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, . . .
Marco Fenucci, University of Belgrade I-CELMECH Seminars 3 / 34
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!
Marco Fenucci, University of Belgrade I-CELMECH Seminars 4 / 34
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.
Marco Fenucci, University of Belgrade I-CELMECH Seminars 5 / 34
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.
Marco Fenucci, University of Belgrade I-CELMECH Seminars 6 / 34
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.
Marco Fenucci, University of Belgrade I-CELMECH Seminars 7 / 34
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 ).
Marco Fenucci, University of Belgrade I-CELMECH Seminars 8 / 34
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).
Marco Fenucci, University of Belgrade I-CELMECH Seminars 9 / 34
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.
Marco Fenucci, University of Belgrade I-CELMECH Seminars 10 / 34
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)).
Marco Fenucci, University of Belgrade I-CELMECH Seminars 11 / 34
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 .
Marco Fenucci, University of Belgrade I-CELMECH Seminars 12 / 34
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.
Marco Fenucci, University of Belgrade I-CELMECH Seminars 13 / 34
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)
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Application to the N-body problem
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
Marco Fenucci, University of Belgrade I-CELMECH Seminars 15 / 34
Application to the N-body problem
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.
Marco Fenucci, University of Belgrade I-CELMECH Seminars 16 / 34
Application to the N-body problem
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
Marco Fenucci, University of Belgrade I-CELMECH Seminars 17 / 34
Application to the N-body problem
Example of computation
Step 2: Start the gradient descent iterations, starting from theprevious first guess.
Initial action:1100.716Final action:477.026
Marco Fenucci, University of Belgrade I-CELMECH Seminars 18 / 34
Application to the N-body problem
Example of computationStep 3: Start the shooting method, using the previous loop asfirst guess.
Action of the solution: 469.487
Marco Fenucci, University of Belgrade I-CELMECH Seminars 19 / 34
Application to the N-body problem
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
Marco Fenucci, University of Belgrade I-CELMECH Seminars 20 / 34
Application to the N-body problem
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).
Marco Fenucci, University of Belgrade I-CELMECH Seminars 21 / 34
Application to the N-body problem
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
Application to the N-body problem
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]
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Application to the N-body problem
Example of computation: continuationCentral mass: m0 = 0
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Application to the N-body problem
Example of computation: continuationCentral mass: m0 = 100
Marco Fenucci, University of Belgrade I-CELMECH Seminars 25 / 34
Application to the N-body problem
Example of computation: continuationCentral mass: m0 = 400
Marco Fenucci, University of Belgrade I-CELMECH Seminars 26 / 34
Application to the N-body problem
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)
Marco Fenucci, University of Belgrade I-CELMECH Seminars 27 / 34
Application to the N-body problem
Central mass: m0 = 400
Marco Fenucci, University of Belgrade I-CELMECH Seminars 28 / 34
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)
Marco Fenucci, University of Belgrade I-CELMECH Seminars 29 / 34
Application to the Coulomb (1 + N)-body problem
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.
Marco Fenucci, University of Belgrade I-CELMECH Seminars 30 / 34
Application to the Coulomb (1 + N)-body problem
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
Marco Fenucci, University of Belgrade I-CELMECH Seminars 31 / 34
Application to the Coulomb (1 + N)-body problem
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
Application to the Coulomb (1 + N)-body problem
Please visit:http://www.stardust-network.eu/
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Application to the Coulomb (1 + N)-body problem
Thank You!
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