Dynamical boundaries in a variety of mechanical systems Shane D. Ross Engineering Science and Mechanics, Virginia Tech Virginia Tech‐Wake Forest Univ. School of Biomedical Eng. and Sciences
Separatrices: dynamical boundaries Coastal flow Transport barriers in state space separating qualitatively different kinds of behavior
Separatrices: dynamical boundaries Region A Coastal flow Region B Transport barriers in state space separating qualitatively different kinds of behavior
Separatrices: high dimensions System with many basins, not necessarily attracting sets or attractors Potential surface with several minima (“bowls”) separated by ridges Basins are “almost‐invariant structures” Coastal flow System known analytically: vector field or map
Separatrices: high dimensions Coastal flow Small body in solar system : Transport from one basin to another controlled by high dimensional separatrix surfaces Geometrically tubes in this case Tubes are attached to practically unobservable periodic orbits or other bound orbits
Realms and tubes � Planetary and sun realms connected by tubes 1 Sun Realm Planetary Realm p y L 1 x L 1 Sun y Position Space Phase Space (Position + Velocity) 1 Conley & McGehee, 1960s, found these locally, speculated use for “low energy transfers” 6
Transport between realms � Asymptotic orbits form 4D invariant manifold tubes ( S 3 × R ), separatrices in 5D energy surface 2 2 Ross [2006] The interplanetary transport network, American Scientist 7
Transport between realms Incoming P Tube Moon Earth Ballistic L 2 Capture Into Moon Outgoing Elliptical Orbit Tube � Tubes in phase space ◦ Objects mediating transport through bottlenecks 8
Tube dynamics Earth Realm Moon Realm p y L 1 x y � Tube dynamics: All motion between realms connected by bottlenecks must occur through the interior of tubes 9
Multi-scale dynamics � Slices of energy surface: Poincar´ e sections U i � Tube dynamics: evolution between U i � What about evolution on on U i ? Poincare Section Poincare Section U 2 U 1 U 2 z 5 f 2 E n t rance z 3 f 2 z 4 L 1 f 1 2 E art h U 1 f 1 z 0 f 1 z 2 z 1 Exit 10
Some remarks on tube dynamics � Tubes are general; consequence of rank 1 saddle – saddle × center × · · · × center – e.g., ubiquitous in chemistry � Tubes persist – in presence of additional massive body – when primary bodies’ orbit is eccentric 11
Tubes in elliptic restricted 3-body problem Poincare Section Poincare Section U 1 U 2 Trajectories about to be captured L 1 x E art h x Consider first cut of stable manifold of L 1 NHIM 12
Tubes in elliptic restricted 3-body problem Gawlik, Marsden, Du Toit, Campagnola [2008] “Lagrangian coherent structures in the planar elliptic re- stricted three-body problem,” submitted to Celestial Mechanics and Dynamical Astronomy. 13
Tubes in elliptic restricted 3-body problem Gawlik, Marsden, Du Toit, Campagnola [2008] “Lagrangian coherent structures in the planar elliptic re- stricted three-body problem,” submitted to Celestial Mechanics and Dynamical Astronomy. 14
Some remarks on tube dynamics � Tubes are general; consequence of rank 1 saddle – saddle × center × · · · × center – e.g., ubiquitous in chemistry � Tubes persist – in presence of additional massive body – when primary bodies’ orbit is eccentric � Observed in the solar system (e.g., Oterma) � Even on galactic and atomic scales! Koon, Lo, Marsden, & Ross [2000], G´ omez, Koon, Lo, Marsden, Masdemont, & Ross [2004], Yamato & Spencer [2003], Wilczak & Zgliczy´ nski [2005], Ross & Marsden [2006], Gawlik, Marsden, Du Toit, Campagnola [2008], Combes, Leon, Meylan [1999], Heggie [2000], Romero-G´ omez, et al. [2006,2007,2008] 15
Multi-scale dynamics � Slices of energy surface: Poincar´ e sections U i � Tube dynamics: evolution between U i � − → What about evolution on on U i ? ← − Poincare Section Poincare Section U 2 U 1 U 2 z 5 f 2 E n t rance z 3 f 2 z 4 L 1 f 1 2 E art h U 1 f 1 z 0 f 1 z 2 z 1 Exit 16
Infinity to capture about small companion in binary pair? � After consecutive gravity assists , large orbit changes 17
Kicks at periapsis � Key idea: model particle motion as “ kicks ” at periapsis Semimajor Axis vs. Time Δ a + m 1 m 2 Δ a − Δ a − Δ a + In rotating frame where m 1 , m 2 are fixed 18
Kicks at periapsis � Sensitive dependence on argument of periapse ω Semimajor Axis vs. Time Δ a + m 1 m 2 Δ a − Δ a − Δ a + In rotating frame where m 1 , m 2 are fixed 19
Kicks at periapsis � Construct update map ( ω 1 , a 1 , e 1 ) �→ ( ω 2 , a 2 , e 2 ) using average perturbation per orbit by smaller mass ( ω 1 , a 1 , e 1 ) 20
Kicks at periapsis � Construct update map ( ω 1 , a 1 , e 1 ) �→ ( ω 2 , a 2 , e 2 ) using average perturbation per orbit by smaller mass ( ω 1 , a 1 , e 1 ) ( ω 2 , a 2 , e 2 ) 21
Not hyperbolic swing-by � Occur outside sphere of influence (Hill radius) – not the close, hyperbolic swing-bys of Voyager 22
Capture by secondary � Dynamically connected to capture thru tubes 23
Capture by secondary � Particle assumed on near-Keplerian orbit around m 1 � In the frame co-rotating with m 2 and m 1 , H rot ( l, ω, L, G ) = K ( L ) + µR ( l, ω, L, G ) − G, in Delaunay variables � Evolution is Hamitlon’s equations: d dt ( l, ω, L, G ) = f ( l, ω, L, G ) � Jacobi constant, C J = − 2 H rot conserved along trajectories 24
Change in orbital elements over one particle orbit � Evolution of G (angular momentum) dG dt = − µ∂R ∂ω, � Picard’s approximation: � T/ 2 ∂R ∆ G = − µ ∂ω dt − T/ 2 � r ��� π � π � �� � 3 = − µ � sin( ω + ν − t ( ν )) dν − sin ω 2 cos( ν − t ( ν )) dν G r 2 − π 0 � ∆ K = Keplerian energy change over an orbit ∆ K = ∆ G − µ ∆ R 25
Energy kick function � Changes have form ∆ K = µf ( ω ) , f is the energy kick function with parameters K, C J 20 10 f 0 −10 −20 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ω/π 26
Maximum changes on either side of perturber Semimajor Axis vs. Time Δ a + ω m a x m 1 m 2 − ω m a x Δ a − Δ a − Δ a + 20 10 f 0 −10 −20 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ω/π 27
The periapsis kick map (Keplerian Map) � Cumulative effect of consecutive passes by perturber � Can construct an update map ( ω n +1 , K n +1 ) = F ( ω n , K n ) on the cylinder Σ = S 1 × R , i.e., F : Σ → Σ where � ω n +1 ω n − 2 π ( − 2( K n + µf ( ω n ))) − 3 / 2 � � � = K n +1 K n + µf ( ω n ) � Area-preserving (symplectic twist) map � Example: particle in Jupiter-Callisto system µ = 5 × 10 − 5 28
Verification of Keplerian map: phase portrait Keplerian map 29
Verification of Keplerian map: phase portrait Keplerian map numerical integration of ODEs ◦ Keplerian map = fast orbit propagator ◦ preserves phase space features — but breaks left-right symmetry present in original system — can be removed using another method (Hamilton-Jacobi) 30
Dynamics of Keplerian map 1.8 1.7 . a 1.6 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ω/π Resonance zone 3 � Structured motion around resonance zones 3 in the terminology of MacKay, Meiss, and Percival [1987] 31
Dynamics of Keplerian map 1.8 1.7 . a 1.6 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 ω/π Resonance zone 4 � Structured motion around resonance zones 4 in the terminology of MacKay, Meiss, and Percival [1987] 32
Large orbit changes via multiple resonance zones � multiple flybys for orbit reduction or expansion P m 1 m 2 33
Large orbit changes, Γ n = F n (Γ 0 ) ω/π 0.10 Γ 0 0.05 Δ a 0 -0.05 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 34
Large orbit changes, Γ n = F n (Γ 0 ) ω/π 0.10 Γ 0 0.05 Δ a 0 -0.05 Γ 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 35
Large orbit changes, Γ n = F n (Γ 0 ) ω/π 0.10 Γ 0 0.05 Δ a 0 -0.05 Γ 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.10 0.05 0 -0.05 Γ 2 -0.10 Γ 10 -0.15 0 0.005 0.01 0.015 0.02 0.025 36
Large orbit changes, Γ n = F n (Γ 0 ) ω/π 0.10 Γ 0 0.05 Δ a 0 -0.05 Γ 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.10 0.05 0 -0.05 Γ 2 -0.10 Γ 10 -0.15 0 0.005 0.01 0.015 0.02 0.025 Γ 10 0 -0.05 -0.10 Γ 13 -0.15 0.006 0.007 0.008 0.009 0.010 0.005 37
Large orbit changes, Γ n = F n (Γ 0 ) ω/π 0.10 Γ 0 0.05 Δ a 0 -0.05 Γ 2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0.10 0.05 0 -0.05 Γ 2 -0.10 Γ 10 -0.15 0 0.005 0.01 0.015 0.02 0.025 Γ 10 0 -0.05 -0.10 Γ 13 -0.15 0.006 0.007 0.008 0.009 0.010 0.005 0.05 Γ 10 0 -0.05 Γ 25 -0.10 -0.15 b -0.20 0.008461 0.008463 0.008465 0.008467 0.008469 38
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