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The Axiom and Laws of Motion The Axiom and Laws of Motion Efthimios Harokopos Abstract The law of inertia and the law of interaction are derived from the axiom of motion, an expression relating the time rate of change of the kinetic energy of a particle to its velocity and time rate of change of momentum. These laws of motion are shown to (i) treat uniform circular orbits as effects of inertia, (ii) encompass Newton’s Laws of Motion and (iii) have a special link to Leibniz’s Laws of Motion. I also discuss some metaphysical issues arising from the axiom and laws of motion regarding the ontology of space-time. 1. Introduction The publication of Newton’s Principia in 1686 marked the beginning of a new era in science. The axioms, or laws, of motion, the law of universal gravitation and the mathematical methods developed by Newton based on the work of Descartes, Galileo and many others established the foundation of classical mechanics. The accomplishments of Newton impacted science and technology and, as a result, many aspects of human life. There were those, however, who criticized Newton for his use of the concept of force in an effort to ground his physics on his metaphysics and there is still considerable interest EXT-2004-018 15/11/2003 regarding both the physical and metaphysical interpretation of Principia. In Science and Hypothesis, Poincaré writes: When are two forces equal? We are told that it is when they give the same acceleration to the same mass, or when acting in opposite directions they are in equilibrium. This definition is a sham. (Poincaré 1952, 98) In Principles of Dynamics, a renowned graduate text, Donald T. Greenwood offers an illuminating account of the issues raised by Newton’s concept of force: The concept of force as a fundamental quantity in the study of mechanics has been criticized by various scientists and philosophers of science from shortly after Newton’s enunciation of the laws of motion until the present time. Briefly, the idea of a force, and a field force in particular, was considered to be an intellectual construction, which has no real existence. It is merely another name for the product of mass and acceleration, which occurs in the mathematics of solving a problem. Furthermore, the idea of force as a cause of motion should be discarded since the assumed cause and effect relationship cannot be proved. (Greenwood 1965, 24, italics added) 1 The philosophical issues surrounding Newton's use of force and absolute space-time are well known to the philosophers of science and will be further discussed in sections 4, 5 and 6. The question I pose for now is the following: is it possible that Newton's Laws are just a (metaphysical) interpretation of a more general principle of motion? Or put in another form, is it possible that there exists a system of laws from which Newton's Laws can be derived as a special case? I will demonstrate in this paper that there is such a system of laws of motion from which Newton's but also Leibniz's Laws of Motion can be derived, and these laws of motion are deduced from single principle, which I call the axiom of motion. I will also show in section 4, where I compare this alternative system of laws to Newton's, that the existence of this more general principle of motion is even acknowledged by Newton, in his own writings. Furthermore, in section 5, the relation of the axiom and laws of motion to Leibniz's Laws of Motion is discussed and in section 6 a space-time structure is proposed, which reconciles the differences of the substantival and relational views of the world and serves as a metaphysical ground. 2. The Axiom of Motion I begin the derivation of the laws of motion by stating the axiom of motion, an expression relating the velocity and the time rate of change of momentum of a particle, to a scalar quantity called the time rate of change of kinetic energy. Axiom of Motion: The time rate of change of the kinetic energy of a particle is the scalar product of its velocity and time rate of change of momentum. Denoting the kinetic energy by Ek and the momentum by P, the axiom of motion is expressed as follows: dE dP dr k = ⋅ (1) dt dt dt where r is the position vector of the particle. The momentum P is defined by the equation dr P = m (2) dt If the mass m of the particle is independent of time t and position r, then by combining equations (1) and (2), the time rate of change of the kinetic energy Ek can be written as follows: dE d 2r dr k = m ⋅ (3) dt dt 2 dt Corollary I: The kinetic energy of a particle with a constant mass m is given by 2 1 E = mv ⋅ v k 2 where the velocity v is dr v = dt Proof: From equation (3) we obtain: dE d 2r dr dr d dr d 1 dr dr k = m ⋅ = m ⋅ = m ⋅ dt dt 2 dt dt dt dt dt 2 dt dt or 1 dr dr 1 E = m ⋅ = mv ⋅ v k 2 dt dt 2 The axiom of motion is the only principle required for deriving the laws of motion, as it will be shown in the next section. 3. The Laws of Motion Law of Inertia: If the time rate of change of the kinetic energy of a particle is zero then the particle will maintain its state of rectilinear or uniform circular motion. Proof: If the time rate of change of the kinetic energy of a particle is zero, then from equation (3) we obtain d 2r dr m ⋅ = 0 (4) dt 2 dt Since the mass of the particle in equation 4 is always a non-zero quantity, there are two cases, A and B: Case A. The velocity and acceleration vectors are not orthogonal. In this case, equation 4 is satisfied if, and only if: 2 d d r r (5) = 0 = v0 dt 2 dt or ⇔ or dr dr = 0 = 0 (6) dt dt 3 where v0 is a constant. Therefore, in this case, the particle either moves with a constant velocity v0, given by equation (5), or it is at rest, given by equation (6). Case B. The velocity and acceleration vectors are orthogonal. From equation (4) we observe that the scalar product: d 2r dr ⋅ dt 2 dt is equal to zero for all t when the velocity and acceleration vectors remain orthogonal for all t and both vectors are not null. This is the case when a particle is on a uniform circular motion. Therefore, a particle of mass m will maintain its state of uniform circular motion as long as the time rate of change of its kinetic energy remains zero, which is mathematically equivalent to the acceleration and velocity vectors remaining orthogonal for all t. The law of inertia is a statement about the tendency of particles to maintain their state of motion, linear or uniform circular, when the time rate of change in their kinetic energy is zero and this tendency is called inertia. The philosophical implications from the inclusion of uniform circular motion in the class of effects due to inertia will be discussed in section 6. Corollary II: If the time rate of change of the kinetic energy of a particle is zero and the particle moves with a uniform linear velocity, its momentum is conserved. Proof: This is a direct consequence of the law of inertia, and specifically case A. If the time rate of change of the kinetic energy is zero and the velocity is denoted by v, then from equation (1) we obtain dP (7) ⋅ v = 0 dt Now using equation (2) and since m is not zero and v is not the null vector by definition, we obtain from equation (7) the result: d(mv) = 0 ⇒ (mv) − (mv) = 0 dt 2 1 or (mv) 2 = (mv)1 = mv = constant (8) 4 Equation (8) (see for example Meirovitch 1970, 12) is the mathematical statement of the theorem of the conservation of linear momentum. Law of Interaction: To every action there is an equal and opposite reaction; that is, in an isolated system of two particles acting upon each other, the mutual time rates of change of kinetic energy are equal in magnitude and opposite in sign. Proof: We denote the two interacting particles as mass m1 and m2, respectively. Furthermore, we denote m1 as the agent causing the action in the system. The total kinetic energy of the interacting system of particles is the sum of the kinetic energies of the two particles: E E E (9) k = k1 + k2 From equations (1) and (9) we obtain dE dP dP k = 1 ⋅ v + 2 ⋅ v (10) dt dt 1 dt 2 where v1 and v2 are the velocities of the two particles with momentum P1 and P2, respectively. Now, we consider the mutual changes in the time rate of change of kinetic energy imposed by the particles upon each other. The time rate of change of kinetic energy of particle m2, denoted as Ek2, is equal to the action imposed on it by particle m1, denoted as Ek12 and given by The time rate of change of the kinetic energy of particle m1 is equal to the sum of the time rate of change of the kinetic energy of the system due to its action as an agent and dE dP dEk k 2 = 2 ⋅ v = 12 (11) dt dt 2 dt that imposed on it by particle m2 in the form of a reaction, denoted as Ek21 dEk dE dEk dP 1 = k + 21 = 1 ⋅ v (12) dt dt dt dt 1 After combining equations (10), (11) and (12), we obtain the result: dE dE k k12 = − 21 (13) dt dt 5 Equation (13) is the mathematical statement of the law of interaction.
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