McCord UFPE minicourse lecture 1

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Integral Manifolds of the N-Body Problem
Christopher McCord
UFPE Mini-Course
Christopher McCord N-Body Integral Manifolds Oct. 2024 1 / 26
Outline
The General Theory & Reduction Formula
Definitions and Problem Statement
Singular Values of Energy
The Reduction to the Mass Ellipsoid
The Homology Reduction Formula
Four Equal Masses
Bifurcations at Infinity
Finite Bifurcations
Four Bodies with Three Equal Masses
Bifurcations at Infinity
Finite Bifurcations/Central Configurations
Christopher McCord N-Body Integral Manifolds Oct. 2024 2 / 26
Integral Manifolds
Consider the classical N-body problem in three dimensions
The integral manifolds M(c,h) are the level sets of the classical
constants of motion
Center of Mass
∑
mi q⃗i = 0⃗
Linear Momentum
∑
mi p⃗i = 0⃗
Angular Momentum
∑
mi q⃗i × p⃗i = ck̂
Energy
∑ 1
2mi
p⃗2
i − U(q) = h
where U(q) =
∑ mi mj
|q⃗i−q⃗j |
There is a rotational symmetry
SO3 symmetry if angular momentum c = 0
SO2 symmetry if angular momentum c ̸= 0
and reduced integral manifolds MR(c,h) = M(c,h)/SOk
Christopher McCord N-Body Integral Manifolds Oct. 2024 3 / 26
Goals
For fixed masses and non-zero angular momentum, the integral
manifolds depend only on the single parameter ν = hc2
In the broadest terms, the goals of a topological study of the integral
manifolds are:
Describe the integral manifolds (in some sense)
Identify all global bifurcations of the manifold structure as a
function of the mass parameters and either ν or h
Use the information obtained to detect presence or absence of
dynamics
Christopher McCord N-Body Integral Manifolds Oct. 2024 4 / 26
Prior Studies
Moulton gave a complete description of the collinear problem
Smale analyzed the planar problem
Hildeberto Cabral analyzed the spatial problem with 0 angular
momentum, and with non-zero angular momentum and positive
energy
Simo, Saari, Iacob, Easton, Chen analyzed aspects of the spatial
three-body problem with negative energy
Alain Albouy found necessary conditions for bifurcations of the
spatial manifolds
Ken Meyer, Don Wang and I analyzed the spatial three-body
problem, providing the complementary analysis to show that all of
the candidate values were indeed bifurcation values
Christopher McCord N-Body Integral Manifolds Oct. 2024 5 / 26
Sampling of Prior Results
For zero angular momentum, the only bifurcation as h varies is at
h = 0.
There are no bifurcations for either planar or spatial manifolds for
positive energy.
For the planar problem, bifurcations can only occur at relative
equilibria, at energy levels associated with the corresponding
central configurations.
There is a formula for the homology groups of the integral
manifolds in terms of the level sets of U on the mass ellipsoid.
Applying this confirms that, for either three arbitrary masses or
four equal masses, bifurcations of the planar manifold do occur at
all relative equilibria.
Christopher McCord N-Body Integral Manifolds Oct. 2024 6 / 26
Singular Values
For the spatial problem with non-zero angular momentum and negative
energy, the work of Albouy and predecessors established:
Let A be the manifold of fixed angular momentum, linear
momentum and center of mass. Bifurcations can only occur at
singular values of energy on A.
The “finite” singular points are the same as those for the planar
problem, namely, the relative equilibria, with the appropriate
energy. In particular, spatial (non-planar) central configurations
play no role.
Additional bifurcation values are possible. These “bifurcations at
infinity” are associated with clusters of “local relative equilibria”:
partition the N masses into clusters; for each cluster σk , put a
relative equilibrium of Nk masses at height zk , scaled so that all
clusters rotate together; then send zk ’s to infinity. The limiting
energy is a singular value of H.
The set of singular values is bounded, and the singular points are
bounded away from collision.
The set of singular values is determined if all central
configurations of all sub-clusters of the masses are known.
Singular values are not necessarily bifurcation values.
Christopher McCord N-Body Integral Manifolds Oct. 2024 7 / 26
Example: Four Equal Masses
Bifurcations at Infinity Bifurcations at Relative Equilibria 
One 
Binary 
Two 
Binary Equilateral Triple 
Collinear Square Isoseles Equilateral Collinear 
-0.25 -2 -4.5 -6.25 -29.31 -33.5879 -33.5885 -45.88 
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 These 8 singular levels, along with h = 0, create 10 parameter
intervals.
We compute the homology groups H∗(MR(c,h)) for each interval.
Changes in the homology groups from one interval to the next
show that the topology of the manifold has changed.
Christopher McCord N-Body Integral Manifolds Oct. 2024 8 / 26
Approach
Project M onto configuration space
S = {q ∈ R3N \∆|
∑
mi q⃗i = 0⃗,
∑
mix2
i = 1}
Recognize integral manifold as
a (singular) sphere bundle
over the set of allowable
configurations K(h, c)
Describe K(h, c) as super-level
set K(h, c) = {D(h) ≥ −2h} of
D : S0 → R+, where
D(q) = U2(q)
Y (q)
Christopher McCord N-Body Integral Manifolds Oct. 2024 9 / 26
Limitations
The function D(q) = U2(q)
Y (q) is at least as complicated as U(q)
On the boundary D(q) = −2h, the fiber collapses to a point
Since collisions are deleted, the set of allowable configurations is
non-compact
but more importantly ...
Christopher McCord N-Body Integral Manifolds Oct. 2024 10 / 26
Limitations
Everything goes to hell at collinear
configurations
Collinear configurations
outside of the invariant plane
are forbidden; collinear
configurations in the invariant
plane are allowed
The dimension of the fiber
jumps by 1 at the allowable
collinear configurations
The function Y (and hence D)
is discontinuous at the
allowable collinear
configurations
Christopher McCord N-Body Integral Manifolds Oct. 2024 11 / 26
Remedy
In 2000, Hildeberto and I revisited the spatial manifolds with positive
energy, and found that we could eliminate the discontinuities by
introducing a blow-up construction B on the set of allowable
configurations, replacing each point with S2N−4 \ S0 – essentially
introducing polar coordinates at the collinear configurations.
This allowed us to globalize coordinates
on the positive energy manifolds, and also
allowed us to compute the homology
groups of those manifolds.
Christopher McCord N-Body Integral Manifolds Oct. 2024 12 / 26
Remedy
The same blow-up construction can be used to eliminate the
discontinuities for h 4, the homology groups of the integral manifold M are given by
Hk(M) ∼=

Hk (B,B0)⊕ Hk−3N+6 (B, ∂B+)
2N − 5,2N − 4,2N − 3
3N − 8,3N − 7,3N − 6
5N − 10,5N − 9,5N − 8
Hk (B)⊕ Hk−3N+6 (B, ∂B) otherwise
For N ≥ 4, the homology groups of the reduced integral manifold MR
are given by
Hk (MR) ∼=

Hk (BR,BR0)⊕ Hk−3N+6
(
BR, ∂B
+
R
) 2N − 5,2N − 4
3N − 8,3N − 7
5N − 10,5N − 9
Hk (BR)⊕ Hk−3N+6 (BR, ∂BR) otherwise
Christopher McCord N-Body Integral Manifolds Oct. 2024 16 / 26
Homology Formulae
Theorem
For N = 4, the homology groups of H∗(M) are given by
Hk (M) ∼=

im (j∗ : H3(B) → H3(B,B0)) k = 3
Hk (B,B0)⊕ Hk−6 (B, ∂B+) k = 4,5,6,10,11,12
Hk (B)⊕ Hk−6 (B, ∂B) k = 0,1,2,7,8,9, k > 12
For N = 3, the homology groups of H∗(M) are given by
Hk (M) ∼=

coker (ι∗ : H1(Bf ) → H1(B)) 1
∂−1(im (ι∗ : H1(Bf ) → H1(B0)) 2
coker (ι∗ : H2(Bf , ∂Bf ) → H2(B, ∂B))⊕ H5(B) 5
∂−1(im (ι∗ : H2(Bf , ∂Bf ) → H2(B0, ∂B0))⊕ H6(B,B0) 6
Hk (B,B0)⊕ Hk−3 (B, ∂B+) 3,7
Hk (B)⊕ Hk−3 (B, ∂B) 0,4,8, . . .
where the boundary maps for k = 2 and k = 6 are
∂ : H2(B,B0) → H1(B0)
and
∂ : H3(B, ∂B+) → H2(B0, ∂B0)
respectively.
Christopher McCord N-Body Integral Manifolds Oct. 2024 17 / 26
Homology Formulae
The projection of M(c,h) onto the position coordinates defines the
Hill’s region H(c,h).
Corollary
For any set of N masses, the projection Π : M(c,h) → H(c,h) yields
an isomorphism Π∗ : Hk (M(c,h)) → Hk (H(c,h)) and
π∗ : Hk (MR(c,h)) → Hk (HR(c,h)) for k ≤ 3N − 6, while
Hk (H(c,h)) = Hk (H(c,h)) = 0 for k > 3N − 6.
Alternatively,
Hk (H(c,h)) ∼=
Hk (B,B0)
2N − 5,2N − 4,2N − 3
3N − 8,3N − 7,3N − 6
Hk (B) otherwise
Christopher McCord N-Body Integral Manifolds Oct. 2024 18 / 26