Book of Abstracts

ヘリシティと時空対称性，古典場から量子場まで

Helicity and space-time symmetry― a new perspective ofclassical and quantum systems
October 5 (Tue)-8 (Fri), 2021 @ Media Center, Osaka City U.

Support:Osaka City University Advanced Mathematical Institute:
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics
JPMXP0619217849
Co-organized by
Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City University(2021-10-05, 12:45 version)

1

Abstracts
Yutaka Matsuo (The University of Tokyo), Akio Sugamoto (Ochanomizu University)
A description of perfect fluid by Kalb-Ramond field and the duality
with Gross Pitaevskii. . . . . . . . . . . . . . . . . . . . . . . . .

3

Z. Yoshida (National Institute for Fusion Science), P. J. Morrison (University of Texas at Austin)
The kinetic origin of the fluid helicity – a symmetry in the kinetic
phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5Keith Moffatt (Trinity College, Cambridge)
Invariance of Helicity and some Topological Implications. . . . .

7

Helicity and topology of a tangle of vortices . . . . . . . . . . . . .

8Carlo F. Barenghi (Newcastle University)
Naoki Yamamoto (Keio University) and Di-Lun Yang (Academia
Sinica)Chiral radiation transport theory of neutrinos. . . . . . . . . . .

9Sota Hanai (Keio University), Naoki Yamamoto (Keio University)
Gravitational waves caused by chiral transport phenomena in neutron stars and supernovae . . . . . . . . . . . . . . . . . . . . . . .

11

Y. Masada (Aichi University of Education), T. Takiwaki (National Astronomical Observatory of Japan), K. Kotake (Fukuoka University)
Convection and Dynamo in Newly-born Neutron Stars . . . . . . .

13

Koya Shimokawa (Saitama University)
Untying pathways of vortex knots. . . . . . . . . . . . . . . . . .

14

Physics of Optical Vortex and its application . . . . . . . . . . . .

15

Ryuji MORITA (Hokkaido University)

2

Xin LIU (Beijing University of Technology)
Detecting topology of fluid vortex knots in terms of polynomial invariants constructed from helicity. . . . . . . . . . . . . . . . . .

17

Minimal unlinking pathways as geodesics in knot polynomial space

18

Renzo L. Ricca (U. Milano-Bicocca)
Kohkichi Konno (National Institute of Technology, Tomakomai College)
Review of Chern-Simons modified gravity: its effects on compact
objects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19Takahisa Igata (KEK)
Energy extraction from a rotating black hole

. . . . . . . . . . . .

21

Surface defects, vortices and integrable systems . . . . . . . . . . .

23

K. Maruyoshi (Seikei University)
Katsunobu Nishihara (Osaka University), Chihiro Matsuoka (Osaka
City University), Takayoshi Sano (Osaka University), Francisco
Cobos-Campos (Universidad de Castilla-La Mancha), Vasily
Zhakhovsky (Rosatom)
Richtmyer-Meshkov instability and vortices . . . . . . . . . . . . .

25

Takayoshi Sano (Osaka University)
Laser-driven interfacial instabilities in magnetized plasmas. . . .

27

Ken ABE (Osaka City University)
Rigidity of Beltrami fields with a non-constant proportionality factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

Chihiro Matsuoka (Osaka City University) and Koichi Hiraide(Ehime University)
Exact solutions in non-integrable systems . . . . . . . . . . . . . .

30

Hiroaki Yoshimura (Waseda University)
Dirac geometry and variational structures in nonequilibrium thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

32

Linyu Peng (Keio University)
Continuous symmetries of semi-discrete equations and conserved
quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Yasuhide Fukumoto (Kyushu University), Rong Zou (University of
Hawai’i)
Particle relabeling symmetry and Noether’s first and second theorems for fluid and MHD equations . . . . . . . . . . . . . . . . . .

435

A description of perfect fluid by Kalb-Ramond field and the duality
with Gross-Pitaevskii
Yutaka Matsuo (The University of Tokyo), Akio Sugamoto (Ochanomizu University)

In this talk, we explain a description of the perfect fluid in terms of a two-form gauge field(Kalb-Ramond field). Nambu proposed such an idea long ago in 1977, and one of the authors(AS) developed the idea further.
In the perfect fluid motion, the vortex keeps the circulation and behaves as a "soliton." In
this sense, it is natural to describe the fluid dynamics in terms of the vortices, which were
explored long ago by Lund and Regge.
The Kalb-Ramond field couples naturally with the vortex degree of freedom through the
integration over the vortex (string) world-sheet. We study the detailed relation between the
dynamical degree of freedom of the Euler equation with the Kalb-Ramond field.
We also compare it with the Gross-Pitaevskii equation, giving an alternative description of
the fluid motion. We explain that the two descriptions are dual, similar to theelectric/magnetic fields in the Maxwell theory. In this sense, we expect to have thestrong/weak duality in the two descriptions.

5A description of perfect fluid by Kalb-Ramond field and the duality with Gross-Pitaevskii
松尾 泰 (東京大学), 菅本晶夫 (お茶の水代大学)

この講演では理想流体の一つの記述法として 2 階テンソル場 (Kalb-Ramond 場)による記述について説明する。このような記述法の発想は古くもともとの提案は南部陽一郎氏による提案があり、その後菅本による研究もなされている。
流体力学の運動において、渦糸の循環が常に保たれる。すなわち渦糸は理想流体においてソ
リトン的な振る舞いを行う。この意味では流体の運動を渦糸を用いて行うことも可能であ
るはずであり、実際そのような定式化は Lund と Regge により７０年代に提案が行われている。
Kalb-Ramond 場は渦糸の世界面（空間的な広がり＋時間発展）と面積分を用いて自然に結合することができる。これをソース項として取り入れることにより流体の Kalb-Ramond 場
による記述が可能となる。この提案では以前行われていた研究をさらに進めて流体の変数と Kalb-Ramond 場の対応関係をあからさまに導いた。
またこの定式化と理想流体のもう一つの記述法としてよく知られている Gross-Pitaevskii 方
程式を比較した。特にこれら二つの力学変数は、電磁気学における電場と磁場のように、互いに双対なものであることを示した。この意味で GP 方程式と KR 場の記述法は互いに強弱双対の関係になることが予想される。

6

The kinetic origin of the fluid helicity -- a symmetry in the kinetic phase space
Z. Yoshida (National Institute for Fusion Science)
P.J. Morrison (University of Texas at Austin)
Helicity, a topological degree that measures the winding and linking of vortex lines, ispreserved by ideal (barotropic) fluid dynamics.

In the context of the Hamiltonian description,

the helicity is a Casimir invariant characterizing a foliation of the associated Poisson manifold.
Casimir invariants are special invariants that depend on the Poisson bracket, not on the particular
choice of the Hamiltonian. The total mass (or particle number) is another Casimir invariant,whose invariance guarantees the mass (particle) conservation (independent of any specific choiceof the Hamiltonian). In a kinetic description (e.g. that of the Vlasov equation), the helicity is nolonger an invariant (although the total mass remains a Casimir of the Vlasov's Poisson algebra).
The implication is that some “kinetic effect” can violate the constancy of the helicity.

Toelucidate how the helicity constraint emerges or submerges, we examine the fluid reduction of the
Vlasov system; the fluid (macroscopic) system is a ``sub-algebra'' of the kinetic (microscopic)
Vlasov system.

In the Vlasov system, the helicity can be conserved, if a special helicity

symmetry condition holds. To put it another way, breaking helicity symmetry induces a change
in the helicity.

We delineate the geometrical meaning of helicity symmetry, and show that, fora special class of flows (so-called epi-2 dimensional flows), the helicity symmetry is written as
𝜕𝛾 = 0 for a coordinate γ of the configuration space.
https://arxiv.org/abs/2103.03990

7

流体ヘリシティーの運動論的起源 —運動論位相空間における対称性

吉田善章（核融合科学研究所）
Philip J. Morrison（テキサス大学オースチン校）

順圧理想流体の保存量であるヘリシティは，渦線の絡みを計る位相幾何学的指数である．ハミルトン力学の文脈では，ヘリシティは，ポアソン多様体の葉状構造を特徴付けるカシミール不変量である．カシミール不変量は，ハミルトニアンの選択に依存せず（すなわち系の対称性に関係せず），ポアソン括弧の代数構造だけに起因する特別な不変量である．ヘリシティーと並んで，全質量（または全粒子数）もカシミール不変量であり，その不変性は（相互作用の在りように関係なく）質量（粒子）の保存を保証している．
興味深いことに，運動論的モデルである Vlasov 方程式系でも，全質量は Vlasov-ポアソン代数のカシミール不変量であるが，ヘリシティはもはや不変量ではない．つまり，何らかの「運動論的効果」がヘリシティの保存を破ることを意味している．ヘリシティ保存による渦線のトポロジー束縛がどのような運動論的起源をもつのかを解明するために、Vlasov 方程式系から流体モデルへのリダクションで起こる構造変化を分析する．流体（巨視的）システムは，運動論的（微視的）Vlasov 方程式系の「部分代数」であることが知られている．Vlasov 方程式系において，「ヘリシティ対称性」が成り立つ場合，
ヘリシティは保存することが示される．逆に言えば，ヘリシティ対称性を破ると，ヘリシティに変化が生じる．特別なクラスの流れ場（いわゆる epi-2 次元流）の場合，ヘリシティ対称性は座標空間の座標γに対して𝜕𝛾 = 0と表すことができる．
https://arxiv.org/abs/2103.03990

8

Invariance of Helicity and some Topological Implications
Keith Moffatt
Trinity College, Cambridge
Abstract
Helicity is a topological invariant of any solenoidal vector field that istransported by (i.e. 'frozen in') a fluid flow. The best known examples are themagnetic field in a perfectly conducting fluid and the vorticity field in an ideal
fluid, the latter providing an important invariant of the classical Euler equationsof fluid flow. In this lecture, I shall first discuss the historical origin of theseinvariants, and I will then comment on two complementary problems in which
helicity plays a key role: (i) the spontaneous generation of magnetic field (i.e.
dynamo action) by turbulence with non-zero mean helicity; and (ii) the
relaxation of knotted magnetic flux tubes in a perfectly conducting fluid tominimum-energy states constrained by their topology. Here there is an analogy
with the 'tight-knot' problem: what is the minimum length of rope of unit
circular cross-section with which a knot of particular type may be tied? There
are differences however in that, for the magnetic relaxation problem, it is thevolume of the flux tube that is conserved under deformation (and not the area of
its cross-section), and the internal twist of the magnetic field affects theminimum-energy state. A particular problem here concerns the possibility of
two or more minimum-energy states for a given knot type, and the manner inwhich a transition from one such state to another may be accomplished. Some
aspects of this problem, under current consideration, will be discussed, insofar
as time permits.

9

Helicity and topology of a tangle of vortices
Carlo F. Barenghi
School of Mathematics, Statistics and Physics,
Newcastle University, United Kingdom
and JQC (Joint Quantum Centre Durham-Newcastle)
We address the question as to whether turbulence contains vortex knots
in a context which is simpler than ordinary fluids: the context of superfluid
helium. Unlike what happens in ordinary Navier-Stokes fluids (where vorticity
is a continuous field), in superfluid helium quantum mechanics constraints thevorticity to individual (discrete) vortex lines. Quantum turbulence is therefore
a disordered tangle of vortex lines. Since the vortex core thickness is manyorders of magnitude smaller than the average distance between vortex lines intypical experiments, the vortex tangle can be modelled using the classical BiotSavart law and the assumption that vortex lines reconnect [1] when they collide(as seen in the experiments), a mesoscale level of description called the Vortex
Filament Model (VFM).
After numerically generating a state of turbulence in a statistical-steadystate, we exploit the discrete nature of the vorticity to instantaneously associate
an Alexander polynomial to each vortex loop. The degrees of the Alexanderpolynomials can thus be used to quantify the complexity of the turbulence,
revealing a highly knotted distribution [3].
In the last part of the lecture I shall discuss the classical definition of helicity[2] in the context of superfluid helium. In particular, I shall show that theclassical definition of helicity is consistent with what we know microscopically
about the superfluid vortex core from N-body quantum mechanics, without thedifficulties which arise in the simpler mean-field approach of Gross-Pitaevskii
theory. I shall show that helicity, compute in the VFM, captures the physicaldifference [4] between the two forms of quantum turbulence which have been
identified in experiments and numerical simulations.

References[1] L. Galantucci & al, Proc. Nat. Sci. USA 116: 12204, 2019[2] K.H. Moffatt, J. Fluid Mech. 35: 117, 1969.
[3] R.G. Cooper & al, Sci. Reports 9: 10545, 2019.
[4] L. Galantucci & al, Phys. Rev. A 103: 144503, 2021.

1

10

Chiral radiation transport theory of neutrinos
Naoki Yamamoto1a and Di-Lun Yangba

Department of Physics, Keio University, Japanb

Institute of Physics, Academia Sinica, Taiwan

Abstract
One of the most important properties of neutrinos in the Standard Model ofparticle physics is the chirality. Although neutrinos are considered to play important roles in the dynamics of core-collapse supernovae, this property has been
completely neglected in the conventional neutrino transport theory. Recently, we
have systematically constructed the radiation transport theory incorporating theeﬀects of chirality of neutrinos based on the underlying quantum ﬁeld theory.
This is the chiral radiation transport theory [1].
In this talk, we ﬁrst give a brief introduction to transport phenomena dueto the chirality of elementary particles, called chiral transport phenomena. One
of its consequences is the chiral plasma instability, which generates a magnetic
ﬁeld with magnetic helicity from the chirality imbalance of fermions. We thendiscuss applications of the chiral radiation transport theory for neutrinos in corecollapse supernovae, and in particular, its possible implications for the origins ofmagnetars and pulsar kicks [2].

References[1] N. Yamamoto and D. L. Yang, Astrophys. J. 895, 56 (2020) [arXiv:2002.11348(astro-ph.HE)].
[2] N. Yamamoto and D. L. Yang, [arXiv:2103.13159 (hep-ph)]; work in preparation.
1nyama@rk.phys.keio.ac.jp

11ニュートリノのカイラル輻射輸送理論
山本 直希*1 （慶應義塾大学）， 楊 廸倫（中央研究院）

概要
素粒子の標準模型におけるニュートリノの最も重要な性質の一つがカイラリティである．
ニュートリノは重力崩壊型超新星爆発のダイナミクスに重要な役割を果たすと考えられているものの，この性質は従来のニュートリノ輸送理論では全く無視されていた．最近，我々は場の量子論に基づき，ニュートリノのカイラリティの効果を取り入れた輻射輸送理論である
カイラル輻射輸送理論を系統的に構築した [1]．
本講演では，まず素粒子のカイラリティによって生じる輸送現象であるカイラル輸送現象
について導入を行う．このような輸送現象の帰結の一つが，素粒子のカイラリティから磁気
ヘリシティをもった磁場を生成するカイラルプラズマ不安定性である．さらに，超新星におけるニュートリノのカイラル輻射輸送理論の応用，特に，マグネターやパルサーキックの起源を説明する可能性 [2] について議論する．

参考文献[1] N. Yamamoto and D. L. Yang, Astrophys. J. 895, 56 (2020) [arXiv:2002.11348 (astroph.HE)].
[2] N. Yamamoto and D. L. Yang, [arXiv:2103.13159 (hep-ph)]; work in preparation.

*1 nyama@rk.phys.keio.ac.jp12

Gravitational waves caused by chiral transport phenomena inneutron stars and supernovae
Sota Hanai, Naoki Yamamoto (Keio University)
Exploring the internal structure of neutron stars is an important issue in astrophysics. Asteroseismology is a method to study the inside of stars by analyzing their seismic oscillations.
These oscillations are classified according to their physical origins, such as p-modes due topressure, g-modes due to buoyancy, and so on. In the case of neutron stars, gravitational
waves are expected to be emitted induced by these seismic oscillations.
In recent years, chiral transport phenomena caused by the chirality of relativistic quarks
have been vigorously discussed in the context of quark-gluon plasmas created in relativistic
heavy-ion collisions experiments. Typical examples are the chiral magnetic eﬀect, which is thecurrent in the direction of the magnetic field, and the chiral vortical eﬀect, which is the current
in the direction of the vorticity. It is known that these chiral transport phenomena induce
collective excitations called chiral magnetic waves and chiral vortical waves, respectively.
In this talk, we will show that chiral magnetic waves are generated in the electron matter
inside neutron stars, and examine their properties. In particular, we will discuss the eﬀect ofelectron mass on the chiral waves. We will estimate the frequency of new types of gravitational
waves produced by these chiral magnetic waves and the possibility of observations. We will
also show the possible existence of the other type of gravitational waves induced by the
propagation of chiral vortical waves in the core of supernovae.

13中性子星・超新星内部のカイラル輸送現象に起因した重力波
花井奏太, 山本直希 (慶應義塾大学)
中性子星の内部構造を解明することは，宇宙物理学において重要な課題である．こうした天体の内部構造を推測する方法に，星震を解析する星震学がある．この振動は，圧力に起因する p モー
ド，浮力による g モードなど，その物理的な起源によって分類されている．特に中性子星のようなコンパクト天体からは，星震に伴って重力波が放出されると期待されている．
ところで，近年，相対論的重イオン衝突実験で実現される超高温物質クォーク ・グルーオン・プラズマにおいて，相対論的なクォークのカイラリティに起因するカイラル輸送現象が精力的に議論されている．その代表的な例が，磁場方向のカレントであるカイラル磁気効果や，渦度方向のカレントであるカイラル渦効果である．そしてこれらのカイラル輸送現象に伴って，それぞれ
カイラル磁気波，カイラル渦波と呼ばれる波が誘起されることが知られている．
本講演では，中性子星内部の電子物質においてカイラル磁気波が生じることを示し，その性質を見る．特に電子質量の効果によって生じるカイラリティのフリップについて考察する．またこ
うしたカイラル磁気波によって生じる新たなタイプの重力波の振動数の評価を行い，その観測可能性についても議論する．超新星のコアが回転している場合には，カイラル渦波が生じ，それに伴う別の新たな重力波が生じることを見る．

14

Y. Masada (Aichi U. Edu.), T. Takiwaki (Natl. Astron. Obs. Japan),
K. Kotake (Fukuoka U.)

政田洋平 (愛教大)，滝脇知也 (国立天文台)，固武慶 (福岡大)15

Untying pathways of vortex knots
Koya Shimokawa
Department of Mathematics, Saitama University
In this talk we will analyze untying pathways of vortex knots by applying knot theory.
The topology of entangled vortices is represented using oriented knots and links, andthe reconnection of vortices is modeled using a coherent band surgery of knots andlinks. For the case where the topology of the vortex is a torus knot or link T(2,p), wecharacterize the untying pathway, and determine where the reconnection occurs.

Reference[1] Proc. Natl. Acad. Sci. USA 110, 20906-20911 (2013)[2] Sci. Rep. 7, 12420 (2017)

Untying pathways of vortex knots
下川 航也
埼⽟⼤学⼤学院理⼯学研究科
渦が作る結び⽬が繋ぎ換え（リコネクション）により解かれることが観察されている。この講演では、その様⼦を結び⽬理論を⽤いて解析する。渦のトポロジーを向きづけられた結び⽬や絡み⽬を⽤いて表し、渦の繋ぎ換えを整合的なバンド⼿術を⽤いてモデル化する。渦のトポロジーがトーラス結び⽬・絡み⽬ T(2,p)の場合について、その解消の経路の特徴付けと、その解消がどのように起こるかを決定する。

16

Physics of Optical Vortex and its application
Ryuji MORITA
Department of Applied Physics, Hokkaido University
Optical vortices (OVs), such as Laguerre-Gaussian modes, whose spatial phase depends on
the azimuthal coordinate, have attracted much attention because of their unique properties. Possessing a phase singularity on their beam center, they carry the orbital angular momentum proportional to the topological charge, which is expressed by the value of the line integral of thespatial phase around the singular point. Thanks to these properties, they are used in the various
applications such as laser processing, laser trapping/manipulation, quantum information processing, classical communications, super-resolution microscopy, and nonlinear spectroscopy. In
the presentation, giving an overview of physical properties of OVs, we will discuss the interaction between OVs and materials from the viewpoint of orbital angular momentum, spin angularmomentum, total angular momentum and helicity.

17光渦の物理とその応用
森田 隆二（北海道大学）「光渦」は波動方程式の近軸近似のもとでの固有解である Laguerre-Gauss モードに代表される，位相特異点を有する光波である．光渦はよく定義された軌道角運動量を持つことが示されて以来，光トラッピング，光マニピュレーション，光通信，超解像顕微鏡，量子情報処理，天文光学，レーザー加工，非線型分光などの分野において多くの応用がなされつつある．本講演では，光渦に関してその性質を概観するとともに，光渦と物質との相互作用による得意な構造形成実験結果に関してヘリシティ，全角運動量，軌道角運動量，ス
ピン角運動量の観点から議論する．

18

Osaka (Zoom) — October 5–8, 2021

Detecting topology of fluid vortex knots in termsof polynomial invariants constructed from helicity
XIN LIU
Institute of Theoretical Physics, Faculty of Sciences,
Beijing University of Technology, P.R. China
E-mail: xin.liu@bjut.edu.cn
ABSTRACT
Helicity is the most important topological invariant in fluid mechanics, a conserved quantity for
the Euler equation and the only integral invariant for volume-preserving diffeomorphic
transformations[1]. From vortex filaments viewpoint, helicity provides access to characterize
fluid knot topology. In the seminal work of Moffatt[2], then extended by Moffatt & Ricca[3][4],the relationship between helicity and knot linking numbers was revealed; since then linking
number interpretation of helicity has played a key role in the study of the topological hierarchyof entangled fluid vortex knots. With the discovery of highly complex knots in superfluid
turbulence[5] linking numbers and crossing numbers are proven to be no longer sufficient todetect topological complexity and changes in topology due to reconnections[6]. More powerfultools from knot theory are thus needed; for this reason we have derived Jones and HOMFLYPT
knot polynomials from helicity[7][8]. In this talk we present a brief introduction to our derivation,emphasizing the role played by writhe and twist helicity in the polynomial skein relations.
This is joint work with Renzo L. Ricca (U Milano-Bicocca, Italy).
[1]

Enciso A., Peralta-Salas D. & Torres de Lizaur F. 2016 Helicity is the only integral invariant ofvolume-preserving transformations. PNAS USA 113, 2035.

[2]

Moffatt H.K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117.

[3]

Moffatt H.K. & Ricca R.L. 1992 Helicity and the Călugăreanu invariant. Proc. R. Soc. A 439, 411.

[4]

Ricca R.L. & Moffatt H.K. 1992 The helicity of a knotted vortex filament. In Topological Aspectsof the Dynamics of Fluids and Plasmas (ed. H.K. Moffatt et al.), pp. 225–236. Kluwer Acad.
Publs.

[5]

Cooper R.G., Mesgarnezhad M., Baggaley A.W. & Barenghi C.F. 2019 Knot spectrum of
turbulence, Sci. Rep. 9, 10545.

[6]

Kleckner D., Kauffman L.H. & Irvine W.T.M. 2016 How superfluid vortex knots untie, Nature
Phys. 12, 650–655.

[7]

Liu X. & Ricca R.L. 2012 The Jones polynomial for fluid knots from helicity. J. Phys. A: Math. &Theor. 45, 205501.

[8]

Liu X. & Ricca R.L. 2015 On the derivation of HOMFLYPT polynomial invariant for fluid knots.
J. Fluid Mech. 773, 34–48.

19

Osaka (Zoom) – October 5-8, 2021

Minimal unlinking pathways
as geodesics in knot polynomial space
RENZO L. RICCA
Department of Mathematics & Applications, U. Milano-Bicocca, Italy
E-mail: renzo.ricca@unimib.it
ABSTRACTDuring evolution fluid structures, such as linked and knotted vortex filaments inclassical or quantum systems, decay naturally by a series of reconnection events
through a process of consecutive topological simplifications. The process bears
similarities with the DNA unlinking scenario studied in recombinant DNA
plasmids [1]. By applying adapted knot polynomials, such as Jones and
HOMFLYPT derived from helicity [2,3], we show that a topological cascade canbe described as a geodesic flow in an appropriate knot polynomial space [4]. Eachpoint of this space coincides with the knot type given by the polynomial value,that for moderate crossing number (£ 9) identifies uniquely knot types. For thesake of example we consider adapted Jones polynomials with metric given by the
inner product of Legendre polynomials. Unlinking pathways can thus be identified
with geodesics on this space, and optimal decay paths measured by associatedprobability. Results are very promising, showing an interesting connection withthe logarithmic behavior associated with the groundstate energy spectrum of tightknots and links [5].
This is joint work with Xin LIU (BJUT, China) and Xinfei LI (Guangxi UST,
China).
[1] Stolz R., Yoshida M., Brasher R., Flanner M., Ishihara K., Sherratt D.J., Shimokawa K. &Vazquez M. 2017 Pathways of DNA unlinking: A story of stepwise simplification. Nature
Sci. Rep. 7, 12420.
[2] Liu X. & Ricca R.L. 2015 On the derivation of HOMFLYPT polynomial invariant for fluidknots. J. Fluid Mech. 773, 34-48.
[3] Liu X. & Ricca R.L. 2016 Knots cascade detected by a monotonically decreasing sequenceof values. Nature Sci. Rep. 6, 24118.
[4] Liu X., Ricca R.L. & Li X-F. 2020 Minimal unlinking pathways as geodesics in knot
polynomial space. Nature Comm. Phys. 3,136.
[5] Ricca R.L. & Maggioni F. 2014 On the groundstate energy spectrum of magnetic knots andlinks. J. Phys. A: Math. & Theor. 47, 205501.

20Review of Chern-Simons modified gravity: its effects on compact objects
Kohkichi Konno
National Institute of Technology, Tomakomai College
In recent years, many modified gravity theories have been investigated to seek the ultimate
gravitational theory. Among them, Chern-Simons modified gravity is strongly suggested as aneffective theory by two promising quantum gravity theories and by an extension of the standard
model. Furthermore, Chern-Simons modified gravity is consistent with the recent results ofgravitational wave detection.
In the present talk, I will briefly review Chern-Simons modified gravity from the fundamental
part of this theory to recent topics. Chern-Simons corrections play an important role aroundrotating compact objects. Thus I focus on the effects of Chern-Simons modified gravity on
compact objects in this talk.

21

Review of Chern-Simons modified gravity: its effects on compact objects
金野 幸吉
独立行政法人国立高等専門学校機構
苫小牧工業高等専門学校
近年、究極的な重力理論を模索する観点から、一般相対論を超えた重力理論が活発に研究されている。その中で、チャーン・サイモン修正重力は、量子重力の２つの候補理論と標準理論の拡張との両者から強く示唆され、最新の重力波検出の結果からも有効場理論として残りえる理論であるということが分かってきた。
本講演では、チャーン・サイモン修正重力の基礎理論の部分から最近の動向までを
簡単にレビューする。特にチャーン・サイモン修正重力は、回転を伴うコンパクトな重力源において、その効果が最も顕著となる。そこで、本講演ではコンパクト天体における影響に焦点を当てる。

22

Energy extraction from a rotating black hole
Takahisa Igata1, ∗1

KEK Theory Center, Institute of Particle and Nuclear Studies,High Energy Accelerator Research Organization, Tsukuba 305-0801, Japan

ABSTRACTNo material body or light ray never can be extracted from a black hole. However, Penrose
found that energy can be extracted from a rotating black hole [1]. In this talk, I will introduce
an energy extraction mechanism from a rotating black hole that reconciles these seemingly contradictory statements (see, e.g., [2] for a review). The energy extraction mechanism mediated bya free electromagnetic ﬁeld, the Blandford–Znajek process [3], has been widely believed to be a
viable mechanism for relativistic jet formation because it achieves a large energy ﬂux. From aspacetime perspective, the dynamics of magnetic ﬁeld lines in a force-free magnetic ﬁeld can be
identiﬁed with the dynamics of strings [4, 5]. Using this fact, we will argue that the physical pictureof the Blandford–Znajek process by a stationary axisymmetric force-free electromagnetic ﬁeld isequivalent to the energy extraction process by a rigidly rotating Nambu-Goto string [6].

[1] R. Penrose, Gravitational collapse: the role of general relativity, Rev. Nuovo Cimento, 1, 252 (1969).
[2] J. P. Lasota, E. Gourgoulhon, M. Abramowicz, A. Tchekhovskoy, and R. Narayan, Extracting black-holerotational energy: The generalized Penrose process, Phys. Rev. D 89, 024041 (2014) [arXiv:1310.7499[gr-qc]].
[3] R. D. Blandford and R. L. Znajek, Electromagnetic extraction of energy from Kerr black holes, Mon.
Not. Roy. Astron. Soc. 179, 433 (1977).
[4] T. Uchida, Theory of force-free electromagnetic ﬁelds. I. General theory, Phys. Rev. E 56, 2181 (1997).
[5] S. E. Gralla and T. Jacobson, Spacetime approach to force-free magnetospheres, Mon. Not. Roy. Astron.
Soc. 445, 2500 (2014) [arXiv:1401.6159 [astro-ph.HE]].
[6] S. Kinoshita and T. Igata, The essence of the Blandford–Znajek process, Prog. Theor. Exp. Phys. 2018,
033E02 (2018) [arXiv:1710.09152 [gr-qc]].

∗

igata@post.kek.jp

23

回転ブラックホールからのエネルギー引き抜き
伊形 尚久 1, ∗1

高エネルギー加速器研究センター 素粒子原子核研究所 理論センター

講演要旨

物体や光は、一度ブラックホールに落ちてしまうと、二度と取り出すことはできない。しかしながら、ペンローズによれば、エネルギーを回転ブラックホールから取り出すことは可能である [1]。本講演では、この一見矛盾した主張を両立させる、回転ブラックホールからのエネルギーを取り出す機
構を紹介する (例えば [2])。フォースフリー電磁場によってこの機構が媒介される Blandford–Znajek
過程 [3] は、大きなエネルギーフラックスを実現し得ることから、相対論的ジェットの形成のメカニズムとして広く受け入れられている。一般相対論の枠組みを用いることで、フォースフリー電磁場の磁力線のダイナミクスは、ストリングのダイナミクスとみなすことができる [4, 5]。この事実を利用して、定常軸対称なフォースフリー電磁場に対する Blandford–Znajek 過程の物理的描像が、剛体回転する南部–後藤ストリングによるエネルギー引き抜き過程と等価であることを議論する [6, 7]。

[1] R. Penrose, Gravitational collapse: the role of general relativity, Rev. Nuovo Cimento, 1, 252 (1969).
[2] J. P. Lasota, E. Gourgoulhon, M. Abramowicz, A. Tchekhovskoy, and R. Narayan, Extracting black-holerotational energy: The generalized Penrose process, Phys. Rev. D 89, 024041 (2014) [arXiv:1310.7499[gr-qc]].
[3] R. D. Blandford and R. L. Znajek, Electromagnetic extraction of energy from Kerr black holes, Mon.
Not. Roy. Astron. Soc. 179, 433 (1977).
[4] T. Uchida, Theory of force-free electromagnetic ﬁelds. I. General theory, Phys. Rev. E 56, 2181 (1997).
[5] S. E. Gralla and T. Jacobson, Spacetime approach to force-free magnetospheres, Mon. Not. Roy. Astron.
Soc. 445, 2500-2534 (2014) [arXiv:1401.6159 [astro-ph.HE]].
[6] S. Kinoshita and T. Igata, The essence of the Blandford–Znajek process, Prog. Theor. Exp. Phys. 2018,
033E02 (2018) [arXiv:1710.09152 [gr-qc]].
[7] 木下俊一郎, 伊形尚久, ブラックホールからエネルギーを引き抜く: Blandford–Znajek 機構の本質, 日本物理学会誌, 2019 年 8 月号 542–550 頁.

∗

igata@post.kek.jp24Surface defects, vortices and integrable systems
K. Maruyoshi (Seikei U.)

25

Surface defects, vortices and integrable systems
丸吉一暢 (成蹊大)26Richtmyer-Meshkov instability and vortices
Katsunobu Nishihara1, Chihiro Matsuoka2, Takayoshi Sano1,
Francisco Cobos-Campos3, Vasily Zhakhovsky4
1 Institute of Laser Engineering, Osaka University,
2 Laboratory of Applied Mathematics, Osaka City University,
3 Instituto de Investigaciones Energeticas and CYTEMA, Universidad de Castilla-La

Mancha,4 Dukhov Research Institute of Automatics, Rosatom

When an incident shock wave hits a corrugated interface, ripples are induced on thetransmitted and reflected shocks. Due to the refraction of fluids crossing the rippled
shock fronts, the velocity shear left at the interface behind the shocks. The RichtmyerMeshkov instability occurs due to the shear flow at the corrugated interface. In linear
regime, i.e., the distances among the interface and shock fronts are not large, the rippled
shock fronts and the corrugated interface interact each other thorough sound wave andentropy wave. However it is known that the shock ripples and sound wave decay
exponentially and the perturbation becomes incompressible. There exists an asymptotic
linear growth rate. But fluids are not irrigational and vortices remain in the bulk
between the shocks [1,2].
We investigated the nonlinear dynamics of RMI by developing a non-uniform vortex
sheet model [3]. In RMI, bulk vortices are generated behind the rippled shocks. We here
consider more generally nonlinear interactions between a non-uniform vortex sheet withpoint vortices. It will be shown that the nonlinear interaction between a non-uniform
vortex sheet and point vortices as unstable and the mixing will be enhanced [4].
[1] R. D. Richtmyer, Commun. Pure Appl. Math, 13, 297 (1960); E. E. Meshkov, Fluid Dyn. 4, 101(1972); Y. Zhou, Phys. Rep. 720-722, 1 (2017).
[2] K. Nishihara et al Philos. Trans. R. Soc. A 368, 1769 (2010)；G. Wouchuk and K. Nishihara, Phys.
Plasmas, 3, 3761 (1996); ibid 4, 1028 (1997).
[3] C. Matsuoka, K. Nishihara and Y. Fukuda, Phys. Rev. E 67, 036301 (2003), C. Matsuoka and K.
Nishihara, Phys. Rev. E, 73, 053304 (2006).
[4] C. Matsuoka and K. Nishihara, Phys. Plasmas, 27. 052305 (2020).

27

リヒトマイヤー・メシュコフ不安定性と渦(Richtmyer-Meshkov instability and vortices)西原

功修 1, 松岡千博 2, 佐野孝好 1, Francisco Cobos-Campos3,

Vasily Zhakhovsky4
1 大阪大学レーザー科学研究所,
2 大阪市立大学数学研究所,
3 Instituto de Investigaciones Energeticas and CYTEMA, Universidad de Castilla-La

Mancha,4 Dukhov Research Institute of Automatics, Rosatom

密度の跳びがある界面を衝撃波が通過すると透過、および反射衝撃波（あるいは反射膨張波）が形成されるが、リヒトマイヤー・メシュコフ不安定性（RMI）は、界面に凹凸のある場合に生成される衝撃波波面のリップルにより、衝撃波波面を横切る流体は屈折を受け、その結果界面に速度シェアーが生じ、界面の凹凸が時間に比例して成長する不安定性である。線形成長過程、すなわち界面と衝撃波波面の距離が小さいときには、リップル衝
撃波波面と凹凸のある界面に挟まれた流体を伝搬すると音波・エントロピー波によって衝撃波波面と界面の凹凸は相互作用することによって波面の形状が時間的に変化する。しかし衝撃波のリップルやこれらの波は指数関数的に減衰し、これらの摂動運動は非圧縮性となり、界面の凹凸は漸近的な成長速度をもつことが知られている。しかしながら流体の回転性は残ります。[1,2]
そこで、我々はこれまで非一様な速度シェアーのある渦シートモデル VSM（Vortex
Sheet Model）を開発し、RMI の非線形ダイナミックスを調べてきました[3]。RMI では衝撃波の後面に渦が生成されますが、ここではより一般的に非一様な渦シートと点渦の非線形相互作用を VSM で取り扱い、その非線形発展を調べた結果を紹介します。例えば、
非一様速度シェアーのある界面と点渦の相互作用は不安定であり、複雑な界面形状が生成されることが明らかになりました[4]。

[1] R. D. Richtmyer, Commun. Pure Appl. Math, 13, 297 (1960); E. E. Meshkov, Fluid Dyn. 4, 101(1972); Y. Zhou, Phys. Rep. 720-722, 1 (2017).
[2] K. Nishihara et al Philos. Trans. R. Soc. A 368, 1769 (2010)；G. Wouchuk and K. Nishihara, Phys.
Plasmas, 3, 3761 (1996); ibid 4, 1028 (1997).
[3] C. Matsuoka, K. Nishihara and Y. Fukuda, Phys. Rev. E 67, 036301 (2003), C. Matsuoka and K.
Nishihara, Phys. Rev. E, 73, 053304 (2006).
[4] C. Matsuoka and K. Nishihara, Phys. Plasmas, 27. 052305 (2020).

28

Laser-driven interfacial instabilities in magnetized plasmas
Takayoshi Sano (Institute of Laser Engineering, Osaka University)
Extreme plasma states observed in astrophysical phenomena can be created in thelaboratory using large laser facilities such as the GEKKO laser at Osaka University. For
example, the verification of the metallic hydrogen state under ultra-high pressurecomparable to that of Jupiter's interior and the formation of collisionless shocks are being
actively pursued. We are conducting laser experiments and MHD simulations on
interfacial instabilities in magnetized plasmas associated with supernova explosions. Suchphenomena are very important processes in laser fusion plasmas to realize ideal implosion.
Therefore, there is an urgent need to understand the mechanisms that suppress the growthof interfacial instabilities. In this talk, I introduce our laser astrophysics experiments andtheoretical understandings on the amplification of magnetic fields by the RichtmyerMeshkov instability [1-3].
References:[1] T. Sano, Alfven number for the Richtmyer-Meshkov instability in magnetizedplasmas, Astrophys. J. in press (2021)[2] T. Sano et al., Laser astrophysics experiment on the amplification of magnetic fields
by shock-induced interfacial instabilities, Phys. Rev. E 104, 035206 (2021)[3] T. Sano et al., Suppression of the Richtmyer-Meshkov instability due to a density
transition layer at the interface, Phys. Rev. E 102, 013203 (2020)

29

レーザー駆動衝撃波を⽤いた磁化プラズマ中での界⾯流体不安定
佐野孝好(⼤阪⼤学レーザー科学研究所)
⼤阪⼤学の激光 XII 号レーザーのような⼤型レーザー施設を使って、天体現象で観測される極限的なプラズマ状態を実験室で作り出すことができます。例えば、⽊星内部に匹敵する超⾼圧下での⾦属⽔素状態の検証や、無衝突衝撃の形成などが積極的に進められています。また、超新星爆発に伴う磁化プラズマの界⾯不安定性について、レーザー実
験や MHD シミュレーションを⾏っています。このような現象は、レーザー核融合プ
ラズマにおいて、理想的な爆縮を実現する上でも⾮常に重要なプロセスです。したがって、界⾯不安定性の成⻑を抑制する機構について理解が急務になっています。本講演では、我々が⾏っているレーザー天体物理学の実験と、リヒトマイヤー・メシュコフ不安定性による磁場の増幅に関する理論的理解を紹介します[1-3]。
参考⽂献:[1] T. Sano, Alfven number for the Richtmyer-Meshkov instability in magnetizedplasmas, Astrophys. J. in press (2021)[2] T. Sano et al., Laser astrophysics experiment on the amplification of magnetic fields
by shock-induced interfacial instabilities, Phys. Rev. E 104, 035206 (2021)[3] T. Sano et al., Suppression of the Richtmyer-Meshkov instability due to a density
transition layer at the interface, Phys. Rev. E 102, 013203 (2020)

30

RIGIDITY OF BELTRAMI FIELDS WITH A NON-CONSTANTPROPORTIONALITY FACTORKEN ABE

Beltrami fields curl u = f u, div u = 0 appear as steady states of ideal incompressible
flows or plasma equilibria. I will discuss existence and non-existence issues on them fornon-constant factor f . In the first half of the talk, I will explain existence of axisymmetric Beltrami fields forming vortex rings and their construction via a variational principle.
In the second half, I will discuss a rigidity problem on symmetry of u for symmetric f
and a relation with Grad’s conjecture. This talk is based on preprints arXiv:2008.09345,
arXiv:2108.03870.

非定数係数を持つベルトラミ場の剛性阿部健
非圧縮理想流, プラズマ平衡で現れるベルトラミ場 curl u = f u, div u = 0 について比例係数 f が非定数となる場合の解の存在と非存在を考える. 講演の前半では軸対称ベル
トラミ場からなる渦輪と変分原理による解の構成について説明する. 後半では比例係数
f に対称性がある場合に u の対称性が得られるかという剛性の問題を考え, グラドの予想との関係を考察する. 本講演はプレプリント arXiv:2008.09345, arXiv:2108.03870 に基づく.
(K. ABE) Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku Osaka, 558-8585, Japan
E-mail address: kabe@osaka-cu.ac.jp

31

Exact solutions in non-integrable systems
Chihiro Matsuoka1 and Koichi Hiraide2
1

Osaka City University and 2 Ehime University

We present exact solutions in non-integrable systems, taking the Henon map
as an example. The solutions are constructed using the Laplace transform.
Unlike integrable systems such as soliton systems, the obtained solutions cannot
be described by any existing functions. We visualize a boundary of KAM tori
and the Henon attractor by use of the obtained functions, in which chaotic orbits
are captured accurately.

32非可積分系における厳密解松岡千博 (大阪市大), 平出耕一（愛媛大）エノン写像を例にとって、非可積分系における厳密解を与える。解はラプラス変換を用いて構成される。ここで用いる構成方法は Resurgent analysis と呼ばれる、変数係数常微分方程式の大局解を導くために開発された手法を拡張したものである。ソリトン系のような完全可積分系とは異なり、得られた解は既存の関数では記述できない。講演では、ハミルトン系に出現する KAM トーラスの境界や、散逸系カオスの代表であるエノンアトラクターを得られた関数を用いて描き、馬蹄構造やホモクリニック点といったカオスを特徴づける構造が正確に捉えられていることを示す。

33

Dirac geometry and variational structures innonequilibrium thermodynamics
Hiroaki Yoshimura
Waseda University, Tokyo, Japan

Abstract
A Dirac structure is a geometric object of unifying symplectic and Poisson structures. In mechanics, for the case of degenerate Lagrangian systems with nonholonomic constraints, which are linear in velocities, given
as a distribution on a configuration manifold Q, it is known that an induced (almost) Dirac structure over the Pontryagin bundle P = T Q⊕T ∗ Q
can be defined from the given distribution.
On the other hand, a general class of nonholonomic systems with nonlinear constraints may be formulated by a general Lagrange-d’Alembertprinciple by introducing variational constraints CV ⊂ T Q ⊕ T Q as well
as nonlinear kinematic constraints CK ⊂ T Q, in which case one cannot
define any appropriate geometric structure such as Poisson, symplectic aswell as Dirac structures for such a general class of nonholonomic systems
with nonlinear constraints.
In this talk, we study a class of nonholonomic systems with nonlinear
constraints of thermodynamic type, in which CK can be mathematicallydeduced from CV . For the cases of nonholonomic systems with nonlinear constraints of thermodynamic type, we show that an induced Dirac
structure DP ⊂ T P ⊕ T ∗ P can be defined from CV and therefore a Diracdynamical system can be introduced by using DP . We also show that
there exists an associated variational structure with the Dirac dynamicalsystem. In thermodynamics, such a class of nonlinear nonholonomic constraints typically appears in the relation of entropy production regarding
the second law of thermodynamics. Finally, we show that We illustrate
our theory with some examples of nonequilibrium thermodynamic systems
for the case of adiabatically closed and open cases.
This is a joint work with François Gay-Balmaz in École Normale
Supérieure in Paris.

34

非平衡熱力学の変分構造とディラック幾何吉村 浩明
早稲田大学 基幹理工学部

概 要
ディラック構造は，シンプレクティック構造とポアソン構造を統一した幾何学的概念であり，力学では，速度に関して線形な性質を有する非ホ
ロノミック拘束を受ける，退化ラグランジュ系において典型的に現れることが知られている．すなわち，配位空間 Q 上の拘束ディストリビュー
ションとして与えられる非ホロノミック拘束と正準シンプレクティック
構造によって，ポントリヤーギン束 P = T Q ⊕ T ∗ Q 上にディラック構造が定義でき，これを用いて非ホロノミック系はディラック力学系として定式化できる．
一般的な非線形かつ非ホロノミックな拘束を有する力学系では，非線形拘束 CK ⊂ T Q とは別に，変分拘束 CV ⊂ T Q ⊕ T Q を導入するによって，一般化ラグランジュ・ダランベール原理の枠組みで定式化できるが，その背後には，ポアソン構造，シンプレクティック構造，さらには，ディラック構造さえも存在しないことが知られている．
本講演では，CK が CV によって規定できる，いわゆる熱力学的なタイプの非線形拘束を受ける非ホロノミック系について考察し，その場合
は，力学系の幾何的構造として，P 上のディラック構造 DP ⊂ T P ⊕ T ∗ P
が定義できることを示す．その上で，DP を用いてディラック力学系を定義し，その背後にある変分構造を明らかにする．このタイプの非線形な非ホロノミック拘束は，熱力学の第２法則に関連する，エントロピー
生成の関係式として典型的に現れる．そこで，具体例として，断熱閉鎖
系及び開放系の非平衡熱力学を幾つか例にとり，本研究で提案したディ
ラック力学系による定式化手法の有効性を示す．
この研究は，パリ高等師範学校 (ENS) の François Gay-Balmaz 博士との共同研究に基づくものである．

35

OCAMI 研究集会
講演題目：半離散方程式の連続対称性と保存量
発表者：彭 林玉（慶應義塾大学）
講演要旨: 1880 年代に，ソフス・リーは，常微分方程式の可解性において連続変換がどのような役割を果たしているかについての研究を開始しました．それから，１世紀以上経ました．現在は微分方程式の可積分性と十分な対称性（ある解を別の解に移す連続変換）または十分な保存量の存在すること関連深いのがよく知っています．変分方程式の場合ではネーターの定理に従ってそれらが一対一で対応しています．
一方，多くの力学および物理現象は，戸田格子などの半離散方程式としてモデル化できます．ただし，ある連続変換の微分と差分両方を含む変数への延長が本質的な課題
であり，何十年にもわたっても半離散方程式の対称性理論は十分に確立されていません．
この講演では，半離散方程式の対称性の一般的な理論を確立して，ネーターの（第一と第二）定理を半離散方程式に拡張できるようにことを示します．本講演の一部は、Peter Hydon 教授（ケント大学）との共同研究に基づいています。

Title: Continuous symmetries of semi-discrete equations and conserved quantities
Speaker: Linyu Peng (Keio University)
Abstract: During the1880s, Sophus Lie started his study on what role continuous transformations
play in the solvability of ordinary differential equations. Nowadays, we know it well thatsolvability or integrability of differential equations is greatly due to the existence of sufficient
symmetries, i.e., continuous transformations moving one solution to another solution, or sufficientconserved quantities. For variational equations, they are connected to each other according to thecelebrated Noether’s theorem in a one-to-one manner.
On the other hand, many mechanical and physical systems can be modeled as semi-discrete
equations, such the Toda lattice. However, symmetries of semi-discrete equations have not been
well established because prolonging a transformation to variables involving both derivatives andshifts has been the essential challenge, that has bothered scholars for decades. In this talk, we willshow how this problem is eventually solved that allows us to extend Noether’s two theorems tosemi-discrete equations. This talk is partially related to joint work with Peter Hydon (Universityof Kent).

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Particle relabeling symmetry and Noether's first and second
theorems for fluid and MHD equations
Yasuhide Fukumoto
Institute of Mathematics for Industry, Kyushu University, Japan

Rong Zou
Department of Civil & Environmental Engineering, University of Hawai'iat Mānoa, USA
For the ideal fluid dynamics and magnetohydrodynamics (MHD), Noether's
theorem states that the topological invariant associated with the particle relabelingsymmetry is the cross helicity, the volume integral of the scalar product of thevelocity field and a frozen-in field. A proof to it is given in terms of variation of the
Lagrangian label as a function of the Eulerian position. According to Noether's
second theorem, the existence of the topological invariant implies redundancy of
the governing equations. The generalized Bianchi identity reflecting it is derived
for MHD.

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Particle relabeling symmetry and Noether's first and second
theorems for fluid and MHD equations
福本康秀 (九州大学マス・フォア・インダストリ研究所)
Rong Zou (ハワイ大学土木・環境工学科)

変分原理において、粒子のラベル付け替え対称性に付随する保存量がトポロジカル不変量で、ネーターの定理は、理想流体と電磁流体（MHD）に対して、トポロジカル不変量はクロス・ヘリシティに限ることをいう。クロス・ヘリシティの定義は、速度場と凍結場(=渦度場)の内積の全領域にわたる体積積分である。これに対し、オイラー的位置の関数としてのラグランジュラベルの変分を用いた証明を与える。ネーターの第 2定理によれば、トポロジカル不変量の存在は、支配方程式に冗長性があることを意味
し、支配方程式間の関係をあらわす一般化されたビアンキ恒等式が導かれる。

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