Lagrange

Joseph-Louis Lagrange (1736~1813), whose full name was Joseph Louis Lagrange, was a famous French mathematician and physicist. Born on January 25, 1736 in Turin, Italy, and died on April 10, 1813 in Paris. He has made historic contributions in the three subject fields of mathematics, mechanics and astronomy, among which his achievements in mathematics are the most outstanding.

Lagrange's father was an officer in the French Army's cavalry. Later, his family fell apart due to business bankruptcy. According to Lagrange's own recollections, if his family had been rich as a child, he would not have done mathematical research because his father wanted to train him to become a lawyer. Lagrange personally had no interest in the law.

The Lagrange Institute of Science covers a wide range of fields. His most outstanding contribution to mathematics was to separate mathematical analysis from geometry and mechanics, to make the independence of mathematics clearer, and from then on mathematics was no longer just a tool for other disciplines.

Lagrange summarized the mathematical achievements of the 18th century and at the same time opened the way for mathematical research in the 19th century. He can be called the most outstanding mathematical master in France. At the same time, his achievements on lunar motion (three-body problem), planetary motion, orbit calculation, two fixed center problems, fluid mechanics, etc. have also played a historic role in mechanizing and analyzing astronomy. The role has promoted the further development of mechanics and celestial mechanics, and has become groundbreaking or foundational research in these fields.

In the first ten years of his work in Berlin, Lagrange spent a great deal of time on the solution of algebraic and transcendental equations, making a valuable contribution to the development of a generation of mathematics. He submitted two famous papers to the Berlin Academy of Sciences: "On the Solution of Numerical Equations" and "On the Study of Algebraic Solutions of Equations". Summarize the various solutions of the three and four algebraic equations of the predecessors into a standard method, that is, convert the equations into equations of lower order (called auxiliary equations or pre-solutions) to solve.

Lagrange was also the founder of analytical mechanics. In his famous book "Analytical Mechanics", Lagrange developed the research results of D'Alembert, Euler and others on the basis of summarizing the basic principles of mechanics in history, introduced the concepts of potential and equipotential surface, and further applied mathematical analysis to particle and rigid body mechanics, proposed universal equations for statics and dynamics, introduced the concept of generalized coordinates, established Lagrange's equations, transformed the equations of motion of the mechanical system from Newtonian form with force as the basic concept. The change to a form of analytical mechanics with energy as the basic concept laid the foundation for analytical mechanics and opened the way for the popularization and application of mechanical theory to other fields of physics.

He also gave a solution to the Euler dynamic equation of a rigid body rotating around a fixed point on the axis of rotational symmetry under the action of gravity (Lagrange's gyroscope), which made an important contribution to the solution of the three-body problem and solved the problem of restricted three-body motion. Lagrange also made important contributions to the theory of fluid motion and proposed a Lagrange method to describe fluid motion.

About half of Lagrange's research work was related to celestial mechanics. He used his principles and formulas in analytical mechanics to establish the equations of motion of various celestial bodies. In the solution of the equations of motion of celestial bodies, Lagrange discovered five special solutions to the equations of motion of the three-body problem, namely the Lagrange translation solution. In addition, he also studied the perturbation of comets and asteroids, and proposed the hypothesis of the origin of comets.

Over the past hundred years, many new achievements in the field of mathematics can be traced directly or indirectly to Lagrange's work. Therefore, he is regarded in the history of mathematics as one of the mathematicians who has had an overall influence on the development of analytical mathematics.