Talk: Symplectic methods in trajectory design

Symplectic methods in space mission design
Prof. Agustin Moreno
IAS Princeton/Heidelberg
based on j.w.w. Dayung Koh (JPL), Urs Frauenfelder (Augsburg), Cengiz Aydin (Neuchâtel).
Agustin Moreno Science Coffee, ESA

What, Why, Where, How.
‚ (The What), i.e. Goal: Study periodic trajectories of Hamiltonian
systems, in families.

‚ (But Why), i.e. Motivation: Placing satellites in orbit around a
Planet–Moon system (a restricted three-body problem).

‚ (And Where), i.e. Problem: Orbits in families undergo bifurcation
ù new families. Need methods to keep track of this data.

‚ (And Where), i.e. Problem: Orbits in families undergo bifurcation
ù new families. Need methods to keep track of this data.
Aim of the talk: The How.

Motivating questions
(Classification) Given two orbits, can we tell if they are
qualitatively different, i.e. there is no regular family between
them?

them?
(Catalogue) Can we refine existing data bases of periodic
orbits?

them?
(Catalogue) Can we refine existing data bases of periodic
orbits?
(Symplectic geometry) Can we use modern mathematical
methods from symplectic geometry to guide the numerical
work?

Goal of the talk: introduce our toolkit
(1) The B-signs: ˘ signs associated to elliptic/hyperbolic
orbits, which help predict bifurcations.

(2) Global topological methods: the GIT-sequence, a
topological refinement of Broucke’s stability diagram, which
encodes bifurcations and stability of orbits.

(3) Conley-Zehnder indices: a number associated to a
(non-degenerate) orbit which only jumps at bifurcation, and
so predicts which families connect to which.

(4) Floer numerical invariants: numerical counts of orbits that
stay the same before and after a bifurcation, and so help
predict existence of orbits.

Preliminaries

Symplectic geometry and Hamiltonian dynamics
Mechanics: classical particles are point-like and massive, and
move in phase-space.

Phase-space: pq, pq “ pposition, momentaq P Rn ‘ Rn. Phase-
space is the collection M “ tpq, pqu Ă R2n.

Phase-space: pq, pq “ pposition, momentaq P Rn ‘ Rn. Phase-
space is the collection M “ tpq, pqu Ă R2n.
Hamiltonian: a system is given by an energy function
H : M Ñ R, H “ Hpq, pq.

A solution to the equations of motion is a curve t Ñ xptq “ pqptq, pptqq P
M which solves Hamilton’s equations:
#
9
q “ BH
Bp
9
p “ ´BH
Bq

A solution to the equations of motion is a curve t Ñ xptq “ pqptq, pptqq P
M which solves Hamilton’s equations:
#
9
q “ BH
Bp
9
p “ ´BH
Bq
The Hamiltonian flow φt : M Ñ M preserves the symplectic form, i.e.
φ˚
t ω “ ω where
ωpv, wq “ vT
Jw, J “
ˆ
0 1
´1 0
˙
.

Restricted three-body problem
Setup. Three massive objects: Earth (E), Moon (M), Satellite (S), un-
der gravitational interaction.
Classical assumptions:
1 (Restricted) mS “ 0, i.e. S is negligible.
2 (Circular) The primaries E and M move in circles around their
center of mass.
3 (Planar) S moves in the plane containing E and M, n “ 2 .
Spatial case: drop the planar assumption, n “ 3 .

Restricted three-body problem
Setup. Three massive objects: Earth (E), Moon (M), Satellite (S), un-
der gravitational interaction.
Classical assumptions:
1 (Restricted) mS “ 0, i.e. S is negligible.
2 (Circular) The primaries E and M move in circles around their
center of mass.
3 (Planar) S moves in the plane containing E and M, n “ 2 .
Spatial case: drop the planar assumption, n “ 3 .
Two parameters: µ (mass of Moon), and c (Jacobi constant =
energy).
Different choices of µ models different systems in our Solar system
(Jupiter–Europa, Saturn–Enceladus, etc).

Monodromy matrix
Notation: Spp2nq “ tsymplectic matricesu.
The monodromy matrix of a periodic orbit x is
Mx “ DφH
T P Spp2nq, where T is the period of x, and φH
t is the
Hamiltonian flow.
Note: 1 appears twice as a trivial eigenvalue of Mx . Can ignore them
if we consider the reduced monodromy matrix Mred
x P Spp2n ´ 2q.

Monodromy matrix
Notation: Spp2nq “ tsymplectic matricesu.
The monodromy matrix of a periodic orbit x is
Mx “ DφH
T P Spp2nq, where T is the period of x, and φH
t is the
Hamiltonian flow.
Note: 1 appears twice as a trivial eigenvalue of Mx . Can ignore them
if we consider the reduced monodromy matrix Mred
x P Spp2n ´ 2q.
A Floquet multiplier of x is an eigenvalue of Mx , which is not one
of the trivial eigenvalues (i.e. an eigenvalue of Mred
x ).
An orbit is non-degenerate if 1 does not appear among its
Floquet multipliers.
An orbit is stable if all its Floquet multipliers are semi-simple and
lie in the unit circle.

Lemma
If µ P C is an eigenvalue of Mx , then so are µ, 1{µ, 1{µ.
μ
1/μ
1
μ 1/μ
-1
H
H
λ
λ
E
C C
τ
1/τ
1/τ
τ
Elliptic (E), positive/negative hyperbolic (H˘
q, nonreal (N).

Bifurcations
"Foundations of Mechanics", Abraham-Marsden.
Period-doubling bifurcation
Period-doubling bifurcation or subtle division.

Bifurcations
Creation or birth/death.

Emission, or k-fold bifurcation (k “ 4).

Symmetries
An anti-symplectic involution is a map ρ : M Ñ M satisfying
ρ2 “ 1;
ρ˚ω “ ´ω.

Symmetries
ρ2 “ 1;
ρ˚ω “ ´ω.
Its fixed-point locus is fixpρq “ tx : ρpxq “ xu.

Symmetries
ρ2 “ 1;
ρ˚ω “ ´ω.
Its fixed-point locus is fixpρq “ tx : ρpxq “ xu. An anti-symplectic
involution ρ is a symmetry of the system if H ˝ ρ “ H.

Symmetries
ρ2 “ 1;
ρ˚ω “ ´ω.
Its fixed-point locus is fixpρq “ tx : ρpxq “ xu. An anti-symplectic
involution ρ is a symmetry of the system if H ˝ ρ “ H.
A periodic orbit x is symmetric if ρpxp´tqq “ xptq for all t.
L=fix(ρ)
symmetric points

Wonenburger matrices
The monodromy matrix of a symmetric orbit at a symmetric point has
special form, a Wonenburger matrix:
M “ MA,B,C “
ˆ
A B
C AT
˙
P Spp2nq, (1)
where
B “ BT
, C “ CT
, AB “ BAT
, AT
C “ CA, A2
´ BC “ 1, (2)

Wonenburger matrices
The monodromy matrix of a symmetric orbit at a symmetric point has
special form, a Wonenburger matrix:
M “ MA,B,C “
ˆ
A B
C AT
˙
P Spp2nq, (1)
where
B “ BT
, C “ CT
, AB “ BAT
, AT
C “ CA, A2
´ BC “ 1, (2)
The eigenvalues of M are determined by those of the first block A:
Lemma
λ e-val of M ù its stability index apλq “ 1
2 pλ ` 1{λq e-val of A.
a e-val of A ù λpaq “ a `
?
a2 ´ 1 e-val of M.

Toolkit

Global topological methods
These methods encode:
Bifurcations;
stability;
eigenvalue configurations;
obstructions to existence of regular families;
B-signs,
in a visual and resource-efficient way.

Broucke’s stability diagram: 2D
Let n “ 2 , λ eigenvalue of Mred P Spp2q, with stability index apλq “
1
2pλ ` 1{λq. Then:
λ “ ˘1 iff apλq “ ˘1.
λ positive hyperbolic iff apλq ą 1;
λ negative hyperbolic iff apλq ă ´1;
λ elliptic (stable) iff ´1 ă apλq ă 1.
-1 1
pos. hyp
neg. hyp
elliptic
λ
1/λ
a(λ)
+
-
Broucke stability
diagram
periodic orbits Symplectic
matrices
stable
region
Krein
signs

Broucke’s stability diagrams: 3D
Let n “ 3 . Given Mred “ MA,B,C P Spp4q, its stability point is p “
ptrpAq, detpAqq P R2.

Broucke’s stability diagrams: 3D
Let n “ 3 . Given Mred “ MA,B,C P Spp4q, its stability point is p “
ptrpAq, detpAqq P R2. The plane splits into regions corresponding to the
eigenvalue configuration of Mred :
2
-2 2
-2
N
H
H
E
E
H
E
H
H
Γ
Γ
d
1
Γ
-1
Γ˘1 = e-val ˘1.
Γd = double e-val.
E2 = doubly elliptic (stable region).
etc.

Bifurcations in the Broucke diagram
An orbit family t ÞÑ xt induces a path t ÞÑ pt P R2 of stability points.

The family bifurcates if pt crosses Γ1.
Γ
d
Γ
-1
Γ
1
Γ
d
Γ
-1
Γ
1
bifurcation
H
E H
E
2
E
period-doubling
bifurcation

Γ
d
Γ
-1
Γ
1
Γ
d
Γ
-1
Γ
1
bifurcation
H
E H
E
2
E
period-doubling
bifurcation
More generally:
Γθ = line with slope cosp2πθq P r´1, 1s = matrices with e-val e2πiθ;
Γλ = line with slope apλq P Rzr´1, 1s = matrices with e-val λ.
A k-fold bifurcation happens when crossing Γl{k for some l.

Γ
d
Γ
-1
Γ
1
Γ
d
Γ
-1
Γ
1
bifurcation
H
E H
E
2
E
period-doubling
bifurcation
More generally:
Γθ = line with slope cosp2πθq P r´1, 1s = matrices with e-val e2πiθ;
Γλ = line with slope apλq P Rzr´1, 1s = matrices with e-val λ.
A k-fold bifurcation happens when crossing Γl{k for some l.
If we know that two points lie in different components, then one should
expect bifurcations in any path between them.

B-signs
Assume n “ 2, 3 . Let x be a symmetric orbit with monodromy
MA,B,C “
ˆ
A B
C AT
˙
at a symmetric point. Assume a is a real, simple and nontrivial eigen-
value of A (i.e. λpaq elliptic or hyperbolic)). Let v satisfy AT v “ a ¨ v.
The B-sign of λpaq is
pλpaqq “ signpvT
Bvq “ ˘.

B-signs
Assume n “ 2, 3 . Let x be a symmetric orbit with monodromy
MA,B,C “
ˆ
A B
C AT
˙
at a symmetric point. Assume a is a real, simple and nontrivial eigen-
value of A (i.e. λpaq elliptic or hyperbolic)). Let v satisfy AT v “ a ¨ v.
The B-sign of λpaq is
pλpaqq “ signpvT
Bvq “ ˘.
Note: Independent of v.
n “ 2 two B-signs 1, 2, one for each symmetric point.
n “ 3 two pairs of B-signs p1
1, 1
2q, p2
1, 2
2q, one for each
symmetric point and each eigenvalue.
Fact: A planar symmetric orbit is negative hyperbolic iff the B-signs of
its two symmetric points differ (Frauenfelder–M. [FM], ’23).

Global topological methods: GIT sequence, 2D
GIT sequence = refinement of Broucke diagram for symmetric orbits.
-1 1
pos. hyp
pos. hyp I
pos. hyp II
neg. hyp II
neg. hyp
elliptic
elliptic
λ
1/λ
a(λ)
+
-
+
-
+
-
+
-
Broucke stability
diagram
neg. hyp I
periodic orbits
symmetric
periodic orbits
Wonenburger
matrices
Symplectic
matrices
stable
region
Krein
signs
B-signs
B-signs “separate” hyperbolic branches, for symmetric orbits.
If two points lie in the same component of the Broucke diagram,
but if B-signs differ, one should also expect bifurcation in any path
joining them.

Global topological methods: GIT sequence, 3D
Γ
d
1
Γ
1
N
H
Γ
d
3
N
H
H E
H
E H
B-signs
Krein signs
E
3
Γ
-1 H
H
E 1
Γ
1 H
H
E 1
Γ
1
2
2
E H
E
Γ
-1
2
2
E Γ
d
2
2
Γ
-1 H
H
E 3
N
Broucke stability
diagram
periodic orbits
symmetric
periodic orbits
Wonenburger
matrices
Symplectic
matrices
stable
region stable
region
The branches are two-dimensional, and come together at the “branching
locus”, where we cross from one region to another.

Conley–Zehnder index
The CZ-index (introduced by Conley and Zehnder) is part of the index
theory of the symplectic group. It assigns a (winding) number to non-
degenerate orbits.

The CZ-index (introduced by Conley and Zehnder) is part of the index
theory of the symplectic group. It assigns a (winding) number to non-
degenerate orbits.
Helps understand which families of orbits connect to which
(CZ-index stays constant if no bifurcation occurs);
Helps determine if orbits are elliptic/hyperbolic.

n=2 x planar orbit with (reduced) monodromy Mred
x , xk k-fold cover.
Elliptic case: Mred
x conjugated to rotation,
Mred
x „
ˆ
cos θ ´ sin θ
sin θ cos θ
˙
,
with Floquet multipliers e˘2πiθ. Then
µCZ pxk
q “ 1 ` 2 ¨ tk ¨ θ{2πu
In particular, it is odd, and jumps by ˘ 2 if the e-val 1 is crossed.

n=2 x planar orbit with (reduced) monodromy Mred
x , xk k-fold cover.
Elliptic case: Mred
x conjugated to rotation,
Mred
x „
ˆ
cos θ ´ sin θ
sin θ cos θ
˙
,
with Floquet multipliers e˘2πiθ. Then
µCZ pxk
q “ 1 ` 2 ¨ tk ¨ θ{2πu
In particular, it is odd, and jumps by ˘ 2 if the e-val 1 is crossed.
Hyperbolic case:
Mred
x „
ˆ
λ 0
0 1{λ
˙
,
with Floquet multipliers λ, 1{λ. Then
µCZ pxk
q “ k ¨ n,
where DXHptq rotates eigenspaces by angle πnt
T , with n even/odd
if x pos./neg. hyp.

CZ-jumps
µCZ jumps by ˘1 when crossing 1, according to direction of bifurcation. If it
stays elliptic, the jump is by ˘2.

n “ 3 , planar orbits. Assume H admits the reflection along the px, yq-
plane as symmetry (e.g. 3BP). If x Ă R2 planar orbit,
Mred
x „
ˆ
Mred
p 0
0 Ms
˙
P Spp4q.
Then
µCZ pxq “ µp
CZ pxq ` µs
CZ pxq,
where each summand corresponds to Mred
p and Ms respectively.
Planar to planar bifurcations correspond to jumps in µp
CZ .
Planar to spatial bifurcations correspond to jumps of µs
CZ .

Floer numerical invariants
A periodic orbit x is good if µCZ pxk q “ µCZ pxqpmod 2q for all
k ě 1.
Note: a planar orbit is bad iff it is an even cover of a negative hyper-
bolic orbit.

Floer numerical invariants
A periodic orbit x is good if µCZ pxk q “ µCZ pxqpmod 2q for all
k ě 1.
Note: a planar orbit is bad iff it is an even cover of a negative hyper-
bolic orbit.
energy
before
after
bifurcation
CZ CZ
CZ
CZ
CZ
bef bef
aft
aft
aft
1
1
2
2
3
degenerate
orbit
x
Given a bifurcation at x, the SFT-Euler characteristic (or the Floer
number) of x is
χSFT pxq “
ÿ
i
p´1qCZbef
i “
ÿ
j
p´1qCZaft
j .
The sum on the LHS is over good orbits before bifurcation, and RHS
is over good orbits after bifurcation.

Invariance
The fact that the sums agree before and after –invariance– follows
from Floer theory in symplectic geometry.
In Memoriam Andreas Floer, 1956-1991.
The Floer number can be used as a test: if the sums do not
agree, we know the algorithm missed an orbit.

Example: symmetric period doubling bifurcation
L L L
t=0 t=1/2
A B C
symmetric points
symmetric points fake points symmetric points
fake points
The simple symmetric orbit x goes from elliptic to negative hyperbolic.
A priori there could be two bifurcations for each symmetric point
(B or C).

L L L
t=0 t=1/2
A B C
symmetric points
fake points
(B or C).
Invariance of χSFT px2q implies only one can happen (note x2 is
bad).

L L L
t=0 t=1/2
A B C
symmetric points
fake points
(B or C).
Invariance of χSFT px2q implies only one can happen (note x2 is
bad).
Bifurcation happens at the symmetric point in which the B-sign
does not jump.

Summary of toolkit
(1) The B-signs: a ˘ sign associated to each elliptic or
hyperbolic Floquet multiplier of an orbit, which helps predict
bifurcations.
(4) Floer numerical invariants: numerical counts of orbits that
stay the same before and after a bifurcation, and so help
predict existence of orbits.

Numerical work

Missions
To find conditions suitable for life,
missions proposed by NASA:
Jupiter-Europa system
(Europa Clipper); and
Saturn-Enceladus system.
This motivates studies of orbits for
these systems.

The power of deformations
Two options:
Fix µ and change c; or
Fix c and change µ.

The power of deformations
Two options:
Fix µ and change c; or
Fix c and change µ.
I.e. to study a system, sometimes it is worthy to study another nearby
system:
Hill’s lunar problem ù Saturn–Enceladus ù Jupiter–Europa ù
Earth–Moon.

Example: Pitchfork bifurcation
g
g
g' g'
pitchfork
birth-death
deformation
before
after
energy
g
LPO1
LPO2
DPO
mass
3
3 3
3
3
2 2
Lunar problem has more symmetry: a (non-generic) pitchfork bifurcation in
lunar problem (Hénon) deforms to a generic situation in Jupiter–Europa.
Birth-death branch might be hard to predict otherwise.

Hill’s lunar problem
Bifurcation graph involving covers of f, g, g1
(Cengiz Aydin, PhD thesis ’23).
Each family has constant CZ-index. Floer invariants are easy to compute.

Numerical work
Period-doubling bifurcation in the Jupiter-Europa system
(µ “ 2.5266448850435E´05
), found via the cell-mapping method of
Koh–Anderson–Bermejo-Moreno [KAB].

GIT plots
2
E
2
E
H
E E
H
E
2
E
H
E
GIT plot of the period-doubling bifurcation of the snitch configuration
(Frauenfelder–Koh–M. [FKM]).

Thank you!

References I
Cengiz Aydin.
A study of the Hill three-body problem by modern symplectic
geometry.
PhD Thesis, Université de Nauchâtel, 2023.
Urs Frauenfelder, Dayung Koh, Agustin Moreno.
Symplectic methods in the numerical search of orbits in real-life
planetary systems.
Preprint arXiv:2206.00627.
Urs Frauenfelder, Agustin Moreno.
On GIT quotients of the symplectic group, stability and bifurcations
of symmetric orbits.
To appear in the Journal of Symplectic Geometry.

References II
Urs Frauenfelder, Agustin Moreno.
On doubly symmetric periodic orbits.
Celestial Mech. Dynam. Astronom. 135 (2023), no. 2, Paper No.
20..
Dayung Koh, Rodney L. Anderson, Ivan Bermejo-Moreno.
Cell-mapping orbit search for mission design at ocean worlds
using parallel computing.
The Journal of the Astronautical Sciences, Volume 68, Issue 1,
p.172-196.

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