Totally conservative integration method for N -body problem andits equilibrium solutions
Yukitaka Minesaki
Tokushima Bunri University
minesaki@tks.bunri-u.ac.jp

Abstract
We design a precise integration method for the general N -body problem (GN BP) which maintains all conserved
quantities. Our method prove to retain the same orbits of equilibrium solutions as the original GN BP. This method
is based on a kind of 2nd-order symplectic integrators, logarithmic Hamiltonian leapfrog ((LogH)2), and involves
an energy-preserving parameter. Until our method, no integration method was known to hold both preservation of
all conserved quantities and the existence of triangular Lagrangian solutions.

Triangular Lagrangian Solutions of (TCI)2
For any integer l, and arbitrary real numbers Θ, and Φ, we assume
(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

(k)

q12 = q(k)R3 (Θ) R1(Φ), q13 = q(k)R3 (Θ±π/3) R1(Φ), q23 = q13 −q12 , k = l, l + 1/2, l + 1,
v12 = v(k)R3 (Θ) R1(Φ), v13 = v(k)R3 (Θ ± π/3) R1(Φ), v23 = v13 −v12 , k = l, l + 1,
(double sign corresponds),

Introduction

where R3(θ), and R1(φ) are respectively the rotational matrix about the 3rd, and the 1st axes. Then,(TCI)2 reduces to the discrete-time general 2-body problem (d-G2BP):
∆σLT∆σLT (l)


v

1

(l+1/2) = t(l) +(l+ 2 ) = q(l) +22

q,t,


(l)
2(l)
2

(v )
(v )
(0)
(0)
− ELT− ELT

221

(l+ 2 )
 (l+1)
1 ∆σq,v= v(l) − α(l+ 2 ) LT 1)(l+2 |2|q∆σLT∆σLT (l+1)
v
1

(l+1) = t(l+1/2) +(l+1) = q(l+ 2 ) +22
,tq,


2
(l+1)
2(v)
(l+1)
(0)
v

− ELT

(0)
2

− ELT2Figure 1: Various integrators for GN BP

Totally conservative integration method
for general 3-body problem (G3BP)
For simplicity, we describe only a 2nd-order totally conservative integration method ((TCI)2) forG3BP
in the relative frame:(l+1)
(l+1)
(l+1)

v12 + v23 − v13 = 0,



(l)

1v1
∆s1(l+ 2 )(l) ∆sij(l+)(l)
qij, 1 ≤ i < j ≤ 3, t 2 = t +,= qij +

(l)
(0)
(l)
(0)
2 TR(v ) − E
2 TR(v ) − E





11(l+ 2 )(l+ 2 )11
j−13(l+)(l+)
X UR(q 2 ) (l+1) (l) M qij  X UR(q 2 ) (l+1) (l) M qji mi(vij −vij )−mi(vji −vji )−=, j = 2, 3,

1
111

(l+ 2 )∆s(l+ 2 ) 3(l+ 2 )∆s(l+ 2 ) 3
αα|||q|qi=1
i=j+1
ijji



(l+1)

1vij1

∆s1
(l+ 2 ) ∆s(l+1)
)
(l+1)
(l+

, 1 ≤ i < j ≤ 3, t=t 2 +,qij = qij + 2
2 TR(v(l+1))−E (0)TR(v(l+1))−E (0)P2 P32mm|v|ijij
i=1 j=i+1
where ∆s: a fictitious time step, the total kinetic energy: TR(v) ≡,2(m1 + m2 + m3)
32 X
Xmimj, the total energy : E (0) ≡ TR(v(0)) + VR(q(0)), thethe total potential : UR(q) ≡ −|qij ||v|2 M
M ∆s, and ELT ≡− .
where ∆σLT ≡m1m2 + m1m3 + m2m3
2|q|
dq
dv
Mq
α(l+1/2) = 1∆s→0= v,=− 3
d-G2BP −−−−−−−−−−−→ (LogH)2 for G2BP −−−−→ G2BP :dt
dt|q|(LogH)2 for G2BP [2] maintains all conserved quantities of G2BP, namely, (i) energy : ELT ,v × (q × v)
q(ii) angular velocity : JLT ≡ q × v, and (iii) Laplace vector : eLT ≡−. ThereM|q|fore, (LogH)2 for G2BP exactly traces the original G2BP. Further, α(l+1/2) is adjusted to con(0)serve ELT . Accordingly, when (TCI)2 holds the triangular Lagrangian solutions, (TCI)2 hasα(l+1/2) = 1, namely, (TCI)2 coincides with (LogH)2 for G2BP. In a similar procedure, we
prove that (TCI)2 has the collinear Lagrangian solutions of G3BP and equilibrium solutions of GN BP.

i=1 j=i+1

total potential : M = m1 + m2 + m3, and energy-preserving parameter : α(l+1/2). In the case of
1l+(α 2 ) = 1, (TCI)2 coincides with (LogH)2 [2].
dqi,j3; v12 +v23 −v13 = 0; !dt = vi,j , 1 ≤ i < j ≤ !
∆s→0
j−1
3(LogH)2 −−−−→ G3BP in the relative frame : X
Xdvj,i M qj,iMqdvi,ji,j=mi, j = 2, 3.
++mi
33
dt
dt|qi,j ||qj,i| i=1
i=j+1
i−1
3
G3BP
in theinertial barycentric frame :X1  Xqi =−mj qji +mj qij i−13XXM(q)dpmk (qi − qk )
m−q
iik k
j=1
j=i+1


−,= mi33−−−−−−−−−−−−−−−−−−−−−−−−−−→dt|qk − qi|
|qi − qk |

k=1
k=i+1
i−13

XXmi 

dqipi


pi =−
mj vji +mj vij=, i = 1, 2, 3.
Mdt
mij=1Figure 3: Orbits of a triangular Lagrangian solution in the inertial barycentric frame

Other Periodic Orbits of (TCI)2(TCI)2 is merely 2nd-order accurate, but it can precisely reproduce some periodic orbits which(LogH)2, and generic geometric integrators cannot do. For example,

j=i+1

Conservation of G3BP ((LogH)2 vs (TCI)2)
Conserved quantities of (TCI)2

Conserved quantities of (LogH)2 = (TCI)2 with α(l+1/2) = 1

Conserved quantities by Coordinate transformation (qij , vij ) → (qi, pi)* the total linear momentum : ` = 0* the position of the center of mass : c = 0





2Figure 4: A Periodic Orbit of m2 in the inertial barycentric frame

3Conclusion

1 X X* the total angular momentum : a ≡mimj qij × vij
M
i=1 j=i+1

* the total energy : E ≡ TR(v) + UR(q) ←− This is accomplished by adjusting α(l+1/2).



We modified (LogH)2 [2] for GN BP to design a new integration method, (TCI)2, which maintains
all conserved quantities and holds equilibrium solutions including Lagrangian solutions. Although(TCI)2 is merely 2nd-order accurate, it can precisely trace some periodic orbits which generic geometric integrators, and (LogH)2 cannot do. Until (TCI)2, no integration method was known tomaintain both preservation of all conserved quantities and the existence of equilibrium solutions.

References[1] D. Greenspan, Discrete Newtonian gravitation and the three-body problem, Fund. Phys. 4 (1974),
299–310.
[2] S. Mikkola and K. Tanikawa, Explicit symplectic algorithms for timetransformed Hamiltonians,Celest. Mech. Dyn. Astron. 74 (1999a), 287–295.
[3] Y. Minesaki, An efficient conservative integrator with a chain regularization for the few-bodyproblem, Astron. J. 150 (2015), 102.

Figure 2: Relative errors of the total energy and total angular momentum for a periodic solution different from Lagrangiansolutions.

The above relation for conservation is fulfilled for GN BP.

Acknowledgments
This study was supported by a grant-in-aid for exploratory research (16K13781) from the Ministry ofEducation, Culture, Sports, Science and Technology of Japan.