Classical Grand Angular Momentum in N-Body Problems
Classical Grand Angular Momentum in N-Body Problems
Classical Grand Angular Momentum in N-Body Problems
sions. We generalized some results from the two-body problem. Furthermore, we derive the general expression for the
scattering angle for the N-body problem.
I. INTRODUCTION
The N-body problem concerns the time evolution of an isolated system consisting of N objects in a 3-dimensional space with
only internal interactions that could be modeled by a potential V dependent only on the distances between them. It is ubiquitous
in physics – first studied in large-scale astronomical contexts as many problems in physics by illustrious names such as Euler
and Lagrange1,2 and later, especially after the advances of quantum mechanics, in relevant scenarios from atomic and nuclear
physics 3–12 . At the same time, it is also well known that an exact solution does not exist for N > 2. Nevertheless, one could still
greatly simplify the problem by making appropriate coordinate transformations and applying conservation laws for any general
N.
One set of such coordinates is Jacobi coordinates, which, after enforcing momentum conservation, can reduce it to an effec-
tive (N − 1)-body problem. The various ways of forming these virtual bodies could be graphically represented by binary Jacobi
trees,13–15 which will be introduced in Part II. By neglecting the trivial center of mass motion, it can further be reduced to a prob-
lem of one virtual body moving in 3N − 3 dimensions.5–7,16 Such a space is usually described in hyperspherical coordinates, first
introduced in nuclear physics17, and showing interesting results in the quantum realm, such as the existence of Efimov states.18
There are also different definitions of hyperspherical coordinates, which can be represented by binary trees. Such details will be
laid out in Part III. Moreover, the grand angular momentum, an extension of angular momentum to higher dimensions, a con-
served magnitude, and as a result, it shows well-defined eigenvalues and associated eigenfunctions in the quantum level.11,16,19
In fact, the concept of grand angular momentum has been more thoroughly studied in a quantum mechanical context, while its
applications in the framework of classical mechanics have received less attention.
Here, in Part IV, we bridge this gap with the help of graphical representations of the coordinates. One can read directly from a
tree an expression for the magnitude of grand angular momentum. Moreover, equipped with this knowledge, a decomposition of
grand angular momentum into regular angular momentum in their magnitudes can be achieved, and a second-order differential
equation that constrains the evolution of the 3N − 4 hyperangles is found.
For the two-body problem in a 3-dimensional space, the only analytically solvable case within the N-body problem, the
Hamiltonian is given by
p21 p2
H= + 2 + V (r2 − r1 ), (1)
2m1 2m2
which describes a system made up from masses m1 and m2 at positions r1 and r2 with conjugate momenta p1 and p2 , respec-
tively, interacting via the interaction potential V (r2 − r1 ). Using the following coordinate transformation
m1 r 1 + m2 r 2
RCM = R12 = (2)
m1 + m2
ρ = ρ 1 = r2 − r1 , (3)
a) https://www.stonybrook.edu/commcms/amotheory/index.php
Grand Angular Momentum 2
where µ = µ1,2 = m1 m2 /(m1 + m2 ) is the two-body reduced mass and M = M12 = m1 + m2 is the total mass of the system. Due
to the nature of the interaction potential, RCM = R12 , does not appear in the interaction potential, and hence the center of mass
momentum, PCM , is a conserved quantity. Therefore, the two-body problem reduces to a single particle with mass µ = µ1,2 ,
placed at ρ = ρ1 with momentum P = µ ρ̇.
For N = 3, after neglecting the center of mass motion, two virtual bodies are required to specify the dynamics. The first
one describes the relative motion between m1 and m2 and is the same as in the case of the two-body problem. The second is
associated with ρ2 = r3 − R12 , joining the center of mass of the first two particles with the third one. In these new coordinates,
the Hamiltonian reads as
p21 p2 p2 P2 P22 P2
H= + 2 + 3 + V (r2 − r1 , r3 − r2 , r1 − r3 ) = 1 + + CM + V (ρ1 , ρ2 ), (5)
2m1 2m2 2m1 2µ1,2 2µ12,3 2M
with
(m1 + m2 )m3
µ12,3 = , (6)
(m1 + m2) + m3
and M = M123 = m1 + m2 + m3 .
It is possible to generalize this approach to the N-body problem, building up Jacobi coordinates in this way as
j
M12... j = ∑ mi , (7)
i=1
M12... j−1 m j
µ12... j−1, j = , (8)
M12... j
j
∑i=1 mi ri
R12... j = , (9)
M12... j
ρ j = r j+1 − R12... j , (10)
after removing the center-of-mass term. Further, by introducing the N-body reduced mass16
∏Ni=1 mi 1/(N−1)
µ ≡( ) , (12)
∑Ni=1 mi
it is possible to define mass weighted Jacobi coordinates as
µ12...i,i+1
r
ρMW,i = ρi . (13)
µ
Next, stacking them on top of each other, namely
ρMW,1
ρMW,2
.
ρMW ≡ , (14)
.
.
ρMW,N−1
2
PMW
H= + V (ρMW ), (15)
2µ
with ρMW and PMW = µ ρ̇MW vectors in 3N − 3 dimensions. The degrees of freedom are thus repackaged.5,7
However, as N increases, there also appear more ways of forming such N − 1 virtual bodies. For N = 2, there is no ambiguity
at all. For N = 3, it is still unique up to labeling. When N = 4, in addition to the recursive way presented above, it is also
permitted to have three virtual bodies as µ1,2 , µ3,4 , and µ12,34 = (m1 + m2 )(m3 + m4 )/M, as shown in Fig. 1. Each particular
choice of Jacobi coordinates has attached to it a binary Jacobi tree, as presented in panels (b) and (d) of Fig. 1.
m4
m1 m1 m2 m3 m4
µ123,4
µ1,2
m3
µ12,3
µ12,3
µ1,2
m2 µ123,4
(a) (b)
m4
µ3,4
m1 m1 m2 m3 m4
µ12,34
µ1,2 µ3,4
m3
µ1,2
m2 µ12,34
(c) (d)
FIG. 1. Visualization of Jacobi coordinates (a) and (c), together with their respective tree representations (b) and (d) for the 4-body problem.
Solid circles represent physical bodies. Hollow circles are for virtual bodies. Unweighted distance vectors are the arrowed lines.
The correspondence between the coordinates and its tree representation is straightforward. Each leaf represents a physical
body, and each node a virtual body of mass µL,R , where L = {l1 , l2 , ..., lm } and R = {r1 , r2 , ..., rn } are collective indices of m
physical masses joining the node from the left and n from the right, respectively, and the ρL,R associated is given by RR − RL ,
the vector from the center of mass of all masses on the left to that of all masses on the right. More explicitly, for each node,
m
ML = Ml1 l2 ...lm = ∑ mli , (16)
i=1
ML MR
µL,R = , (17)
ML + MR
∑m ml rl
RL = Rl1 l2 ...lm = i=1 i i , (18)
Ml1 l2 ...lm
ρL,R = RR − RL , (19)
Grand Angular Momentum 4
could be defined, and there exist N − 1 such nodes13,14 . By the same procedures described by Eq. 13, again a one-body problem
in 3N − 3 dimensions is at hand.
Beyond this point in all later sections, the "MW" subscript will be dropped but implicitly assumed so that every vector, unless
otherwise noted, will be mass-weighted.
After converting an N-body problem in 3 dimensions to an effective one body in 3N − 3 dimensions, hyperspherical coor-
dinates with one hyperradius and 3N − 4 hyperangles are then applied to describe the system to better incorporate rotational
symmetries. There are different ways of defining hyperangles and they could again be represented by binary trees graphically.
There will be 3N − 3 leaves, corresponding to 3N − 3 Cartesian components, and 3N − 4 nodes, to which accord the hyperangles.
By convention, a line joining a node γi from the left represents a factor of cos γi and that from the right sin γi . Then the definition
of each Cartesian component could be read directly from the tree representation by simply tracing from its corresponding leaf to
the root. The range of each hyperangle also ought to be specified - [0, 2π ) if its branches contain no more nodes, [0, π ] if there
are further nodes attaching to one of the branches, and [0, π /2] if both branches contain further nodes11,20 . Two different trees
associated with N = 3 are displayed in Fig. 2. The hyperspherical coordinates are given by
(a) (b)
FIG. 2. Two definitions of hyperangles for N = 3. (a) is a straightforward extension of spherical coordinates, and (b) is defined such that ρ1
and ρ2 are very much living in their own 3-dimensional space.
We have discussed the impact of Jacobi coordinates and its representation in hyperspherical coordinates in the N-body Hamil-
tonian, reducing the N-body problem into a single virtual body in a 3N − 3-dimensional space. Next, we study the generalization
Grand Angular Momentum 5
of angular momentum into the N-body problem. First, the original definition of angular momentum needs to be extended to
higher dimensions, and, therefore, become a tensor. It is called grand angular momentum in the literature and defined as
Λ = ρ∧P, (22)
and each of the components are given by
Λi j = ρi Pj − ρ j Pi , (23)
and
1
Λ2 = (Λi j )2 .
2∑
(24)
i, j
is conserved in the usual setup where there is no external torque acting on the system16,21 .
In hyperspherical coordinates, ρ = ρ ρ̂, the momentum is given by
˙
P = µ ρ̇ = µ (ρ̇ ρ̂ + ρ ρ̂), (25)
and Λ2 can be calculated with the Lagrange identity as follows
˙ 2 = µ 2 ρ 4 [(ρ̂ · ρ̂)(ρ̂˙ · ρ̂)
Λ2 = µ 2 (ρ ∧ ρ̇)2 = µ 2 ρ 4 (ρ̂ ∧ ρ̂) ˙ − (ρ̂ · ρ̂)
˙ 2 ] = µ 2 ρ 4 ρ̂˙ · ρ̂.
˙ (26)
It could be observed that the dependence of Λ2 on ρ and the hyperangles are separated, showcasing the advantage of hyperspher-
ical formulation. One can also see that it recovers the two-body result where ρ̂˙ = θ̇ and L = µρ 2 θ̇ 2 nicely.
Further, the exact dependence of ρ̂˙ · ρ̂˙ on the hyperangles is directly readable from the hyperspherical trees.
Theorem 1. In the tree representation of hyperspherical coordinates, for every hyperangle γi , tracing its path to the root, label
the jth node it reaches from the right αi j , and the kth node reached from the left βik , then ρ̂˙ · ρ̂˙ = ∑i γ̇i2 ∏ j sin2 αi j ∏k cos2 βik .
The proof is by induction.
Proof. For the simplest tree possible with only the root and two leaves, the theorem holds true. Explicitly,
sin γ γ̇ cos γ
ρ̂ = , ρ̂˙ = , ρ̂˙ · ρ̂˙ = γ̇ 2 .
cos γ −γ̇ sin γ
Assume the theorem holds true for trees with all N − 1 nodes on the rightmost branch and add the Nth node as the new root
joining the old tree (now as a branch) with a bare branch on the left, such as shown in FIG. 2a, then
α̇N ρ̂N cos αN + ρ̂˙ N sin αN
ρ̂N sin αN ˙
ρ̂N+1 = , ρ̂N+1 = ,
cos αN −α̇N sin αN
and
ρ̂˙ N+1 · ρ̂˙ N+1 = α˙N 2 + ρ̂˙ N · ρ̂˙ N sin2 αN ,
which has the desired form.
The same applies to trees with nodes all on the left-most branch, only this time
sin βN β̇N cos βN
ρ̂N+1 = , ρ̂˙ N+1 = ,
ρ̂N cos βN −β̇N ρ̂N sin βN + ρ̂˙ N cos βN
and
2
ρ̂˙ N+1 · ρ̂˙ N+1 = β˙N + ρ̂˙ N · ρ̂˙ N cos2 βN .
Next, for trees with nodes on both main branches - M − 1 on the right and N − 1 on the left - and they are joined by the root γ
form a tree for hyperspherical coordinates in M + N dimensions,
γ̇ ρ̂R cos γ + ρ̂˙ R sin γ
ρ̂R sin γ
ρ̂M+N = , ρ̂˙ M+N = ,
ρ̂L cos γ −γ̇ ρ̂L sin γ + ρ̂˙ L cos γ
and
ρ̂˙ M+N · ρ̂˙ M+N = γ̇ 2 + ρ̂˙ R · ρ̂˙ R sin2 γ + ρ̂˙ L · ρ̂˙ L cos2 γ ,
which, again, has the form given by the theorem. Lastly, the same proof applies to two trees of any configuration joining together
through a new root.
Grand Angular Momentum 6
A. Angular momentum
Theorem 1 applies to every possible tree. For instance, when applied to panel (b) of Fig. 2, we find
γ3,2
γ32,1
FIG. 3. Hyperspherical tree associated with panel (b) Fig. 7. ρ1 characterizes the motion of µ12,34 , ρ2 that of µ3,4 , and ρ3 that of µ1,2
B. Scattering angle
It is well-known that angular momentum generates a repulsive barrier responsible for the stability of the motion, or in the case
of scattering problems, it provides the reaction barrier for chemical processes. In the case of more than two bodies, the barrier
appears as well in the dynamics, but it is produced due to the grand angular momentum as
P2 1 Λ2
E= + V (ρ) = µ ρ̇ 2 + + V (ρ), (28)
2µ 2 2µρ 2
and it has the same effect of avoiding any close approach ρ → 0 when Λ2 6= 0.
The impact parameter b essential in all calculations of scattering and reactive processes22 also deserves a generalization - as
the projection of initial position ρ0 onto the hyperplane perpendicular to the initial momentum vector P0 . For N = 2, b is reduced
to a single number whereas for general N, b is a vector living in a (3N − 4)-dimensional hyperplane5,7 . With this definition of b,
Λ2 can be calculated as21
Λ2 = (ρ0 ∧ P0 )2 = (b ∧ P0 )2 = b2 P02 − (b · P0 )2 = 2µ Eb2 . (29)
From the above relations, we find
d ρ̂ d ρ̂ ρ̂˙ · ρ̂˙ b2
ρ̂′ · ρ̂′ = · = 2 = 4 , (30)
dρ dρ ρ̇ ρ (1 − b /ρ 2 − V (ρ)/E)
2
where ρ̂′ · ρ̂′ could be read from the tree just as stated in Theorem 1 with a simple replacement of d/dt by d/d ρ .
Similarly, it is possible to transform the above expression using the chain rule of second derivatives to arrive at a second-order
expression
d 2 ρ̂ b2
ρ̂ · 2
=− 4 . (31)
dρ ρ (1 − b /ρ 2 − V (ρ)/E)
2
Eq. (31) resembles the expression for the scattering angle in the two-body problem. Hence, it can be interpreted as the scattering
angle equation for the N-body problem.
V. CONCLUSION
Using hyperspherical coordinates, we have developed a graphical approach for treating classical grand angular momentum for
the N-body problem. Similarly, we have been able to demonstrate the relationship between the relative angular momentum of the
bodies of a system and the grand angular momentum, which is essential to understand the N-body problem. Specifically, it has
been shown that the exact form of Λ2 is directly readable from the associated graphics, that the magnitude of higher-dimensional
grand angular momentum can be decomposed into magnitudes of 3-dimensional angular momenta, and that the hyperangles
necessarily satisfy a differential equation in Eq. (31). In deriving these results, the only assumption made on the potential is that
it depends only on interparticle distances, making them widely applicable and a valuable asset in analysis.
On the other hand, our results lead us to a formulation of the scattering angle equation for the N-body problem as the general-
ization of the well-known expression in two-body scattering. This result establishes a link between our knowledge of two-body
physics and how it could be generalizable to the N-body problem.
Grand Angular Momentum 8
ACKNOWLEDGMENTS
AUTHOR DECLARATIONS
Conflict of Interests
Author Contributions
Zhongi Liang: Conceptualization (equal); Formal analysis (lead); Investigation (equal); Visualization (lead); Writing – origi-
nal draft (equal); Writing – review & editing (equal). J. Pérez-Ríos: Conceptualization (equal); Investigation (equal); Supervision
(lead).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
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