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![Page 1: Exact Solutions for 3-body and 4-body Problems in 4-dimensional Space Hideki Ishihara Osaka City University.](https://reader031.fdocuments.in/reader031/viewer/2022012402/56649edc5503460f94bed7f0/html5/thumbnails/1.jpg)
Exact Solutions for Exact Solutions for 3-body and 4-body 3-body and 4-body Problems Problems in 4-dimensional Spacein 4-dimensional Space
Hideki IshiharaOsaka City University
![Page 2: Exact Solutions for 3-body and 4-body Problems in 4-dimensional Space Hideki Ishihara Osaka City University.](https://reader031.fdocuments.in/reader031/viewer/2022012402/56649edc5503460f94bed7f0/html5/thumbnails/2.jpg)
1944 Research Institute for Theoretical Physics, Hiroshima
University was founded1948 RITP was re-build after the world war II at Takehara,
Hiroshima 1990 RITP Hiroshima University was closed and merged
together with Yukawa Institute, Kyoto University
![Page 3: Exact Solutions for 3-body and 4-body Problems in 4-dimensional Space Hideki Ishihara Osaka City University.](https://reader031.fdocuments.in/reader031/viewer/2022012402/56649edc5503460f94bed7f0/html5/thumbnails/3.jpg)
Research Institute for Theoretical Physics
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Journal of Science of Hiroshima University,Series A5 (1935)
•P. A. M. Dirac,
"Generalized Hamiltonian dynamics". Can. J. Math. 2: 129–48 (1950).
•R.Arnowitt, S.Deser and C.W.Misner,
"Canonical variables for general relativity,'' Phys. Rev. 117, 1595 (1960).
•B. S. DeWitt,
"Quantum theory of gravity. I. The canonical theory". Phys. Rev. 160: 1113–48 (1967).
![Page 5: Exact Solutions for 3-body and 4-body Problems in 4-dimensional Space Hideki Ishihara Osaka City University.](https://reader031.fdocuments.in/reader031/viewer/2022012402/56649edc5503460f94bed7f0/html5/thumbnails/5.jpg)
三村 剛昂 19 35
岩付 寅之助 19 20
細川 藤右衛門 19 20
森永 覚太郎 19 26
佐久間 澄 19 26
藤原 力 19 20
柴田 隆史 19 26
原田 雅登 22
( )高久 浩俊 旧姓 熊川 22
佐伯 敬一 22
竹野 兵一郎 21 47
池田 峰夫 23 38
木村 利栄 23 1
占部 實 23 26
伊藤 誠 23 26
宮地 良彦 24 35
上野 義夫 25 56
庄野 直美 26 27
中井 浩 26 27
脇田 仁 28 40
成相 秀一 28 61
冨田 憲二 38 2
田地 隆夫 41 55
横山 寛一 41 2
永井 秀明 41 59
久保 禮次郎 41 2
寺崎 邦彦 41 2
冨松 彰 48 60
佐々木 隆 57 2
藤川 和男 58 2
上原 正三 59 2
佐々木 節 61 2
中澤 直仁 63
細谷 暁夫 62 1
須藤 靖 1 2
二宮 正夫 1 2
Staff history of RITP
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Early era
三村 剛昂 19 35
岩付 寅之助 19 20
細川 藤右衛門19 20
森永 覚太郎 19 26
佐久間 澄 19 26
藤原 力 19 20
柴田 隆史 19 26
竹野 兵一郎 21 47
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Middle era 木村 利栄 23 1
上野 義夫 25 56
成相 秀一 28 61
冨田 憲二 38 2
田地 隆夫 41 55
横山 寛一 41 2
永井 秀明 41 59
久保 禮次郎 41 2
寺崎 邦彦 41 2
冨松 彰 48 60
佐々木 隆 57 2
藤川 和男 58 2
上原 正三 59 2
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Students in the middle era 前川 敬好 35 38
青木 正典 41
江沢 康生 41 45
登谷 美穂子 44 46
田辺 健茲 44 58
南方 久和 45 49
岡 隆光 46 50
小野 隆 47 58
新谷 明雲 49 56
堀内 利得 49 58
東 孝博 50 55
遠藤 龍介 51 58
原田 和男 51 56
中澤 直仁 55 63
石原 秀樹 55 59
矢嶋 哲 58 62
葛西 真寿 58 62
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Late era 冨田 憲二 38 2
横山 寛一 41 2
久保 禮次郎 41 2
寺崎 邦彦 41 2
佐々木 隆 57 2
藤川 和男 58 2
上原 正三 59 2
佐々木 節 61 2
中澤 直仁 63
細谷 暁夫 62 1
須藤 靖 1 2
二宮 正夫 1 2
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Students in the late era
( )山本 寿 同志社女子大学 生活科学部 60 62
( )杉山 直 国立天文台 61 63
( )南部 保貞 名古屋大学理学部 60 1
( )早田 次郎 京都大学理学部 61 2
( )中尾 憲一 大阪市立大学理学研究科 61 2
( )鈴木 博 理化学研究所 62 2
( )渡辺 一也 新潟大学理学部 62 2
( )山本 一博 広島大学理学部 1 2
( )上田 晴彦 秋田大学 教育文化学部 1 2
( )松原 隆彦 名古屋大学大学院理学研究科 2
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Quantum field theory in the expanding universe (H.Nariai and T.Kimura)
• ADM formalism in expanding universes
H.Nariai and T.Kimura, PTP 28(’62) 529. [L.Abbot and S. Deser, (’82)]
• Quantization of gravitational wave and mater fields in expanding universes
H.Nariai and T.Kimura, PTP 29(’63) 269; PTP 29(’63) 915; PTP 31(’64) 1138. [A.Penzias and R.Wilson (’63)] [L.Parker PRL 21 (’68) 562 ] [S.W.Hawking, Nature 248 (’74) 30 ]
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Development• Gravitational anomaly T.Kimura, PTP 42 (‘69)1191; PTP 44 (‘70)1353
• Removal of the initital singularity in a big-bang universe
H.Nariai, PTP 46 (‘71)433, H.Nariai and K.Tomita, PTP 46
(‘71) 776
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• In theoretical physics, “unrealistic and non-urgent work” happens to turn to a cardinal issue.
• We should not ask a physically reasonable motivation so urgently.
In the special issue for 60th anniversary of prof. Nariai
But, it would be also necessary to keep a sort of soundness at each stage of research.
Humitaka Sato says
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Exact Solutions for Exact Solutions for 3-body and 4-body Problems 3-body and 4-body Problems
in 4-dimensional Spacein 4-dimensional Space
Hideki IshiharaOsaka City University
Shall we start
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3-dim Gravity
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Introduction
Gravitational phenomena depend on spacetime dimension
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Kepler motion in 3-dim. v.s. 4-dim.
0 .5 1 .0 1 .5 2 .0 2 .5 3 .0
10
5
5
0 .5 1 .0 1 .5 2 .0 2 .5 3 .0
10
5
5
V3(r) V4(r)
Stable bound orbits appear only in the 3-dimensional gravity
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Black holes in general relativity
Black Ring
We shoud study Kerr black hole only
Myers & Perry (1986)
Emparan & Reall (2002)
Black Hole
(4+1)-dimensions
(3+1)-dimensions
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N-body problemN-body problem under the gravitational under the gravitational interactioninteraction
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3-body problem in 3-spatial 3-body problem in 3-spatial dimensionsdimensions
• 2-body (Kepler problem) : integrable → bound orbits are given by ellipses• 3-body : not integrable in general
small numbers of special solutions are known1765 Euler, 1772 Lagrange,
2000 Eight figure choreography
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N-body problem in 4-dim. space
Equations of motion
Lagrangian , Energy
Potential is homogeneous in order -2.
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Bounded orbits
Constant inertial moment
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Examples
Exact solutions for 4-body problem 3-body problem in 4-dimensional space.
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4-body problem
Special configuration with the same mass
Lagrangian
Graviational potential
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Effective LagrangianLagrangian
Effective Lagrangian
Constants of motion
integrable !
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Bounded solutionsEquations of motion
For bounded orbits
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Exact solutions
For closed orbits
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1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
=4/1, =3/1
Closed orbits
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
= 2/1 , = 2/1
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1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
=6/5, =4/3
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
= 4/3 , = 5/3
Closed orbits 2
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=3/2, =5/3
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
1 .0 0 .5 0 .5 1 .0
1 .0
0 .5
0 .5
1 .0
=3/2, =5/2
Closed orbits 3
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3-body problem in 4-dimensions
Special configuration with the same mass
Lagrangian
Graviational potential
Effective Lagrangian
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Bounded solutionsEquations of motion
For bounded orbits
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Exact solutions
Elliptic integral of the second kind
Elliptic integrals of the first kind and third kind
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Condition for closed orbits
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Closed orbit 1
1 .0 0 .5
0 .0
0 .5
1 .0
0 .5 0 .0 0 .5
0 .5
0 .0
0 .5
1 .0 0 .5
0 .00 .5
1 .0
0 .5
0 .00 .5
0 .5
0 .0
0 .5
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Closed orbit 2
1 .0 0 .5 0 .0 0 .5 1 .0
1 .0 0 .5
0 .00 .51 .0
1 .0
0 .5
0 .0
0 .5
1 .0
1 .0
0 .5
0 .0
0 .5
1 .0 1 .0
0 .5
0 .0
0 .5
1 .0
1 .0
0 .5
0 .0
0 .5
1 .0
1 .0 0 .50 .00 .51 .0 1 .0
0 .5
0 .0
0 .5
1 .0
0 .2 0 .0 0 .2
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Closed orbit 3
1 .0
0 .5
0 .0
0 .5
1 .0
0 .5
0 .0
0 .5
1 .0
0 .5
0 .0
0 .5
1 .0
1 .0 0 .5
0 .00 .51 .0
1 .0 0 .5 0 .0 0 .5 1 .0
0 .5
0 .0
0 .5
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Constrained system
Constant of motion on the constraint
System admits conformal Killing vector
Killing hierarchy(T.Igata,T.Koike,and H.I.)
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Conclusions
We consider systems of particles interacting by Newtonian Gravity in 4-dimensional space.
There exists a special class of solutions: vanishing total energy and constant moment of inertia
We obtain exact special solutions for 3-body and 4-body problems