exact solutions for 3-body and 4-body problems in 4-dimensional space hideki ishihara osaka city...
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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
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
Research Institute for Theoretical Physics
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).
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
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
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
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
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
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
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 ]
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
• 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
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
3-dim Gravity
Introduction
Gravitational phenomena depend on spacetime dimension
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
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
N-body problemN-body problem under the gravitational under the gravitational interactioninteraction
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
N-body problem in 4-dim. space
Equations of motion
Lagrangian , Energy
Potential is homogeneous in order -2.
Bounded orbits
Constant inertial moment
Examples
Exact solutions for 4-body problem 3-body problem in 4-dimensional space.
4-body problem
Special configuration with the same mass
Lagrangian
Graviational potential
Effective LagrangianLagrangian
Effective Lagrangian
Constants of motion
integrable !
Bounded solutionsEquations of motion
For bounded orbits
Exact solutions
For 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
=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
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
=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
3-body problem in 4-dimensions
Special configuration with the same mass
Lagrangian
Graviational potential
Effective Lagrangian
Bounded solutionsEquations of motion
For bounded orbits
Exact solutions
Elliptic integral of the second kind
Elliptic integrals of the first kind and third kind
Condition for closed orbits
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
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
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
Constrained system
Constant of motion on the constraint
System admits conformal Killing vector
Killing hierarchy(T.Igata,T.Koike,and H.I.)
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