numerical methods for the computation of symmetric periodic … · 2020. 6. 11. · marco fenucci,...

Numerical methods for the computation ofsymmetric periodic orbits of the N-body problem

M. FenucciMathematics Department, University of Belgrade

2020 June 8

Summary:

(1) Introduction

(2) Numerical methods

(3) Application to the N-body problem

(4) Application to the Coulomb (1 + N)-body problem

Marco Fenucci, University of Belgrade I-CELMECH Seminars 2 / 34

Introduction

Introduction

In the last 20 years the existence of new and unexpected periodicsolutions have been proved.

Choreographies: N equal masses follow the same path with thesame time law with a phase shift of T/N.

Authors: V. Barutello, A. Chenciner, D. Ferrario, R. Montgomery,C. Simò, S. Terracini, A. Venturelli, . . .


Introduction

Introduction

Lagrangian: the masses interact through the gravitational force

L =N∑

i=1

|ui |2

2 +∑

1≤h<k≤N

1|uh − uk |α

, α ≥ 1.

Variational formulation: periodic orbits with fixed period T arestationary points of the Lagrangian action

Aα(u) =∫ T

0L(u, u) dt.

Hint: search for minimizers!


Numerical methods

Gradient descent

Fix FM ∈ N and consider

u(t) = a02 +

FM∑k=1

[ak cos

(2πkT t

)+ bk sin

(2πkT t

)], ak , bk ∈ R3.

Hence

A : R3(2FM+1) → R, A(a0, . . . , aFM , b1, . . . , bFM ) := A(u).

Gradient descent: denoting xk ∈ {ak , bk}, the step is defined as

x ′k = xk − η∂A(x)∂xk

, η > 0.


Numerical methods

Gradient descent

Note that

∂A∂ak

= (2πk)2

T ak +∫ T

0

∂L∂u(u(t)

)cos

(2πkT t

)dt, k ≥ 0,

∂A∂bk

= (2πk)2

T bk +∫ T

0

∂L∂u(u(t)

)sin(2πk

T t)

dt, k > 0.

Warning: The derivatives may grow up at high frequencies.

Modify the step as:

x ′k = xk − δτk∂A∂xk

, δτk = T(2πk)2 η.


Numerical methods

Multiple shooting

Aim: Solve {x = f (x),x(T/M) = Sx(0),

where x ∈ Rn, S ∈ O(n), M ∈ N.

Multiple shooting: Fix 0 = τ0 < τ1 < · · · < τm = T/M, anddefine {

Gi = φτi−τi−1(xi−1)− xi , i = 1, . . . ,m − 1,Gm = φτm−τm−1(xm−1)− Sx0.

Then solveG(x0, . . . , xm−1) = 0

with the Newton method.


Numerical methods

Multiple shooting

Denote X = (x0, . . . , xm−1), the general step of the Newtonmethod is defined by

∂G(X )∂X (X − X ′) = −G(X ),

where

∂G∂X =

M1 − I

M2. . .. . . − I

−S Mm

, Mi = ∂

∂x φτi−τi−1(xi ).


Numerical methods

Multiple shooting

Warning: The Jacobian matrix of G is singular at zeros. Addalso the condition

f (x0) · (x0 − x ′0) = 0.

Remark: G and its Jacobian matrix are obtained integrating theequation of motion coupled with the variational equation

x = f (x),

ddt A(t) = ∂f (x(t))

∂x A(t).


Numerical methods

Continuation method

Suppose that f = f (x , λ), where λ ∈ R.

Suppose that (X , λ) is such that

G(X , λ) = 0.

The couple (X , λ) is displaced to another zero of G by solvingG(X , λ) = 0,

|X − X |2 + (λ− λ)2 − δ2 = 0,

where δ > 0.

Remark: The system is solved again with the Newton method.


Numerical methods

Local minimizers for periodic problemsLet T > 0, consider

A(u) =∫ T

0L(t, u, u)dt,

defined on X ⊆ V := C1T ([0,T ],Rn).

The second variation of a periodic solution u0 is

δ2A(u0)(v) =∫ T

0

(v · Pv + 2v · Qv + v · Rv

)dt,

where

P = Luu(t, u0(t), u0(t)),Q = Luu(t, u0(t), u0(t)),R = Luu(t, u0(t), u0(t)).


Numerical methods

Local minimizers for periodic problems

Jacobi Differential Equation{Y = AY + BZ ,Z = CY − AT Z ,

where A = R−1Q, B = R−1, C = P − QT R−1Q.

Notation: (Y0,Z0), (YT ,ZT ) solutions of (JDE) such that{Y0(0) = 0,Z0(0) = I,

{YT (T ) = 0,ZT (T ) = −I,

and W0 = Z0Y−10 , WT = ZT Y−1

T .


Numerical methods

Local minimizers for periodic problems

Theorem: δ2A(u0) > 0 if and only if- det Y0(t) 6= 0 for t ∈ (0,T ] and- the matrix

W0(T )−WT (0)− Y−10 (T )− Y−T

0 (T )

is positive definite.

Remark: the above conditions can be checked with numericalcomputations.


Application to the N-body problem

N-body problem: symmetry of Platonic Polyhedra1

Setting: R ∈ {T ,O, I} symm. group of a Platonic polyhedron.

We consider N bodies where- N = |R|;- m1 = · · · = mN = 1;

and a potential of the form

1rα , α ≥ 1.

1Fusco, Gronchi, Negrini: Platonic polyhedra, topological constraints andperiodic solutions of the classical N-body problem, Invent. Math. 185 (2011)



Assume that(a) the motion of the particle R ∈ R \ {I} is given by

uR = RuI ;

(b) the trajectory of the generating particle uI([0,T ]) belongs toa given free homotopy class of R3 \ Γ, where

Γ = ∪R∈R\{I}r(R);

(c) there exist M ∈ N and a rotation R ∈ R such that

uI(t + T/M) = RuI(t).

⇒ Aα(uI) = N∫ T

0

(12 |uI |2 + 1

2∑R 6=I

1|(R − I)uI |α

)dt



Encoding the free-homotopy classes

To each group R we canassociate an Archimedeanpolyhedron QR: the rotationaxes pass through the centerof the faces of QR.

5

11

1

14

2

19

16

23

21

22

3

7

9

17

10

20

6

24

8

18

4

12

13

15

ν =[5, 1, 16, 10, 3, 8, 18,7, 20, 23, 14, 11, 5]

A sequence of vertexes νidentifies a free-homotopyclass of R3 \ Γ.



Example of computation

Fix a sequence: ν = [1, 3, 8, 15, 6, 10, 3, 7, 18, 8, 10, 16, 1]

Step 1: Compute the Fourier coefficients of the first guess



Example of computation

Step 2: Start the gradient descent iterations, starting from theprevious first guess.

Initial action:1100.716Final action:477.026



Example of computationStep 3: Start the shooting method, using the previous loop asfirst guess.

Action of the solution: 469.487



Example of computationStep 4: Add the other particles.

Videos athttp://adams.dm.unipi.it/~fenucci/research/nbody.html

http://adams.dm.unipi.it/~fenucci/research/gammaconv.html


http://adams.dm.unipi.it/~fenucci/research/nbody.html

http://adams.dm.unipi.it/~fenucci/research/gammaconv.html


Example of computation: stability2

Compute the eigenvalues of the monodromy matrix

∂φT (x0)∂x .

In this example:

µ1 = 200.131µ2 = 0.093 + 0.995iµ3 = 0.093− 0.995iµ4 = 1.000µ5 = 1.000µ6 = 0.005

|µ1| = 200.131|µ2| = 1.000|µ3| = 1.000|µ4| = 1.000|µ5| = 1.000|µ6| = 0.005

2M. F., Gronchi: On the stability of periodic N-body motions with thesymmetry of Platonic polyhedra, Nonlinearity 31 (2018).



Example of computation: minimality propertyThe determinant of Y0(t)

0 0.2 0.4 0.6 0.8 1

time

10-10

10-5

100

105

det

The eigenvalues of the additional matrix:

27.63, 8.78, 0.00104Marco Fenucci, University of Belgrade I-CELMECH Seminars 22 / 34


Example of computation: continuationAdd a central particle with mass m0 > 0:

Aα(uI) = N∫ T

0

(12 |uI |2 + 1

2∑R 6=I

1|(R − I)uI |α

+ m0|uI |α

)dt.

Continuation: use m0 as a parameter, starting from m0 = 0.

Example: ν = [1, 3, 7, 20, 18, 8, 15, 4, 6, 10, 16, 5, 1]



Example of computation: continuationCentral mass: m0 = 0



Calculus of Variation point of view

Set ε = 1/m1/(2+α)0 , hence

Aαε (v) =∫ T

0

(|v |22 + 1

|v |α + ε

2

N∑i=2

1|(Ri − I)v |α

)dt.

Theorem3: For every α ∈ [1, 2) we have(i) Γ- limε→0Aαε = Aα0(ii) if {v∗ε }ε>0 is a sequence of minimizers, then it converges to a

minimizers of Aα0 .Moreover, a minimizer of Aα0 is composed by circular Keplerianarcs joined at some rotation axes.

3M.F., Gronchi: Symmetric constellations of satellites moving around acentral body of large mass, Preprint (2020)



Central mass: m0 = 400


Application to the Coulomb (1 + N)-body problem

The Coulomb (1 + N)-body problem4

Atomic units: k = 1, me = 1, qe = −1.System: q1 = · · · = qN = qe , Q > 0.

L =N∑

i=1

|ui |2

2−

∑1≤h<k≤N

1|uh − uk |

+N∑

i=1

Q|ui |

.

Rutherford atomic model

Hint: we do not take intoaccount quantum mechanics inthis model.

4M. F., Jorba: Braids with the symmetries of Platonic polyhedra in theCoulomb (N+1)-body problem, CNSNS 83 (2019)



Scheme for numerical computations

Impose the symmetry of Platonic polyhedra.

Scheme of the computation:

i) Generate a first guess in the desired free homotopy class ofR3 \ Γ. Note: gradient method does not work.

ii) Compute a periodic solution for a large value of Q, using athe shooting method.

iii) Use the solution computed at step (ii) as starting point for acontinuation method with respect to Q, and reduce its value.

Common behaviour:(i) there is always a turning point in the continuation w.r.t. Q;(ii) the orbits are unstable for all values of Q.



Example: R = O

Before the turning point, Q = 24 After the turning point, Q = 24

Videos athttp://adams.dm.unipi.it/~fenucci/research/coulomb.html


http://adams.dm.unipi.it/~fenucci/research/coulomb.html


Minimality property

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

time

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

det

Q=15

Q=18

Q=21

Q=24

Q=27

0 0.05 0.1 0.15 0.2 0.25 0.3

time

-10

-8

-6

-4

-2

0

2

4

6

8

de

t

10-3

Q=24

Q=25

Q=30

Q=36

Q=40

→ in general det Y0(t) has a zero;→ when det Y0(t) is positive, then the additional matrix has a

negative eigenvalue.

⇒ these are all saddle points

aMarco Fenucci, University of Belgrade I-CELMECH Seminars 32 / 34


Please visit:http://www.stardust-network.eu/


http://www.stardust-network.eu/


Thank You!


numerical methods for the computation of symmetric periodic … · 2020. 6. 11. · marco fenucci,...

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