fors>21.1471229401...weobtainnon-degeneratesaddlepointsinthesense
of Morse-Bott with two strictly negative eigenvalues, which shows that a bifurcation occurs in this type
of critical configuration.
2. Square right pyramid configuration. As in the case of Thomson problem s = 1 discussed
in the previous section we obtain that the equation (4.2) is equivalent for this configuration with the
following equation:
∂G (c (λ))=0 ⇔ T (α)=0, (4.3)
s α s
where
Ts(α)=(2−2α)−2s +2 (cid:0)2−2α2(cid:1)−2s α+α (cid:0)4−4α2(cid:1)−2s +(2−2α)−2s α.
WeobservethatwehaveT (α)=Q(α),whereQ(α)hasbeendefinedintheprevioussection. Forevery
1
s ≥ 1 we obtain that the equation (4.3) has a solution α in the interval [−0.2432010309...,0). When
s become bigger the base of the pyramid approaches the equator. For s ∈ (0,1] the solution α of the
equation (4.3) is in the interval (−0.25,−0.2432010309...] meaning that when s becomes very small the
base of the pyramid is at the distance close to 0.25 from the equatorial plane. The value of the Riesz
s-energy on this critical configuration is given by
2 4 4
G(cid:101)s(square right pyramid)= √ + √ + √ ,
( 4−4α2)s ( 2−2α2)s ( 2−2α)s
where α is the solution of the equation (4.3).
ThestudyoftheHessianmatrixshowsthattherightsquarepyramidconfigurationundergoesvarious
bifurcation phenomena. More precisely for s < 13.5204990011... we have non-degenerate saddle points
in the sense of Morse-Bott having one eigenvalue strictly negative. For s > 13.5204990011... the right
square pyramid configurations are non-degenerate local minima in the sense of Morse-Bott.
For s between 13.5204990011... and 15.048077392... we have that both bi-pyramidal configuration
and square right pyramid configuration are local minima with
G(cid:101)s(bi-pyramid)<G(cid:101)s(square right pyramid).
For s between 15.048077392... and 21.1471229401... we have that both bi-pyramidal configuration and
square right pyramid configuration are local minima with
G(cid:101)s(bi-pyramid)>G(cid:101)s(square right pyramid),
which is a phenomenon discovered in [21].
For s>21.1471229401... the above inequality remains true.
3. Regular pentagon. The regular pentagon configuration discussed in the previous section is a
solutionfortheequation(4.2)andconsequently, remainscriticalconfigurationofRieszs-energyforany
s > 0. The regular pentagon configuration remains non-degenerate saddle in the sense of Morse-Bott
with two strictly negative eigenvalues for any s > 0. The value of Riesz s-energy on these critical
configuration has the formula
3 3 2 2
G(cid:101)s(regular pentagon)= (cid:16)(cid:113)2−2cos2π(cid:17)s + (cid:0)(cid:112)2+2cosπ5(cid:1)s + (cid:0)(cid:112)2−2sin1π0(cid:1)s + (cid:16)(cid:113)2+2sin3π(cid:17)s.
5 10
10