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The Investigations on the Formation of Fundamental Principles in Classical Mechanics－from Kepler’s Laws of Planetary Motion to the Law of Conservation of Energy
Conservation of Energy
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。
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.
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