古典力学における運動法則の歴史性 : ニュートンの第二法則をめぐって

Abstract

Now we understand that Newton's laws of motion -- the law of inertia, the equation of motion ‘F = ma, ' and the law of ‘action = reaction' -- are the basic principles of classical mechanics. But the recent study of history of mechanics has revealed that the contents of the laws of motion which Isaac Newton articulated in Principia are different with those which we recognize as them. In this study, I examine historically the developments of the laws of motion and the evaluation of Newton's work from Newton's Principia to Mach's Die Mechanik. In Principia, Newton proposed his second law of motion in a less evident form. When he sought for the pursuits of the body attracted by a central force, he didn't solve the problem analytically by integrating differential equations, but he did it geometrically by using ‘the ultimate ratio' and the properties of conic sections. After him, the mathematicians like Leibniz, Varignon, Hermann, and the Bernoullis tried to formulate analytically the equation of motion. Colin Maclaurin succeeded in it by deriving differential equations on the orthogonal coordinate system. Leonhard Euler formulated them in a more general way and transformed them onto the polar one. Thus we can think that the equation of motion F = ma has been formulated at first half of the eighteenth century. Euler further insisted that his formula of the second law was the fundamental and general principle of the entire mechanics, which can apply to all kinds of mechanical systems like those of mass points, rigid bodies, elastic bodies, and fluid bodies. For him it includes all principles of mechanics and the first law i. e. the law of inertia is its particular case when F =0. But this assertion was objected by Jean le Rond d'Alembert who didn't recognize the second law as a principle of mechanics, but as the definition of motive force. As the principles of mechanics, he alternatively proposed the conservation of force of inertia, the composition of motion, and the equilibrium of bodies having same momentum. During the eighteenth century, the mathematicians neither recognized the significance of the second law as the basis of mechanical theory, nor attributed the discovery of the second law of motion to Newton. D'Alembert wrote that we owed the theory of acceleration to Galilei, the law of central force to Huygens, and only the expansion of this law to other curves and the world system to Newton. Joseph Louis Lagrange stated likewise that Newton's contribution was only the extension of the measure theory of accelerating force which Galilei had founded in treating the projectile and which Huygens had developed as the theory of centrifugal force. In Principia, Newton himself stated that the first two laws of motion - the law of inertia and the equation of motion - had been found by Galilei.

In the nineteenth century, two great philosophers of science, William Whewell and Ernst Mach, treated Newton's laws of motion in their histories of mechanics. In History of the Inductive Sciences, Whewell reconstructed the historical processes of scientific discoveries by three stages of ‘Epoch, ' ‘Prelude, ' and ‘Sequel.' For history of mechanics, he attributed its ‘Epoch' to Galilei, not to Newton. Galilei firstly discovered the laws of motion for simple cases, which Huygens, Newton, and the eighteenth century mathematicians generalized. Whewell proposed his own laws of motion. The first law is that of inertia, but the other two are different from Newton's and similar to d'Alembert's. The second law is the composition of motion which Galilei established in treating the projectile motion. The third law is ‘the concept of momentum' which relates statistic force to dynamic one by the mass of a body. How did Whewell think of Newton's second law and third law? In The Philosophy of the Inductive Sciences, he likewise explained his own laws as ‘the principles of dynamics, ' while he reduced Newton's laws of motion to ‘the axioms which relate to the idea of cause.' The law of inertia is an example of the first axiom: Nothing can take place without a cause. The equation of motion is that of the second axiom: Effects are proportional to their causes, and causes are measured by their effects. And the law of action = reaction is that of the third axiom: Reaction is equal and opposite to action. When Whewell treated the relation between external force and change of motion, he didn't take into consideration the mass of a body, but he did only in treating the third law. During the eighteenth century, this disregard of mass in arguing the relation between force and acceleration has been general. D'Alembert defined the acceleration force j = du/dt (u is velocity), from which Lagrange developed his dynamic theory. Thus Whewell's views of the laws of motion followed those of the eighteenth century mathematicians who mostly neither considered Newton's laws as the real principles of mechanics nor regarded him as the founder of mechanical theory. C. Truesdell, the pioneer historian of analytic mechanics, claimed that Ernst Mach had firstly conceived of Newton as the father of classical mechanics. In Mechanik in ihrer Entwickelung historisch-kritisch dargestellt, Mach insisted that Newton had completed the principles of mechanics and that after him no new principle had been proposed. He indicated four achievements of Newton: generalization of the concept of force, proposal of the concept of mass, precise and general formulation of the law of parallelogram of force, and proposal of the principle of equality between action and reaction. Mach reduced the first two laws to the definition of force, while he emphasized the significance of the third law, by which he defined the obscure physical quantities like mass and force. He recognized that Newton had established the fundamental concepts of mechanics, but he denied the laws of motion as the principles of mechanics. It is no doubt for us that Newton's second law defines the relation between force and acceleration, which the eighteenth century's mathematicians didn't admit. Since force was the obscure concept indicating pressure, gravity, impulse, energy and so on, they regarded the second law as a definition of force. To recognize the second law as the basis of mechanics, the concept of force had to be defined more precisely. It is only at the second half of the nineteenth century that Newton's laws of motion were understood as the basis of mechanics and that Newton was recognized as the founder of mechanics.