Abstract Isaac Newton (1642-1727) was a well-known physicist and mathematician who made great contributions to natural science. His great work Mathematical Principles of Natural Philosophy is an important part of classical mechanics theory and an important milestone in the history of human development. This paper discusses his contribution to classical mechanics from Newton’s law of motion and the universal law of gravitation.

Key Words Newton, Classical mechanics, Newton’s law of motion, Law of universal gravitation

Introduction

   Classical mechanics is an ancient subject. Since the ancient Greek philosopher Aristotle, people have been studying the motion of objects. Its theoretical system can be divided into two periods, vector mechanics and analytical mechanics. Galileo made a pioneering and great contribution to the establishment of vector mechanics, the most important of which is that he established the classical mechanics’ theory with solid and reliable scientific experiments. He came up with the correct theory of accelerated motion by putting a small ball on a ramp for quantitative tests. Newton, on the other hand, continued to study the scientific theories pioneered by Galileo, summarizing, and developing Newton’s laws of motion and universal gravitation. In Newton’s time, the “helio-centric” theory was first proposed by Copernicus. Based on Tycho’s observations, Kepler summarized Kepler’s laws of planetary motion. Force and acceleration were also defined by Galileo, and the physical laws related to inertia and free fall were proposed. Except, at that time, each physics and law of physics is independent of each other. Newton “stood on the shoulders of these great men”, comprehensively observed and studied the motion of the planets and objects on the earth and turned it into a complete system by mathematical methods, and was able to express the causal logic. As Newton said, “Natural philosophy should be called ‘physics,’ and its goal is to find the structure and behavior of nature, and to reduce it as far as possible to a universal law and law, to determine the law by observation and experiment, and from this to conclude the causal relationship.” This paper mainly summarizes Newton’s main contribution to classical mechanics, including his book The Mathematical Principles of Natural Philosophy (hereinafter referred to as Principia) and Newton’s laws of motion and universal law of gravitation.

An epoch-making masterpiece – Principia

   Principia is considered by scientists to be one of the greatest books in the history of science (Steele). Newton built on the work of his predecessors in his magnum opus Principia, which was published in 1687. The title “natural philosophy” is because physics was called natural philosophy at the time, and the classical mechanics he was discussing was an important part of physics. The word “mathematics” is used to emphasize the strict mathematical relations in mechanics. This book is not the culmination of the scientific revolution of the 16th and 17th centuries, but also a milestone in the progress and development of human civilization. This book not only summarizes and promotes Newton and all the major achievements of physics before him but also becomes the model of scientific research in the future. It is the product and crystallization of the historical development of astronomy, mathematics, and mechanics. At the same time, he also inherited and developed the previous scientific method, the “induction-deduction” method. The publication of the book marked the establishment of the classical mechanics’ system and established an immortal monument in the history of science.

   Principia is divided into two main parts. In the first part, Newton first elaborated several important concepts, such as mass, external force, centripetal force, inertia, time, space, and so on, and on this basis gave Newton’s three laws of motion, and on this basis put forward such as the synthesis and decomposition of force, momentum conservation, movement of the center of mass and other theorems. In this section, Newton’s second and third laws are given proper scientific and mathematical treatment. This is different from previous attempts to explain similar phenomena, where experiments were either incomplete, incorrect, or without precise mathematical expressions. But Newton was different, he gave a more complete expression. Newton also showed how the law of conservation of momentum and angular momentum works. This part is not much, but it is the foundation of the whole book. The second part is about the application of these theorems and laws. He used deductive reasoning to get the very important law of gravitation; The damping motion, hydrodynamics, and hydrostatics are discussed. According to the discovered laws, the cosmic system was explained, and the observed celestial data were analyzed, such as the determination of the orbit of the satellite around the planet, the planet around the sun, and the comet. At the same time, the free fall and projectile motion of the object on the ground were observed and analyzed, and the movement of the surface and the celestial body were combined for the first time. The combination of Newton’s laws of motion and gravitation provided the most complete and accurate description of classical mechanics, and he proved that these laws apply to celestial bodies as well as everyday objects.

   Although Principia has many limitations, such as the introduction of “absolute time”, “absolute space” and other concepts, and only applies to the macro slow-speed world, it laid the foundation for the development of physics, it is a great scientific work.

Newton’s three laws of motion

Newton’s First Law of Motion: A body at rest persists in its state of rest, and a body in motion remains in constant motion along a straight line unless acted upon by an external force.

Newton’s Second Law of Motion: A body’s acceleration is directly proportional to the force exerted on it and is in the same direction as the force.

Newton’s Third Law of Motion: To every action, there is an equal and opposite reaction.

    Newton’s three laws of motion have now become widely accepted by scientists as macroscopic laws of nature, and they are the basis for inferences.

Newton’s first law of motion

    The first law is based on the experiments of Galileo and Descartes on the inertial motion. Galileo used the inclined plane experiment, if a small ball from any place on the inclined plane rolled down, under ideal conditions, it can never stop rolling. Later, Descartes further studied this ideal experiment, developing the “never-ending roll” into a straight-line motion in space. Based on the above research, Newton came up with Newton’s first law and proposed that inertia is the original property of objects. For physicists today, it is almost the foundation of mechanics.

Newton’s second law of motion

   The second law is a development of Galileo’s idea of dynamics after defining the concepts of mass, velocity, and acceleration. The second law is closely related to the first law. While the first law states that an object moves at rest or in a straight line with constant velocity with no external force or zero combined force, the second law states that an external force only changes the state of motion of the object. This overturned the previous misconception that force was required to maintain a body in motion. Newton’s statement of the second law shows that his original statement of the second law was not about the relationship between force and acceleration, but about the relationship between force and “change of motion”. The “change of motion” in this case is actually the change in momentum over time, while the acceleration is the rate of change of velocity. Newton’s second law plays an important role in Newton’s laws of motion and has a wide range of applications. Newton’s second law holds true for a force, whether electromagnetic or mechanical, for the mass of an object, whether it is the mass of a nuclear particle or a star, and for small or large accelerations.

Newton’s third law of motion

   It has been proven many times that “two objects, when colliding head-on, both move in the same way and, therefore, their movements and reactions are the same.” From this it can be seen that the third law is based on the phenomenon of collision, and its quantified results are obtained by the equal amount of motion of the collision phenomenon.

   The third law summarizes the concept of force comprehensively and states that each force has its own reaction. Collision is the basic phenomenon on which the third law is based. His collision experiment assumes that air resistance has been excluded, and through repeated arguments, it is concluded that when “two objects collide head-on, they produce a change of motion of the same magnitude in the opposite direction of their own motion, so that their action and reaction are always the same”. It can be seen that the collision phenomenon is the basis of Newton’s third law, and its quantitative results are derived from the equal amount of motion of the collision phenomenon. In other words, because two objects touch each other after the change in their respective momentum is the same, and because the length of time used is the same, so the conclusion is drawn that the action and reaction forces are equal. The formulation of the third law has led to a better understanding of the interaction between objects and revealed the internal structure of the natural unity of opposites.

Newton’s law of universal gravitation

“Every particle attracts every other particle in the universe with a force that is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.”

    Newton’s proof of gravity theory indicates that Gravity is an interaction between two objects, and the magnitudes of the forces between these two objects are the same. Moreover, the context of Newton’s third law of motion is that If two bodies exert forces on each other, these forces have the same magnitude but in opposite directions (Thornton and Marion). To some degree, Newton’s proof of gravity theory proves Newton’s third law of motion.

   The law of gravitation is an important historic discovery. It laid the scientific foundation of astronomical motion, used the theory of gravitation, people discovered Neptune, Pluto, and other objects, and explained many terrestrial and celestial phenomena such as Halley’s comet. It also reveals one of the most basic forms of interaction between objects – gravity. Based on the previous inverse square theorem of gravity, he connected the force with the mass of an object and applied the law of reaction to generalize the law of universal gravitation. This law has been proved by numerous experiments, is the world-recognized scientific law, and is one of the most important laws.

Conclusion

   The classical mechanical system takes absolute space, absolute time, mass, and force as the basic elements, centers on the three laws of motion, take the law of gravity as a whole, and describes the motion law of objects in the form of calculus. His logic is strict, the structure is tight, and through observation and experiment, formed a complete scientific system. The establishment of classical mechanics laid a solid foundation for the development of the whole natural science, established the basic concepts and basic laws of mechanics, made mechanics become a systematic and theoretical knowledge system, and gradually mature and perfect. It makes people’s description of the motion state of the object improve from the result of a change to the understanding of the process of change. The unification of the motions of heaven and earth brought about the first great synthesis of man’s knowledge of nature; Let people’s understanding of mechanical motion from kinematics to dynamics. Classical mechanics not only describes the motion of objects but also reveals the causes of their motion, thus allowing people to understand how things move and explain why they move, thus deepening their understanding of nature. However, it also has inherent limitations: it is only effective under macro low-speed conditions. Once the phenomenon of microparticles or objects moving close to the speed of light is produced, classical mechanics cannot make an effective and reasonable explanation. Although the motion of objects in the microcosmic world is explained by quantum mechanics, and the description of objects in high-speed motion is solved by the use of relativity, classical mechanics can still explain most of the daily phenomena in life. At the same time, classical mechanics did not launch the stage of history because of the proposal of relativity and quantum mechanics. Today, satellites, space travel, and so on are inseparable from classical mechanics. Ancient classical mechanics in the new era, are once again full of vitality.

   Newton was a well-known mathematician, physicist, and astronomer. He collected the achievements of all the pioneers in the field of science in the sixteenth and seventeenth centuries. With the publication of Principia as the mark, he used a unified theory to explain the motion state and law of all objects and gave a correct and clear explanation of how objects move under the condition of macroscopic low speed. This is the first major synthesis of natural science in human history. In addition, differential equations and differential equations created by Newton in mathematics also provided important tools for the development of natural sciences in the future, thus opening a new era of physics and mathematics. Of course, although Newton made great contributions in the early stage, later, he classified these inexplicable phenomena into the field of theology, and spent the second half of his life, completing 1.5 million words of theological works. However, Newton’s establishment of classical mechanical proposals and his methods of doing research propelled the rapid development of physics in the 18th and 19th centuries, helping to better the study of the natural sciences, until quantum mechanics and relativity were established. His scientific achievements and ideas not only greatly promoted the academic and ideological circles at that time, but also changed society to some extent, profoundly affected the development of modern science and technology and social development, made great progress in modern society, and made outstanding contributions to the development of all mankind in the world.

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【摘要】 艾萨克·牛顿（1642-1727）是一位家喻户晓的物理学家和数学家，为自然科学做了巨大的贡献。他的巨著《自然哲学的数学原理》是经典力学理论的重要组成部分，是人类发展史上一块重要的里程碑。本文从他所提出的牛顿运动定律和万有引力定律来讨论他对经典力学的贡献。

【关键词】 牛顿 经典力学 牛顿运动定律 万有引力定律

导言

   经典力学是一门古老的学科。从古希腊哲学家亚里士多德起，人们就开始对物体的运动进行研究。其理论体系可划分为两个时期，矢量力学和分析力学。伽利略在建立矢量力学方便做出了做出了开创性的、巨大的贡献，其中最重要的就是他以牢固可靠的科学实验建立了经典力学理论。他把一颗小球放到斜坡上去做定量试验，由此得出了正确的加速运动理论。而牛顿则继续研究伽利略所开创的科学理论，总结并发展出了牛顿运动定律和万有引力定律。在牛顿所处的年代，“日心说”理论被哥白尼率先提出，在第谷的观测结果之上，开普勒归纳出了关于行星运动规律的开普勒定律，力和加速度还被伽利略定义，并提出了和惯性以及自由落体相关的物理规律。只不过，在那时每一个物理和物理定律之间是彼此独立的。牛顿则“站在这些伟人的肩头”，全面地观测和研究了星球和地球上的物体的运动，并用数学的方法，把它变成了一个完整的系统，并且能够表达因果逻辑性。就像牛顿所说的，“自然哲学应该被称为“物理”，其目标是找到自然界的构造和行为，并尽可能的将其归纳成一种普遍的规律和规律，通过观察和试验来确定其规律，由此得出其因果关系。”本文主要总结了牛顿对经典力学的主要贡献，包括他所著的《自然哲学的数学原理》（以下简称《原理》）一书和他所提出的牛顿运动定律、万有引力定律。

划时代的巨著——《原理》

   牛顿在前人工作的基础上写出了《原理》这本巨著并于1687年出版。书名中的“自然哲学”是因为在那个年代人们将物理学称之为自然哲学，而他所探讨的经典力学又是物理学的重要组成部分之一。使用“数学”一词则是为了强调在力学中严格的数学关系。这本书不是十六世纪和十七世纪科学革命的顶峰，同时也是人类文明进步发展的一个里程碑。这本书不但概括并且推进了牛顿及其之前的所有重大的物理成就，同时也成为以后的科学研究的典范，是天文学、数学和力学历史发展的产物与结晶。同时，他还继承并发展了前人的科学方法——“归纳——演绎”法。该书的出版，标志着经典力学体系的建立，在科学史上建立了一个不朽的丰碑。

   《原理》一书主要分成两个部分。在第一部分中，牛顿首先阐述了几个重要的概念，如：质量、外力、向心力、惯性、时间、空间等等，并在此基础上给出了牛顿三大运动定律，并以此为基础提出了如力的合成与分解、动量守恒、质心运动等定理。在这一部分中，牛顿的第二定律和第三定律都进行了适当的科学与数学处理。这和以前试图解释类似现象的情况不大一样，那些实验要么不完整，要么不正确，要么没有精确的数学表达式。但是牛顿不一样，他给出了一个较为完整的表达式。牛顿还展示了动量和角动量守恒定律是如何工作的。这一部分虽然内容并不多，但是是整本书的根基。第二部分讲述的内容是对这些定理定律的应用。他使用演绎推理得到了非常重要的万有引力定律；对阻尼运动、流体动力学和流体静力学进行了论述；根据已经发现的规律，对宇宙系统进行了阐释，并且分析了观测到的天体数据，比如卫星围绕行星，行星围绕太阳，彗星轨道的确定等内容，同时还对地面上物体的自由落体和抛体运动进行观测与分析，第一次把地表的运动与天体的运动结合在一起。牛顿运动定律和万有引力定律的结合为经典力学提供了最完整、最准确的描述，并且他证明了这些规律对天体和日常物体都适用。

   虽然这本书还有许多局限性，比如引入了“绝对时间”、“绝对空间”等概念，并且只适用于宏观低速世界，但它为物理学的发展奠定了基础，是一部伟大的科学著作。

牛顿三大运动定律

Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.

定律一 每一个物体都保持它自身的静止的或者一直向前均匀地运动的状态，除非有外加的力迫使它改变它自身的状态为止。

Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

定律二 运动的改变与外加的引起运动的力成比例，并且发生在沿着那个力被施加的直线上。

Lex III: Actioni contrariam semper et æqualem esse reactionem: sive corporum duorum actiones in se mutuo semper esse æquales et in partes contrarias dirigi.

定律三 对每个作用存在总是相反的且相等的与反作用：或者两个物体彼此的相互作用总是相等的，并且指向对方。

   牛顿三大运动定律现在已经成为使科学家们普遍接受的宏观自然法则，并以其为推论的依据。

   第一定律是基于伽利略、笛卡尔两位科学家对惯性运动的实验得来的。伽利略通过斜面实验，若一个小球从斜面上的任意一个地方滚下来，在理想条件之下，它就可以永不停息地滚动下去。随后，笛卡尔对这个理想实验进行了进一步的研究，把“永不停息的滚动”发展成了在空间内的直线运动。基于以上研究，牛顿得出了牛顿第一定律，提出惯性是物体原有的性质。对当今的物理学家来讲，它几乎成为了力学的基础。

   第二定律是在定义了质量、速度和加速度的概念后，对伽利略动力学思想的发展。第二定律与第一定律紧密相连。第一定律指出，物体在静止状态下或在直线上以恒定的速度运动，没有外力或合力为零，而第二定律指出，外力只会改变物体的运动状态。这推翻了以前认为要维持一个物体的运动状态需要力的错误观念。牛顿关于第二定律的陈述表明，他对第二定律的原始说法并不是关于力和加速度的关系，而是关于力和“运动的改变”之间的关系。在这里所说的“运动的改变”，其实就是动量跟着时间的推移而变化，而加速度却是速度的变化率。牛顿第二定律在牛顿运动定律中起着重要作用，同时有着广泛的应用范围。一个力无论是电磁力还是机械力，一个物体的质量无论是核粒子质量还是星体质量，并且无论加速度的是小是大，牛顿第二定律均成立。

   经过多次的证明，“两个物体在正面相撞时，都会发生相同的移动，因此，它们的动作和反动都是一样的。”从这一点可以看出，第三定律建立在碰撞现象的基础上，它的量化结果是由碰撞现象的运动量相等得到的。

   第三定律全面地总结了力的概念，并指出每个力都有它自己的反作用力。碰撞是第三定律所依据的基本现象。他的碰撞实验假设空气阻力已被排除，通过反复的论证，得出当“两个物体在正面相撞时，他们会向自己运动的相反方向产生一个相同大小的运动的变化，所以他们的作用与反作用总是一样的”。由此可以看出，碰撞现象是牛顿第三定律提出的基础，它的量化结果是因为相等的碰撞现象运动量中得出结论的。换句话说，因为两个物体在相互碰之后，他们各自动量的改变量是一样的，又因为所用的时间长度是一样的，因此得到结论，作用力与反作用力相等。第三定律的提出，让人们更好地理解了物体之间的相互作用，揭示了自然的对立统一的内部构造。

   以上三条看起来很简单的定律，就是整个动力学的基本原理，同时也是牛顿关于物体运动规律的最伟大的发现。

万有引力定律

任意两个质点由通过连心线方向上的力相互吸引。该吸引力的大小与它们的质量乘积成正比，与它们距离的平方成反比，与两物体的化学本质或物理状态以及中介物质无关。

   万有引力定律是一项重大的历史性发现。它奠定了天文运动的科学基础，利用万有引力理论，人们发现了海王星、冥王星等星体，解释了了许多诸如哈雷彗星等地面与天体现象。同时它还揭示了物体之间相互作用的最基本形式之一——引力。他根据先前的引力平方反比定理，将力与物体质量联系起来，运用反作用定律，推广出了万有引力定律。这个定律已被无数次的实验证明，是世界公认的科学定律，是最重要的定律之一。

总结

   经典力学系统以绝对空间、绝对时间、质量、力为基本要素，以三大运动法则为中心，以重力法则为整体，以微积分的形式描述物体的运动规律。他的逻辑严密，结构严密，通过观察和试验，形成了一套完整的科学体系。经典力学的建立，为整个自然科学的发展打下了坚实的基础，确立了力学的基本概念和基本规律，使力学成为一个系统化、理论化的知识体系，并逐步走向成熟和完善。它使人们对物体运动状态的描述从变化的结果提高到对变化过程的认识；把天上和地面上的运动统一起来，使人对自然的认识发生了第一次大综合；让人们对机械运动的理解由运动学发展到了动力学。经典力学不但描绘了物体的运动，还揭示了其产生的原因，从而让人们理解物体是如何移动的，并且解释了它们为何移动，从而加深了对自然的理解。但是它也存在着固有的局限性：只在宏观低速条件下有效，一旦有关微观粒子或者关于接近光速运动的物体所产生的现象，经典力学便无法进行有效的、合理的解释。尽管微观世界的物体的运动都是按照量子力学来解释的，描述高速运动的物体都是运用相对论来解决的，但是经典力学还是可以解释生活当中的绝大部分日常现象，同时也没有因为相对论和量子力学的提出，经典力学从而推出历史的舞台。现如今，人造卫星、宇宙航行等等都与经典力学密不可分。古老的经典力学在新时代，又一次焕发出勃勃生机。

   牛顿是一位家喻户晓的数学家、物理学家和天文学家。他集合了十六世纪和十七世纪世界上所有科学领域的先驱者的成就，以《原理》一书出书作为标志，使用一个统一的理论解释所有物体的运动状态和运动规律，对在宏观低速条件下物体的运动如何运动给出了正确清晰的解释。这是人类有史以来首次在自然科学领域的大综合。此外，牛顿在数学上所创立的微分和微分方程，也为以后的自然科学发展提供了重要的工具，从而开启了物理学和数学的新时代。当然，牛顿虽然在前期做出了巨大的贡献，但到了后来，他将这些难以解释的现象都归类到了神学领域，并用了他的后半生时间，完成了一百五十万字的神学作品。然而，牛顿所建立的经典力学提议以及他进行研究的方法推动了十八世纪和十九世纪物理学飞速的发展，帮助人们更好地进行自然科学研究，直到量子力学和相对论的出现与建立。他的科学成就与观念不但极大地促进了当时的学术界和思想界，也在某种程度上改变了社会，深刻地影响了近代科学技术的发展和社会的发展，让近代社会有了极大的进步，为全世界全人类的发展做出了杰出的贡献。

参考文献

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[12] 摘读《自然哲学的数学原理》, http://www.wendangku.net/doc/f16785267.html，2020