古典力學中主要基本原理形成過程的探討－從克卜勒行星運動定律到能量守恆定律

本研究所要探討的是在古典力學中，功、動能、位能、力學能守恆以及能量守恆理論的發展歷程。為了完整描述整個脈絡發展，我們首先由克卜勒面積律與橢圓律的重建出發，接著再以牛頓的運動定律及萬有引力定律的提出作為揭開古典力學發展的序幕。研究結果發現，功與動能的數學雛型最早是出現在牛頓的《原理》一書中，他是為了想知道行星或自由落體在受向心力作用，而非人為或機械的推或拉時，物體於任意位置的速率而引入，與工程上的需要無關。牛頓的想法隨後由白努利加以擴充，他除了將力與位移的關係重新以內積的方式表示外，他還將這種具有正、負值的物理量命名為「能」(即現在的「功」)，而成為史上第一個提出完整功概念的物理學家。此外，我們也發現在歷史上首次將牛頓第二定律改寫成f=ma的物理學家也是白努利，而非原先科學史家M. Jammer所認為是由歐拉最早寫下簡潔的f=ma表示式。此外，我們也發現重力作功與路徑無關的正合微分條件，是克來若於1743年以偏微分方程式首次明確提出。之後，拉格朗日不僅於1773年寫下了歷史上第一個位能函數，而且力學能守恆律也是他在1780年首次以分析力學推導而成。然而，能量的形式到最後並不是僅有動能和位能而已，因為從十九世紀開始，便已經有多位科學家注意到光、熱、電、磁與化學親和力似乎彼此間有互相轉換的現象，從而開始逐漸建立其自然力量普遍具有轉換性的自然哲學觀。後來，德國物理學家梅爾於1842年以因果等價原理的關係來說明能量的不可毀滅性，並寫下史上第一個由功轉換成熱的熱功當量關係(1 cal等於3.58 J)。英國物理學家焦耳則是於1843年始得知1 cal為4.82 J，之後再經過實驗改良，最後他才於1849年得到1 cal為4.15 J的更精確結果。雖然上述由作功轉成熱的熱功當量已經由實驗得知，不過當時卻還沒有可靠的實驗證據支持熱可轉換成作功。因此除了亥姆霍玆於信念上支持外，當時科學家們普遍因為支持熱質說，其實並不承認熱與功可互相轉換的熱功當量關係。後來於1850年，熱機運作的正確解釋由克勞修斯率先提出。他認為當熱機作功時，除了部份的熱會由高溫往低溫物體傳播之外，也會有部份的熱會轉化為功。他由上述想法提出具有內能概念的熱力學第一定律後，才讓克耳文及大部份的物理學家放棄熱質說，而接受熱與功可互相轉換的概念。從此之後，包含“能量具有不同形態”、“能量不可被創造與毀滅”及“熱與功彼此可互相轉換”三大特性的能量守恆定律，就成為古典力學的重要定律。

關鍵字

功；動能；力學能；內能；能量守恆

並列摘要

The purpose of this study is to investigate the theoretical developments of work, kinetic energy, potential energy, the conservation of mechanical energy and the conservation of energy in classical mechanics. In order to clarify the whole process of development, the reconstructions of Kepler’s first two laws for planets are introduced as the first reference in this study. Subsequently, Newton's law of motion and his law of universal gravitation will be utilized as a prelude to the development of classical mechanics. The results showed that the mathematical prototypes of work and kinetic energy are initially published in Newton’s "Principia". These concepts were proposed due to the fact that Newton wanted to find the speed of a planet or freefall at any position under the action of centripetal force, rather than the external force exerted by the machine. Newton’s hypothesis is eventually expanded by Johann Bernoulli, who was considered to reconstruct force and displacement relationship based on the representation of the dot product and in either a positive or negative physical quantity named "energy" (now is called the "work"), therefore became the first proposed the complete concept of work by physicists. In addition, the results also indicated that it is the physicist, Johann Bernoulli, for the first time in history rewrote Newton's second law as "f = ma" rather than Euler, the one whom the scientific historian M. Jammer perceived the first to write the concise expression "f = ma". We also found that the condition, which is fulfilled by the exact differential about the work of the gravitational force and does not depend on the trajectory of the body, is first explicitly determined in partial differential equation by Clairaut in 1743. Lagrange first proposed the model of potential energy function in history in 1773, and then he successfully established the law of conservation of mechanical energy by using the Analytical Mechanics in 1780. Besides, the energy end up possessing not only two forms of energy such as the kinetic and potential energy. Since the beginning of the nineteenth century, a number of scientists have noted the phenomenon seems to have mutual conversion among light, heat, electricity, magnetism and chemical affinity. Consequently, they began to establish their universal convertibility of natural powers in natural philosophy. After that, German physicist Mayer has illustrated that the principle of “causa aequat effectum” can be used to explain why the energy can’t be destroyed. Furthermore, he has proposed the first mechanical equivalent of heat in the history in 1842, which is related to SI units as shown below: 1 calorie is equal to 3.58 J. However, 1 calorie is equal to 4.82 J in SI units by the British physicist Joule in 1843. Eventually, in 1849, the further results from physicist Joule provide the more accurate values by using improved experiments; the final results showed that the 1 calorie is equal to 4.15 J. Even though the relationship of the convertibility of work into heat has been demonstrated as described above, there was no experimental evidence to support heat can be converted into work. At that time, most of scientists actually support the caloric theory rather than mutual convertibility of heat and work aside from support from Helmholtz’s faith in the convertibility of energy. The correct interpretation of the heat engine operating by Clausius first proposed in 1850. He mentioned that during the process of heat engine starting to work, some portion of heat is not only normally transferred from a high temperature object to a low temperature object, but also converted to work. Consequently, he proposed the first law of thermodynamics incorporating the concept of internal energy and persuaded the opponents including Lord Kelvin and physicists who support the caloric theory to accept the concept regarding mutual convertibility of heat and work. Since that, three characteristics about conservation of energy have become an important law in classical mechanics, including energy that can be changed to many different forms, and that can neither be created nor destroyed as well as the heat and work that are interchangeable.