鲁道夫·利普希茨Rudolf Lipschitz

鲁道夫·利普希茨的父亲是一位地主，利普希茨出生在他父亲位于柯尼斯堡附近的Bönkein庄园。他年幼时就开始大学学习，进入柯尼斯堡大学，师从恩斯特·弗朗茨·诺伊曼。遵循当时在不同大学学习的习俗，利普希茨从柯尼斯堡前往柏林，师从约翰·彼得·古斯塔夫·勒热纳·狄利克雷。这对利普希茨来说并非特别轻松的时期，他的健康状况相当差，导致他休学一年以恢复。然而，他于1853年8月9日完成了博士研究，获得博士学位。

利普希茨没有立即获得大学教职，他在柯尼斯堡的文理中学和埃尔宾的文理中学教了四年书。然而，1857年，利普希茨成为柏林大学的Privatdozent。同年，他娶了Ida Pascha，她是与他父亲庄园相邻的一位地主的女儿。然后1862年，他成为布雷斯劳的编外教授。

在布雷斯劳的两年里，利普希茨写了两篇不太重要的论文。他与Heinrich Schröter和M Frankenheim共同创立了一个数学和数学物理讨论班。论文[5]考察了利普希茨在这两年间的职业生涯。他被波恩大学提名为正教授，并于1864年复活节离开布雷斯劳。

波恩大学是利普希茨度过余下职业生涯的地方。这并非因为他没有机会调动。恰恰相反，在阿尔弗雷德·克莱布什于1872年11月去世后，次年他便获得了哥廷根大学讲席的聘任。然而，利普希茨在波恩十分愉快，于是他拒绝了来自哥廷根的聘任。

菲利克斯·克莱因于1868年在波恩大学获得博士学位。他的导师是尤里乌斯·普吕克，答辩考官是利普希茨。或许，如果在利普希茨获得哥廷根讲席聘任时菲利克斯·克莱因仍在哥廷根，他可能会更倾向于接受。

也许利普希茨工作最引人注目之处在于他贡献的领域广泛多样[1]：-

他在数论、弗里德里希·威廉·贝塞尔函数论与Fourier series、常微分方程与偏微分方程、分析力学与potential theory中进行了许多重要而富有成果的研究。

他研究二次微分形式和力学。在论文[4]中，作者令人信服地表明，利普希茨对波恩哈德·黎曼的微分几何的力学解释将如何被证明是通往阿尔伯特·爱因斯坦狭义相对论道路上的关键一步。利普希茨表明[4]：-

……几何陈述可以被解释为力学定律[但这些正是]使深化相应几何关系成为可能的力学概念。

利普希茨关于用威廉·哈密顿-卡尔·古斯塔夫·雅各布·雅可比方法积分一般动力系统运动方程的工作，导致了在天体力学中的重要应用。

利普希茨因“利普希茨条件”而被铭记，这是一个保证微分方程 $y' = f (x, y)$有唯一解的不等式。朱塞佩·皮亚诺给出了该微分方程的一个存在性定理，给出了保证至少有一个解的条件。

他在代数数论中的工作引导他研究四元数以及诸如威廉·金顿·克利福德代数这样的推广。事实上，利普希茨重新发现了威廉·金顿·克利福德代数，并且是第一个将它们应用于表示欧几里得空间旋转的人，从而引入了自旋群 Spin($n$)。

Rudolf Lipschitz's father was a landowner and Rudolf was born his father's estate at Bönkein which was near Königsberg. He began his university studies at a young age, entering the University of Königsberg and studying there under Franz Neumann. Following the custom of that time to study at different universities, Lipschitz went from Königsberg to Berlin where he studied under Dirichlet. This was not a particularly easy time for Lipschitz whose health was rather poor and caused him to take a year away from his studies to recover. However, he completed his doctoral studies with the award of a doctorate on 9 August 1853.

There was no immediate university teaching post for Lipschitz who spent four years teaching at the Gymnasium in Königsberg and at the Gymnasium in Elbing. In 1857, however, Lipschitz became a Privatdozent at the University of Berlin. In this same year he married Ida Pascha, the daughter of one of the landowners with an estate near to his father's. Then in 1862 he became an extraordinary professor at Breslau.

During his two years in Breslau, Lipschitz wrote two not very important papers. Jointly with Heinrich Schröter and M Frankenheim, he founded a seminar in mathematics and mathematical physics. The paper [5] looks at Lipschitz's career during these two years. He was nominated an ordinary professor by the University of Bonn and he left Breslau at Easter 1864.

The University of Bonn was where Lipschitz spent the rest of his career. This was not because he did not have the opportunity to move. Quite the reverse, after Clebsch died in November 1872 he was offered his chair at Göttingen in the following year. Lipschitz was quite happy at Bonn, however, and he turned down the offer from Göttingen.

Klein received his doctorate from the University of Bonn in 1868. He was supervised by Plücker, and examined by Lipschitz. Perhaps if Klein had still been in Göttingen when Lipschitz was offered the chair there, he may have been more inclined to accept.

Perhaps the most remarkable fact about Lipschitz's work was the widely different topics on which he contributed [1]:-

He carried out many important and fruitful investigations in number theory, in the theory of Bessel functions and of Fourier series, in ordinary and partial differential equations, and in analytical mechanics and potential theory.

He worked on quadratic differential forms and mechanics. In the paper [4] the author shows convincingly how Lipschitz mechanical interpretation of Riemann's differential geometry would prove to be a vital step in the road towards Einstein's special theory of relativity. Lipschitz showed that [4]:-

... the geometrical statements could be interpreted as mechanical laws [but these were] the very mechanical concepts that made it possible to deepen the corresponding geometrical relations.

Lipschitz's work on the Hamilton-Jacobi method for integrating the equations of motion of a general dynamical system led to important applications in celestial mechanics.

Lipschitz is remembered for the 'Lipschitz condition', an inequality that guarantees a unique solution to the differential equation $y' = f (x, y)$. Peano gave an existence theorem for this differential equation, giving conditions which guarantee at least one solution.

His work in algebraic number theory led him to study the quaternions and generalisations such as Clifford algebras. In fact Lipschitz rediscovered Clifford algebras and was the first to apply them to represent rotations of Euclidean spaces, thus introducing the spin groups Spin($n$).