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Eigenvalues and eigenfunctionsfor second derivative of the action
at the figure-eightand slalom solutions
Toshiaki Fujiwara, Hiroshi Fukuda, and Hiroshi OzakiSymposium on Celestial Mechanics 2018
天体力学Ｎ体力学研究会NAO, Mitaka
国立天文台，三鷹
!1
Figure-eight and slalom solutions
!2
S1 S2
S3 figure-eight(5)
figure-eight: Moore 1993, Chenciner & Montgomery 2000slalome: Šuvakov & Dmitrašinović 2013
same homotopy classfive iterations
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Three-body choreography
!3
figure-eight solutionC. Moore 1993,
A. Chenciner and R. Montgomery 2000
Slalom Solutions• Šuvakov M.
• Numerical search for periodic solutions in the vicinity of the figure-eight orbit: slaloming around singularities on the shape sphere, Celest. Mech. Dyn. Astron. 119, 369–377 (2014)
• Šuvakov M., Dmitrašinović V.• Three classes of Newtonian three-body planar periodic orbits,
Phys. Rev. Lett. 110(11), 114301 (2013)• Šuvakov M., Dmitrašinović V.
• A guide to hunting periodic three-body orbits, Am. J. Phys. 82, 609–619 (2014)
• Šuvakov M., Shibayama M• Three topologically nontrivial choreographic motions of three
bodies, Celest. Mech. Dyn. Astron. 124, 155–162 (2016)
!4
Continuation of solutions
!5
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S1
S3
S5S2
S4
S6
SX
S2
figure-eight(5)
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Bifurcations aroundfigure-eight(5)
!6
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S1S2S3
figure-eight(5)
SX
Bifurcations aroundfigure-eight(5)
!7
S1S2
S3figure-eight(5)
SX
Linear stability
!8
Hessian of action
!9
Equal mass planar three-body problem
Eigenvalue problem at a critical point
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: Hessian
Decomposition of Hessianby symmetries
!10
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CC+: choreographic cos +CC-: cos –
CS+: choreographic sin +
CS–: sin –
zero-choreographic that don’t have
choreographic components
trivial the same summitry as figure-eight and slaloms
For detail, see appendix: “Decomposition of the Hessian matrix for action at choreographic three-body solutions” that is enclosed in the same proceedings.
Extra symmetry of Hessianfor figure-eight(5)
!11
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12
34
0<latexit sha1_base64="8jpcVGopBCFNezhAx5t4ely4SWU=">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</latexit>
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<latexit sha1_base64="VcPGsVz6LnB7Qhh9njJRj42DzN4=">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</latexit>
<latexit sha1_base64="TzjxnJxr7talNe9pwYAafvpFTA4=">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</latexit>
eigenfunctions are labeled by<latexit sha1_base64="Mx3FQsjsMK/3y7OS84niE540Gbo=">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</latexit>
Saddle-node bifurcation
!12
<latexit sha1_base64="DQcpjelXypqV2ovhkioc6HzRKqE=">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</latexit>
Eigenvalues of H
!13
S1
S2<latexit sha1_base64="5odyXstUEcN8eKiCquZXqiwqN70=">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</latexit>
<latexit sha1_base64="vrzWHtNlXrBHsaJ1m4LZ+j8IGVs=">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</latexit>
<latexit sha1_base64="BmH1B5YdHjHDWWdHHotNB6yMrJU=">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</latexit>
<latexit sha1_base64="N5FM3RcqjLAgrUBJRsvVoK6umLs=">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</latexit>
Eigenvalues of H for CS+
!14
<latexit sha1_base64="Jh3c5gDDssUHA8ilQ1j/+JmZ5Mg=">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</latexit>
<latexit sha1_base64="vTNGXnNJjaLUP37K0TccBWmzlug=">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</latexit>
<latexit sha1_base64="jtxmRrYaHj0SSZRXzLzAv4S/Os8=">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</latexit>
no solution
action
Euler characteristic:<latexit sha1_base64="IYVIW0edRJwXVRSTGwDdTL1c1uY=">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</latexit>
<latexit sha1_base64="IvxwjvvX4uRtK9Ny6k2zajhggOU=">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</latexit>
Eigenvalues of H for CS+
!15<latexit sha1_base64="Jh3c5gDDssUHA8ilQ1j/+JmZ5Mg=">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</latexit>
<latexit sha1_base64="vTNGXnNJjaLUP37K0TccBWmzlug=">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</latexit>
action
<latexit sha1_base64="Gb0jHpeA1UkKQpaXEX4v9n6N8H0=">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</latexit>
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Euler characteristic
!16
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5
Pitchfork bifurcation
!17
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Eigenvalues of H
!18
S3
SX
figure-eight(5)
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♣1.1031 1.1032 1.1033 1.1034 1.1035
-2.0
-1.5
-1.0
-0.5
0.5figureEight(5), S3 and SX: negative and almost zero engenvalues
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Eigenvalues of H for CS+, CC+
!19
Choreographic Sin +
S3SX
figure-eight(5)
Choreographic Cos +
figure-eight(5)
S3
SX
the eigenvalues for CS+ and CC+ of figure-eight(5) shown here are degenerated
(1-4 doublet)
linear stability of figure-eight(1)
!20
0.5 1.0 1.5 2.0 2.5
-π
- 4π5
- 2π5
2π5
4π5
π
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period-5 bifurcation: S3, SX
choreographic sin+ and cos+ (1-4 doublet)
Euler characteristics
!21
Choreographic Sin + Choreographic Cos +
CS+ CC+
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<latexit sha1_base64="RKXKEyFWQWCIkGjsgT3xE2nk2B0=">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</latexit>
CS+ :CC+ :CS+ and CC+ :
A qualitative model for action
!22
Choreographic Sin + Choreographic Cos +
SXS3
figure-eight(5)
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A qualitative model for action
!23
SXS3
figure-eight(5)
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0.5
1
2.
2.5
1
2
3
4
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<latexit sha1_base64="WF9NQuyuygNRbn8xSnexzqiX+8c=">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</latexit>
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Bifurcations aroundfigure-eight(5)
!24
Bifurcations aroundfigure-eight(5)
!25
S1S2
S1 and S2 are isolated from figure-eight(5)
and other slaloms ???
… and other bifurcationsperiod doubling, non-figure-eight, non-choreographic,
other periods (7,11,…), …
S3
![ESTIMATES ON NEUMANN EIGENFUNCTIONS AT THE …ahb/papers/... · NEUMANN EIGENFUNCTIONS AT THE BOUNDARY 3 This article is a report on completed work on the Dirichlet case [5], and](https://img.pdfslide.tips/doc/110x75/600eb4a04e0432198f2ea1a8/estimates-on-neumann-eigenfunctions-at-the-ahbpapers-neumann-eigenfunctions.jpg)
