查看“︁几何力学”︁的源代码

'''几何力学'''将特定的几何方法应用于许多[[力学]]领域，从质点力学和[[刚体力学]]到[[流体力学]]和[[控制论]]。

几何力学主要适用于[[构型空间]]为[[李群]]或[[微分同胚]]群的系统，更一般地说，适用于构型空间的某些方面具有此种群结构的系统。例如卫星之类刚体的构型空间是欧氏运动群（空间中的平移与旋转），而液晶的构型空间则是与内部状态（规范对称性或有序参数）耦合的微分同胚群。

== 动量映射与还原 ==
几何力学的主要思想之一是还原，可追溯到雅可比在三体问题中对节点的消除；现代形式由K. Meyer (1973)与J.E. Marsden、A. Weinstein (1974)分别独立诠释，都受到Smale (1970)的启发。根据[[诺特定理]]，哈密顿或拉格朗日系统的对称性会表现为守恒量，就是[[动量映射]]'''J'''的分量。若''P''是相空间，''G''是对称群，则动量映射为<math>\mathbf{J}:P\to\mathfrak{g}^*</math>，还原空间是'''J'''的水平集对保相关水平集的''G''子群的商：对<math>\mu\in\mathfrak{g}^*</math>，定义<math>P_\mu=\mathbf{J}^{-1}(\mu)/G_\mu</math>，若<math>\mu</math>是''J''的正则值，则此还原空间是辛流形。
== 几何积分器 ==
力学的重要几何方法是将几何融入数值方法，尤其是辛与变分积分器，特别适于哈密顿和拉格朗日系统的长期积分。

== 历史 ==
“几何力学”偶尔指17世纪的力学。<ref>Sébastien Maronne, Marco Panza. [https://hal.archives-ouvertes.fr/hal-00415933/document "Euler, Reader of Newton: Mechanics and Algebraic Analysis".] {{Wayback|url=https://hal.archives-ouvertes.fr/hal-00415933/document |date=20180725122258 }} In: Raffaelle Pisano. ''Newton, History and Historical Epistemology of Science'', 2014, pp. 12–21.</ref>

作为现代学科，其源于1960年代的四部著作：Vladimir Arnold (1966)、Stephen Smale (1970)、Jean-Marie Souriau(1970)、Abraham & Marsden《力学基础》第一版(1967)。Arnold的基础研究表明，自由刚体的欧拉方程是旋转群SO(3)上的测地流方程，并将这一几何见解应用于理想流体动力学中。当中，旋转群内被保体积微分同胚群取代。Smale关于拓扑学与力学的论文研究了对称性的李群作用于动力系统时，诺特定理产生的守恒量，并定义了现在所谓动量映射（Smale称之为角动量），还提出了能量-动量水平面的拓扑及其对动力的影响。Souriau在书中也考虑了由对称群作用产生的守恒量，但他更关心涉及的几何结构（如该动量在一大类对称中的等差性质），而较少涉及动力学问题。
这些观点，尤其Smale的观点是《力学基础》第二版(Abraham & Marsden, 1978)的核心内容。

== 应用 ==
* 计算机图形学
* 控制论 &mdash; 见Bloch (2003)
* 液晶 &mdash; 见[https://link.springer.com/article/10.1007/s00205-013-0673-1 Gay-Balmaz, Ratiu, Tronci (2013)] {{Wayback|url=https://link.springer.com/article/10.1007/s00205-013-0673-1 |date=20231012195728 }}
* 磁流体动力学
* 分子振荡
* 非完整约束 &mdash; 见Bloch (2003)
* 非线性稳定
* 等离子体 &mdash; 见Holm, Marsden, Weinstein (1985)
* 量子力学
* 量子化学 &mdash; 见[https://link.springer.com/article/10.1007/s10440-019-00257-1 Foskett, Holm, Tronci (2019)] {{Wayback|url=https://link.springer.com/article/10.1007/s10440-019-00257-1 |date=20231012195722 }}
* 超流体
* 热力学 &mdash; 见[https://www.mdpi.com/1099-4300/21/1/8 Gay-Balmaz, Yoshimra (2018)] {{Wayback|url=https://www.mdpi.com/1099-4300/21/1/8 |date=20231117074047 }}
* 空间探索的轨迹规划
* 水下航行器
* 变分积分器；见[https://doi.org/10.1017/S096249290100006X Marsden and West (2001)]

== 参考文献 ==
{{Reflist}}
*{{Citation | last1=Abraham | first1=Ralph| author1-link=Ralph Abraham (mathematician) | last2=Marsden | first2=Jerrold E.| author2-link=Jerrold E. Marsden | title=Foundations of Mechanics| year=1978 | edition=2nd| publisher=Addison-Wesley}}
*{{Citation | last=Arnold | first=Vladimir | author-link=Vladimir Arnold | title=Sur la géométrie différentielle des groupes de Lie de dimension infine et ses applications a l'hydrodynamique des fluides parfaits | journal=Annales de l'Institut Fourier | volume=16 | pages=319–361 | year=1966 | doi=10.5802/aif.233 | url=http://www.numdam.org/article/AIF_1966__16_1_319_0.pdf | doi-access=free | accessdate=2023-12-01 | archive-date=2023-10-12 | archive-url=https://web.archive.org/web/20231012195721/http://www.numdam.org/article/AIF_1966__16_1_319_0.pdf | dead-url=no }}
*{{Citation | last=Arnold | first=Vladimir | author-link=Vladimir Arnold | title=Mathematical Methods for Classical Mechanics | publisher=Springer-Verlag | year=1978}}
*{{cite book | last=Bloch | first=Anthony | title=Nonholonomic Mechanics and Control | publisher=Springer-Verlag | year=2003 }}
*{{cite journal |last1=Foskett |first1=Michael S. |last2=Holm |first2=Darryl D.| last3=Tronci|first3=Cesare |title=Geometry of Nonadiabatic Quantum Hydrodynamics |journal=Acta Applicandae Mathematicae |year=2019 |volume=162 |issue=1 |pages=63–103 |doi= 10.1007/s10440-019-00257-1|arxiv=1807.01031|s2cid=85531406 }}
*{{cite journal|last1=Gay-Balmaz|first1=Francois|last2=Ratiu|first2=Tudor|author2-link=Tudor Ratiu| last3=Tronci|first3=Cesare|title=Equivalent Theories of Liquid Crystal Dynamics| journal=Arch. Ration. Mech. Anal. |volume=210|year=2013|issue=3|pages= 773–811|doi=10.1007/s00205-013-0673-1|arxiv=1102.2918|bibcode=2013ArRMA.210..773G|s2cid=14968950 }}
*{{cite journal|last1=Holm|first1=Darryl D.|last2=Marsden|first2=Jerrold E.|author2-link=Jerrold E. Marsden|last3=Ratiu|first3=Tudor S.|author3-link=Tudor Ratiu|last4=Weinstein|first4=Alan|author4-link=Alan Weinstein|title=Nonlinear stability of fluid and plasma equilibria|journal=Physics Reports|volume=123|year=1985|issue=1–2|pages=1–116|doi=10.1016/0370-1573(85)90028-6|bibcode=1985PhR...123....1H|url=http://infoscience.epfl.ch/record/129451|access-date=2023-12-01|archive-date=2022-01-24|archive-url=https://web.archive.org/web/20220124174412/http://infoscience.epfl.ch/record/129451|dead-url=no}}
*{{cite book| last1=Libermann| first1=Paulette| author1-link=Paulette Libermann| last2=Marle| first2=Charles-Michel| title=Symplectic geometry and analytical mechanics| series=Mathematics and its Applications| volume=35| publisher=D. Reidel| location=Dordrecht| year=1987| isbn=90-277-2438-5| doi=10.1007/978-94-009-3807-6| url-access=registration| url=https://archive.org/details/symplecticgeomet0000libe}}
*{{Citation | last1=Marsden | first1=Jerrold | author1-link=Jerrold E. Marsden| last2=Weinstein| first2=Alan | author2-link=Alan Weinstein|  title=Reduction of Symplectic Manifolds with Symmetry | journal=Reports on Mathematical Physics | year=1974 | pages=121–130 | volume=5 | issue=1 | doi=10.1016/0034-4877(74)90021-4| bibcode=1974RpMP....5..121M}}
*{{cite book| last1=Marsden | first1=Jerrold | author1-link=Jerrold E. Marsden| last2=Ratiu | first2=Tudor S. | author2-link=Tudor Ratiu| title=Introduction to mechanics and symmetry | publisher=Springer-Verlag | location=New York | series=Texts in Applied Mathematics | year=1999 | edition=2| isbn=0-387-98643-X}}
*{{Cite book | last=Meyer|first=Kenneth|chapter=Symmetries and integrals in mechanics |title=Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971)|year=1973|pages=259–272|publisher=Academic Press|location=New York}}
*{{cite book| last1=Ortega| first1=Juan-Pablo|last2=Ratiu| first2=Tudor S.| author2-link=Tudor Ratiu| title=Momentum maps and Hamiltonian reduction|publisher = Birkhauser Boston|series=Progress in Mathematics|volume = 222|year = 2004|isbn = 0-8176-4307-9}}
*{{Citation | last=Smale | first=Stephen | author-link=Stephen Smale| title=Topology and Mechanics I | journal=Inventiones Mathematicae | year=1970 | pages=305–331 | doi= 10.1007/bf01418778| volume=10| issue=4 | bibcode=1970InMat..10..305S| s2cid=189831616 }}
*{{Citation | last=Souriau | first=Jean-Marie | author-link=Jean-Marie Souriau | title=Structure des Systemes Dynamiques | publisher=Dunod | year=1970}}

[[Category:经典力学]]
[[Category:哈密顿力学]]
[[Category:动力系统]]
[[Category:辛几何]]
[[Category:拉格朗日力学]]