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神部 勉
Tsutomu
KAMBE, Tokyo, Japan
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Today's Weather 今日の天気
by the Satellite View at the tenki.jp site
日本気象協会，気象衛星画像
tenki.jp/satellite/
天候は時々刻々変化して, 雲の運動は止ることがない
Weather
is changing from time to time. It never stops.
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Twin typhoons over the Pacific Ocean
Satellite photograph on 29th June, 2004
(Courtesy of Japan Weather Association)
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Vortex ring, made
by a half-spherical shock wave
emerging from a nozzle
(the dark object on the left).
[Shadow-graph]
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Collision of two
vortex rings:
Waves are emitted from two pairs of
colliding dark cores (upper and lower ends of
vertical dark bands [vortex rings]).
渦音 Vortex Sound
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渦音 Sound
generated by vortex motions (Review paper, 2009)
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Vortex
sound : IJA-TK2009a,
International Journal of Aeroacoustics
(2009)
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Vortex sound with special reference to vortex rings:
theory, computer simulations, and experiments
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By Tsutomu Kambe
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Abstract: This is a review paper on vortex sound with special
reference to vortex rings and sound emissions detected in experimental
tests. Any unsteady vortex motion excites acoustic waves. From the
fundamental conservation equations of mass, momentum and energy of fluid
flows, one can derive a wave equation of aerodynamic sound. The
wave equation of acoustic pressure can be reduced to a compact form,
called the equation of vortex sound. This equation predicts sound
generation by unsteady vortex motions. On the other hand, based on the
matched asymptotic expansion (using the multipole
expansions), one can derive a formula of wave pressure excited by
time-dependent vorticity field localized in space.
The theoretical predictions are compared with experimental observations
and direct computer simulations. The systems considered are, head-on
collision or oblique collision of two vortex rings, vortex-cylinder
interaction, vortex-edge interaction, and others. Scaling laws of the
pressure of emitted sound are predicted by the theory and compared with
experimental observations. Comparison between theories and observations
shows excellence of the theoretical predictions. Direct numerical
simulations are reviewed briefly for sound generation by collisions of
two vortex rings and that by a cylinder immersed in a uniform stream (aeolian tones). Vortex sound in superfluid is also reviewed about experimental
observations and computational studies on the basis of the Gross-Pitaevski equation.
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Stories of Bodhi-Dharma
菩提達磨の物語 (2009)
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Personal
memoir "Bodhi-Dharma"
2009/03/01
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Bodhi-Dharma 菩提達摩 (440? -
528?)
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A collection of stories from Chinese literatures
中国の歴史資料からの達摩の物語集
By Tsutomu Kambe 神 部 勉
(Tokyo, Japan; March 2009)
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Abstract:
In August 2002, the author (T. Kambe)
has been to China as a
personal Buddhist pilgrim, visiting the Shao-lin Temple
(少林寺)
and the Dharma
Cave (達摩洞) where an Indian monk Bodhi-Dharma
(菩提達摩) is said to have stayed for nine years. It
was about a thousand and five hundred years ago, and the place is east of
Luo-yang (洛陽) of He-nan Province (河南省), which was
once a glorious capital for several dynasties. Subsequently, the author
visited the Bear-Ear Mountain (熊耳山), west of Luo-yang, recorded as the Bodhi-Dharma's
final resting place. When the author visited there, he had an impression
that Bodhi-Dharma had been trying to return to India
after he had accomplished a great mission of transmitting Dharma to his
Chinese disciples. Since then, the author has been thinking about his
home land. In Chinese documents, most historical articles describe that
he was born in a kingdom
of South India.
Documents published just after Tang dynasty (唐朝
618-907) describe that the name of the Kingdom is expressed with two
Chinese characters 香至.
Very recently (November, 2007), the author came
across to identify it to be Kancheepuram. The
Chinese name 香至 means "fragrance
extreme". At the time of Tang dynasty, it is likely that 香至 is pronounced as Kang-zhi. It is
learned that Kanchee means ‘a radiant jewel’ or 'a luxury belt
with jewels', and Puram means a town. Then, it
is understood that 香至-Kingdom corresponds to Kancheepuram.
The author visited Kancheepuram
in January 2008, wishing to find any further evidence kept in India which shows that Kancheepuram
is in fact the home land
of Bodhi-Dharma.
Before visiting the assumed home land, this article was prepared by
collecting stories of Bodhi-Dharma recorded in
Chinese documents, listed at the end of this article. Most of them were
published more than a thousand years ago.
To a surprise of the author, it was informed that
the Institute
of Asian Studies in
Chennai owns already a land in Kancheepuram and
keeps it to commemorate Bodhi-Dharma and
construct there a monument and an institute. It is hoped that this
article is of use anyhow for the great project. The depth and broadness
of Bodhi-Dharma’s
influence is to be emphasized which he brought to the East land. This is
exemplified by the following fact. Most frequently asked question among
hundreds of Koans (公案)
of Zen schools is “如何是祖師西来意 (Ru-he
shi zu-shi xi-lai yi)“: What is the intention
of Bodhi-Dharma coming from the West-land
(India)? This Koan proposes encouragingly a
fundamental quest for 如是 (Suchness)
which Bodhi-Dharma tried to transmit to the
people of East-land. This brought about revolutionary change and
enlightenment in the culture of the East-land including China, Japan and other countries. “如何是祖師西来意” remains an unchanged Koan to our
present age living in the 21th century.
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Bodhi-Dharma took an Oceanic Silk Road from India
to China !?
達摩大師は海のシルクロードをたどって中国に至った !?
(2009)
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大師の海路についての一考察
2009/02/01
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達摩大師と海のシルクロード
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神 部 勉
元東京大学教授 （物理学）
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中国禅宗の祖、達摩大師がインドから中国に至るのにどのような海路をたどった
のだろうか？
それは詳らかではないが、海のシルクロードをたどった可能性について
一説をここに提示したい。
インドと中国との間には、西暦紀元前からインド洋を横断する海洋交易が盛んで あった。
達摩大師もそのコースをたどったに違いない。
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A
study of sea routes of Bodhi-Dharma
2009/02/01
Bodhi-Dharma and Oceanic Silk Road
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Tstuomu Kambe
Former Professor (Univ. of Tokyo, Physics)
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Bodhi-Dharma is the First Patriarch of
Zen-Buddhism in China.
However, in regard to his route from India
to China,
its details are not known.
Which sea-route did Bodhi-Dharma
take ?
In this article, a possible route is proposed to be an Oceanic Silk
Road.
Already, before the Christian Era, there were active trade routes
across the Indian Ocean
between India and China.
It is very likely that Bodhi Dharma chose
this course.
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Preprint : TK2010a: presented at the International Workshop
on
Mathematical Fluid Dynamics (Waseda University, Tokyo), March, 2010,
Variational formulation of
an ideal fluid and fluid Maxwell equations
by Tsutomu Kambe
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Part I. Variational formulation
There
are two ways in description of fluid flows: Lagrangian
description and Eulerian description.
Variational formulations and Lagrangian functionals are reconsidered from the viewpoint
of guage theory, i.e.
expressed in a form independent of any particular coordinate system.
Transformations between the two spaces are examined, and its
uniqueness is considered. Importance of vorticity equation is emphasized in this regard.
Part II. Fluid Maxwell Equations
There is analogy between the
equations of Fluid mechanics and Electromagnetism. Defining two
vector fields analogous to the electric and magnetic fields, fluid
Maxwell equations are proposed. Vorticity corresponds to the magnetic
field. Equation of sound wave is derived. Hence, the sound wave is
analogous to the electromagnetic
wave. Forces acting on a test particle moving in a flow field are
expressed with the same form as of the electromagnetism.
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Introductory
textbook on Fluid Mechanics (2007)
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Elementary Fluid Mechanics
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By Tsutomu Kambe
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World Scientific,
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This
textbook describes the fundamental physical aspects of fluid flows for
beginners of fluid mechanics in physics, mathematics and engineering,
from the point of view of modern physics.
It emphasizes dynamical aspects of fluid motions rather
than static aspects, illustrating vortex motions, waves, geophysical
flows, chaos and turbulence. Beginning with fundamental concepts of
nature of flows and properties of fluids, the book presents fundamental
conservation equations of mass, momentum and energy, and the equations
of motion for both inviscid and viscous
fluids.
In addition to the fundamentals, this book also covers
water waves and sound waves, vortex motions, geophysical flows,
nonlinear instability, chaos, and turbulence. Furthermore, it includes
the chapters on superfluids and the gauge
theory of fluid flows. The material in the book emerged from the
lecture notes for an intensive course on Elementary Fluid Mechancis for both undergraduate and postgraduate
students of physics given at University
of Tokyo (Japan) and at Nankai
Institute of Mathematics (Tianjin, China).
Hence, each chapter in the second half of the book may be presented
separately as a single lecture.
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Contents:
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Chap 1.
Flows
Chap 2. Fluids
Chap 3. Fundamental Equations of Ideal
Fluids
Chap 4. Viscous Fluids
Chap 5. Flows of Ideal Fluids
Chap 6. Water Waves and Sound Waves
Chap 7. Vortex Motions
Chap 8. Geophysical Flows
Chap 9. Instability and Chaos
Chap 10. Turbulence
Chap 11. Superfluid
and Quantized Circulation
Chap 12. Gauge Principle for Ideal-Fluid
Flows
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Website (World
Scientific) : Elementary Fluid Mechanics
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Geometry and Dynamical systems
Revised edition of the introductory textbook (January
2010)
on Geometry
of dynamical systems, fluid flows, and integrable
systems.
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Geometrical Theory of Dynamical
Systems and Fluid Flows
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By Tsutomu Kambe
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Advanced Series in
Nonlinear Dynamics - Vol. 23
World Scientific, 2010
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This
is an introductory textbook on the geometrical theory of dynamical
systems, fluid flows, and certain integrable
systems. The subjects are interdisciplinary and extend from
mathematics, mechanics and physics to mechanical engineering, and the
approach is very fundamental. The underlying concepts are based on
differential geometry and theory of Lie groups in the mathematical
aspect, and on transformation symmetries and gauge theory in the
physical aspect. A great deal of effort has been directed toward making
the description elementary, clear and concise, so that beginners will
have an access to the topics.
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Contents:
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Part I. Mathematical Bases
1. Manifolds, Flows, Lie Groups and Lie Algebras
2. Geometry of Surfaces in R3
3. Riemannian Geometry
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Part II. Dynamical Systems
4. Free Rotation of a Rigid Body
5. Water waves and KdV
Equation
6. Hamiltonian systems: Chaos, Integrability
and Phase Transition
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Part III. Flows of Ideal Fluids
7. Flows of an Ideal Fluid: Variational
Formulation and Gauge Principle
8. Volume-Preserving Flows of an Ideal Fluid
9. Motion of Vortex Filaments
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Part IV. Geometry of Integrable Systems
10. Geometric Interpretations of Sine-Gordon
Equation
11. Integrable Surfaces:
Riemannian Geometry and Group Theory
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Gauge Theory of Fluid Flows (4 papers)
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Gauge theory in
theoretical physics has been successfully applied to flows of an ideal
fluid, and a new variational formulation is
proposed. The theory is compactly summarized by the paper EE250-TK (Gauge
theory 1). Detailed accounts are given by the two articles: FDR-TK2008
(Gauge theory 2) and FDR-TK2007 (Gauge theory 3) below. A new
comprehensive formulation of the theory is given by the fourth paper
Preprint-TK2008.
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Gauge
theory 1 : EE250 paper,
published in Physica
D: Nonlinear Phenomena, 237 (2008), 2067-2071.
Proceedings of Euler Conference EE250: "Euler
Equations: 250 Years On",
held in June 2007 at Aussois, France.
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Variational formulation of the motion of an ideal fluid on the
basis of gauge principle
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By Tsutomu Kambe
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Abstract: On the basis of gauge principle in the field theory, a
new variational formulation is presented for
flows of an ideal fluid. The fluid is defined thermodynamically by mass
density and entropy density, and its flow fields are characterized by
symmetries of translation and rotation. A structure of rotation
symmetry is equipped with a Lagrangian L
including vorticity, in addition to Lagrangians
of translation symmetry. From the action principle, Euler's equation of
motion is derived. In addition, the equations of continuity and entropy
are derived from the variations. Equations of conserved currents are
deduced as the Noether theorem in the space
of Lagrangian coordinate a. It is
shown that, with the translation symmetry alone, there is freedom in
the transformation between the Lagrangian a-space
and Eulerian x-space. The Lagrangian L provides non-trivial topology
of vorticity field and yields a source term of the helicity.
The vorticity equation is derived as an equation of the gauge field.
Present formulation provides a basis on which the transformation
between the a space and the x space is determined
uniquely.
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Gauge
theory 2 : FDR-TK2008,
published in Fluid Dynamics Research (2008)
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Variational formulation of ideal fluid flows according to
gauge principle
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By Tsutomu Kambe
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Fluid Dynamics
Research
vol.40 (2008), 399 - 426.
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Abstract: On the basis of gauge principle in the field theory, a
new variational formulation is presented for
flows of an ideal fluid. The fluid is defined thermodynamically by mass
density and entropy density, and its flow fields are characterized by
symmetries of translation and rotation. Rotational transformations are
regarded as gauge transformations. In addition to the Lagrangians representing translational symmetry, a
structure of rotational symmetry is equipped with a Lagrangian
L including vorticity. Euler's equation of motion is derived
from variations according to the action principle. In addition, the
equations of continuity and entropy are derived from the variations.
Equations of conserved currents are deduced as the Noether
theorem in the space of Lagrangian coordinate
a. Without L, the action principle results in the Clebsch solution with vanishing helicity.
The Lagrangian L yields non-vanishing
vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an
equation of the gauge field and the L characterizes topology of
the field. The present formulation is comprehensive and provides a consistent
basis for a unique transformation between the Lagrangian
a space and the Eulerian x
space. In contrast, with translation symmetry alone, there is an
arbitrariness in the transformation between these spaces.
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Gauge
theory 3 : FDR-TK2007,
published in Fluid Dynamics Research (2007)
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Gauge principle and variational
formulation for ideal fluids with reference to translation symmetry
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By Tsutomu Kambe
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Fluid Dynamics
Research
vol.39 (2007), 98 - 120.
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Abstract: Following the gauge principle in the field theory
of physics, a new variational formulation is
presented for flows of an ideal fluid. In the present gauge-theoretical
analysis, it is assumed that the field of fluid flow is characterized
by a translation symmetry (group) and in addition that the fluid itself
is a material in motion characterized thermodynamically by the mass
density and entropy (per unit mass). Local gauge transformation is
equivalent to local Galilean transformation. In complying with the
requirement of local gauge invariance of Lagrangians,
a gauge-covariant derivative with respect to time is defined by
introducing a gauge term. Galilean invariance requires that the
covariant derivative should be the {\it convective} derivative, \ie so-called the Lagrange derivative. Using this
gauge-covariant operator, a free-field Lagrangian
and a Lagrangian associated with the gauge
field are defined under the gauge symmetry. Euler's equation of motion
is derived from the action principle. Simutaneously,
the equation of continuity and equation of entropy conservation are
derived from the variational principle. It is
found that general solution thus obtained is equivalent to the
classical Clebsch solution. If entropy of the
fluid is non-uniform, the flow will be rotational. However, if the
entropy is uniform throughout the space (\ie homentropic), then the flow field reduces to that
of a potential flow. Discussions are given on the issue. From global
gauge invariance of the Lagrangian with
respect to translational transfomations,
conservation equation of momentum is deduced as Noether's
theorem.
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Gauge
theory 4 : Preprint-TK2008 (New paper)
New comprehensive formulation of the
Gauge theory
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Variational formulation and gauge symmetries of an ideal
fluid
--- Lagrangian description and Eulerian description ---
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By Tsutomu Kambe
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Abstract: On the basis of gauge principle in the field theory, a
new variational formulation is presented for
flows of an ideal fluid. The fluid is defined thermodynamically by a
mass density and an entropy density invariant along particle
trajectories (by definition of ideal fluid). Flow fields are
characterized by symmetries of translation and rotation. A Lagrangian functional is defined in terms of
kinetic energy and internal energy, and the action is defined by its
integral with respect to time. Noether's
theorem leads to Euler's equation of motion and an energy equation.
Requirement of the Lagrangian with respect to
rotational gauge transformations of particle coordinates (\ie in Lagrange space) results in the invariance of
vorticity transformed to the Lagrange space. This implies that the
vorticity is a gauge field.
Taking into account invariances of mass,
entropy and vorticity in the Lagrange space, one can introduce three
additional Lagrangians of the form of total
time derivative. The variational principle of
the extended action leads to the the
continuity equation, the entropy conservation equation, and the
vorticity equation. Rotational component of velocity is defined
naturally with this formulation. In addition, there exits a close
relation between the helicity and the Lagrangian associated with the vorticity, both of
which are regarded as Chern-Simons
invariants.
Present formulation provides a basis on which the transformation
between the Lagrangian space and the Eulerian space is determined uniquely. In most of
traditional formualtions, the continuity
equation and the entropy equaiton are taken
into account as constraints by using Lagrange multiplier whose physical
meaning is not clear. Thus, present formulation is consistent as a
whole for description of flows of an ideal fluid.
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日本物理学会の企画『日本の物理学100年とこれから』 の依頼を受けて、 日本の物理学の歴史を、物理学の一般の分野について、その歴史 を振り返ってみる。明治初期から1930年あたりまでは 近代科学の黎明期といえる。しかしそのようなときでも、 独創的でしかも現在 でも国際的にその影響が萎えることのない研究が存在していた。
日本近代物理の私的回顧 :
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「独自性の展開」 ― 物理学一般
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神 部 勉 Tsutomu
Kambe
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「日本の物理学100年とこれから」
日本物理学会誌 2006, VOL.61, NO.3,
156-164.
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Abstract: 日本の物理学の歴史100年を振り返るとき、明治初期から1930年あたりまでは黎明期
ともいえる。しかしそのようなときでも、独創的な研究が存在していた。 さらに戦後の1945年以後、日本の物理学の研究が
爆発的な勢いで質と量を増したことは、我々のよく知るところである。
物理学一般の分野において、独創的で、現在でもその影響が衰えることのない研究はいくつもある。 独自性のある研究をテーマにして、それらに注目してみたい。個人レベルの研究に
加えて、大型プロジェクト研究も大きな役割を果たしてきた。わが国での 「物理学一般」の研究の展開は、なかなかすてたものではないことがわかる。
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Development of Originalities: Physics General in Japan
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Tsutomu Kambe
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In the series "
A Hundred Years of Japan Physics and Future "
Butsuri (Physics),
Phys. Soc. of Japan,
2006, VOL.61, NO.3, 156-164.
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Abstract: When we look back one hundred years of
development of macroscopic physics in Japan with eyes on
uniqueness, we witness a lot of studies which were original and keep
its international influence even now. In addition to studies of
personal level, big research projects played important parts. Those
developments are reviewed in the various fields of physics general.
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Presented at IUTAM Symposium: One HundredYears of Boundary Layer Research,
August 12 - 14, 2004, Germany.
Review
paper :
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VORTICITY IN FLOW FIELDS
in relation to Prandtl's work and subsequent
developments
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By Tsutomu Kambe
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Preprint TK2004b.
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Abstract: One of the important properties of boundary
layer is, from the point of view of Stroemungslehre,
that it pumps vorticity into the flow field. Vorticity in fluid flows
plays diverse roles and produces tremendous variety of flow phenomena.
We consider four aspects of vorticity in fluid flows: (A) Kinematical
aspect of a vortex sheet, (B) mechanical aspect of hydrodynamic impulse
of a vortex system, (C) dynamical aspect: vorticity dynamics,
excitation of acoustic waves and formation of dissipative structure in
turbulence, and (D) gauge field associated with local rotational
symmetry.
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SPP講義 (Science Partnership Program
Lecture)
川越高等学校 (Kawagoe High
School), 2004年6月12日
講義ノート（Lecture Note in Japanese） :
高校生のための物理学講義 (Physics lecture for high-school students)
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物理学とはどういう学問なのか
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What kind of science is the Physics ?
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神部 勉 (Tsutomu Kambe)
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1. はじめに
宇宙ニュートリノ, 自然科学の歴史
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2. 宇宙ニュートリノの検出と超新星爆発のシナリオの検証
星の進化、超新星、元素合成のシナリオ
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3. ニュートンの運動方程式はどの時刻どの場所でも成りたつ
不変性の原理
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4. 膨張宇宙、赤方偏移、ハッブルの法則
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5. ビッグバン名残りの宇宙背景放射
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6. 重力の法則、ブラックホール、光線の湾曲
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7. 気象現象と決定論的カオス
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Essay :
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その発見はどのように役立つでしょうか？
What is the use of this discovery ?
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神部 勉 (2001)
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『学術の動向』2001年5月号
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2001年2月に公開されたヒトゲノム解析は、この年を21世紀の単なる 最初の年以上の新時代的意味をもたせることになるであろう。
The genome sequencing published open in February
2001 would make the year a new age, more than the first year of the
21st century.
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Full pdf-text
in Japanese : What.pdf
What is the
use of this discovery ?
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Gauge theory of physics has been applied
to flows of an ideal fluid.
Original
paper on gauge theory :
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Gauge principle and variational
formulation for flows of an ideal fluid
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By T. Kambe
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Acta Mechanica
Sinica, vol.19, No.5, 437-452 (2003).
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1. Introduction
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2. Conceptual scenario of the gauge
principle
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3. Preliminary study of velocity
field
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4. Free-field Lagrangian
of fluid flows
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5. Local gauge transformation
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6. Hamilton's principle under isentropic
material variations
6.1 Flows of an ideal fluid
6.2 Lagrangian and variational formulation
6.3 Material variation
6.4 Noether's theorem
6.5 Particle-permutation symmetry and conservation
equation
6.6 Equation of vorticity and Kelvin's circulation
theorem
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7. Summary and conclusions
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What is gauge principle in fluid flows ?
Brief
Communication :
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Gauge principle for flows of an ideal fluid
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By T. Kambe
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Fluid Dynamics
Research, vol.32, 193-199 (2003).
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Abstract: A gauge principle is applied to flows of a compressible
ideal fluid. First, a free-field Lagrangian
is defined with a constraint condition of continuity equation. The Lagrangian is invariant with respect to global
SO(3) gauge transformations as well as Galilei
transformation. From the variational principle,
we obtain the equation of motion for a potential flow. Next, in order
to satisfy local SO(3) gauge invariance, we define a gauge field and a
gauge-covariant derivative. Requiring the covariant derivative to be Galilei-invariant, it is found that the gauge field
coincides with the vorticity and the covariant derivative is the
material derivative for the velocity. Based on the gauge principle and
the gauge-covariant derivative, the Euler's equation of motion is
derived for a homentropic rotational flow. Noether's law associated with global SO(3) gauge
invariance leads to the conservation of total angular momentum. This
provides a gauge-theoretical ground for analogy between acoustic-wave
and vortex interaction in fluid dynamics and the electron-wave and
magnetic-field interaction in quantum electrodynamics.
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Reviews on physics of flow-acoustics,
vortex sound and sound scattering.
解説（Reviews
in Japanese） :
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流れと音の物理
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Physics of Flow-Acoustics
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神部 勉
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『ながれ』日本流体 力学会誌 第20巻3号 (2001年6月)
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1. はじめに
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2. 空力音の理論
2.1 ライトヒル方程式
2.2 遠方場
2.3 スケール則
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3. 流れによる音の発生（非線形機構）
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4. 粘性効果
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5. 渦音：
Vortex sound
5.1 理論
5.2 実験
5.3 直接数値シミュレーション
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6. ２重極音
6.1 エオルス音 Aeolian tone
6.2 実験
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7. エッジ音 Edgetone
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8. 音のファラデーの法則
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9. 音波の散乱
Scattering
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10. おわりに
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Tsutomu KAMBE
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神 部 勉
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kambe@ruby.dti.ne.jp
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HigashiYama 2-11-3, Meguro-ku
Tokyo 153-0043, Japan
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〒153-0043 東京都目黒区東山２−１１−３
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Visiting
Professor, Chern Institute of
Mathematics (Tianjin, China)
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南開大学（天津、中国）
陳省身数学研究所
客員教授 （2003 -
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Former Professor (Physics), University of Tokyo
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(2000年退官）東京大学教授（物理学、流体力学）
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川越高等学校卒業（埼玉県立）
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Former
Member of IUTAM Bureau (2004 - 2008)
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元国際理論応用力学連合理事 (2004 - 2008)
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Former Editor-in-Chief : Fluid Dynamics Res.
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Fluid
Dynamics Research 編集長
（1991 - 2002）
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