There is a wide-spread misconception about Newton’s laws of motion. Many people believe that all mechanics can be deduced logically from Newton’s laws in a manner similar to how geometry can be deduced from Euclid’s axioms. This is incorrect. By no means Newton’s laws constitute a self-sufficient set of logical axioms. It was not until Leonhard Euler, when the laws of mechanics, including rotational motion, were formulated.
Here is what prof. Walter Noll wrote in his paper “On the concept of force” (April 2007):
“Engineering students often take a course called “Statics”, which deals with forces in systems having no moving parts at all, and hence accelerations are completely absent. The beginning of a textbook on statics often contains a statement of Newton’s laws, but this functions like a prayer before a business meeting; it is almost totally irrelevant to the substance of the subject. The substance of statics consists in singling out parts of the system under consideration by drawing “free-body diagrams”.
For suﬃciently many of such parts, one writes down two equations: The ﬁrst states that the sum of all the forces acting on the part is zero, and the second that all the torques acting on the part is zero. In this way, one obtains suﬃciently many linear equations to determine the force acting on each structural member of the system. This information is then used to decide whether the system may or may not collapse.
Engineering students often also take a course called “Dynamics”. Its basic structure diﬀers from the course in statics only by including the inertial forces among the forces considered. (I have taught courses on Statics and Dynamics in the late 1950s, and this experience has inﬂuenced my analysis of the foundations of mechanics.) The two basic principles of classical mechanics are these: 1) Balance of forces: The total force acting on a physical system and each of its parts is zero. 2) Balance of torques: The total torque acting on a physical system and each of its parts is zero.”
Isaac Newton (1642-1727) Leonhard Euler (1707-1783)
Newton’s Second law.
The above-mentioned paper by Noll is a part of an argument with Frank Wilczek about the concept of force and about the second Law of Newton, F=ma. This discussion represents the difference in the approach of a mechanician and that of a physicist. This is what Wilczek said:
“Newton’s second law of motion, F = ma, is the soul of classical mechanics. Like other souls, it is insubstantial. The right−hand side is the product of two terms with profound meanings. Acceleration is a purely kinematical concept, defined in terms of space and time. Mass quite directly reflects basic measurable properties of bodies (weights, recoil velocities). The left−hand side, on the other hand, has no independent meaning…
The paradox deepens when we consider force from the perspective of modern physics. In fact, the concept of force is conspicuously absent from our most advanced formulations of the basic laws. It doesn’t appear in Schrödinger’s equation, or in any reasonable formulation of quantum field theory, or in the foundations of general relativity. Astute observers commented on this trend to eliminate force even before the emergence of relativity and quantum mechanics.” ( F. Wilczek, Whence the Force of F = ma? Physics Today, 2004)
Indeed, there were several attempts to formulate mechanics without the concept of force. The best known are by Jean-Baptiste d’Alembert and by Heinrich Hertz.
The key issue of the argument is how we see the law F = ma. Whether this is a falsifiable statement in the Popper sense (i.e., a statement that could, in principle, be refuted one day by new experimental data), or it is just a definition of the concept of force, which is always true by definition.
There are other laws of classical physics, which tell us what a force should be in any particular situation. For example, the Hooke’s law of elasticity states that, in an elastic situation, the force is proportional to the extension, F = kx. The law of gravity states that the gravity force is dependent upon interacting masses and a distance between them, F = GMm/r2.
Therefore, one way to define things is to say that a real (falsifiable) law of nature would be, say, in the case of gravity, GMm/r2 = ma, while in the case of the elasticity kx = ma. With this approach we do not need the concept of force at all!
A different approach would be to state that the force is defined essentially by the laws like the Hooke’s law or the gravity law, whereas the 2nd Newton’s law constitutes a falsifiable statement.
To some extent, as you can see also from the above argument of Noll and Wilczek, this difference of approaches represents a difference between the way of thinking of a physicist and a mechanician about fundamental issues.
Newton’s First law.
The issue is more striking with Newton’s First law or the law of inertia. Essentially it states that there is no acceleration when there is no external force. Isn’t this just a consequence of the Second law in a very special case when F=0? Yes it is, but the meaning of the first law is that it defines an inertial frame of reference and, furthermore, states that such frames do exist in nature.
Unlike velocities, the acceleration is absolute in Newtonian mechanics, and it is defined relative to the certain absolute frames of reference called the inertial frames. Their existence is postulated by Newton’s First law. In physics, there is a so-called Mach’s principle, which states that local inertial frames are determined by the large scale distribution of matter, such as remote stars and galaxies. Mach’s principle is an imprecise hypothesis, and it is more metaphysical than physical. However, there is no better physical explanation of why the inertial frames exist.
Every physical quantity should be defined through some measurable properties. And the latter are typically related to the human ability to observe and measure them using human-built equipment. Therefore, no matter what length scale is discussed by a physical theory, from the microscale to the galaxies, it should be related to properties measurable by a human on a human length scale. In most cases this means that physical properties are eventually reduced to mechanical properties observable by a human with appropriate instruments.
There are interesting attempts to relate the inertial frames to the human nature. The same Walter Noll introduces a Chomskyan approach towards the frames of reference in his essay “On the Illusion of Physical Space“:
“Thus, it seems that the predisposition to ﬁxate on a particular frame of reference at any given situation is hardwired into our brain at birth, just as is the ability to acquire language. Which particular frame we ﬁxate on (or which particular language we learn) depends on the environment. Usually, it is the background that determines this ﬁxation. When we talk about motion we mean motion relative to the ﬁxated frame, without being consciously aware that we do so. Our brain chooses the ﬁxation of the frame of reference in such a way that it facilitates our ability to understand our environment with as little mental computation as possible. Occasionally, we may ﬁxate on a frame that is less than appropriate.“
This comparison between the language acquisition and the frames of references may be a bit far-fetched in extending the already limited “Machist” philosophical view of mechanics. Ernst Mach believed that every aspect of a physical theory at its every step should relate to properties observable by humans. This was productive in many situations, for example, for the development of Schrodinger’s theory of colors (which is inherently Machist). However, it also prevented Mach from recognizing the existence of atoms.
On the other hand, the attempts to answer metaphysical questions such as “why do inertial frames exist?” may lead to new insights. Another famous metaphysical question is “what is time?” In mechanics, it is reduced to the question of how time is measured or “what is a clock?” Two clocks should not be identical to each other, so that they may be placed in different parts of space; however, at the same time, they should be similar enough so that we are sure that their time-behaviors (e.g, oscillations of the pendula) are the same, so that we can assume they are synchronous.
Why such object as clocks, which are different enough to place them at different point but similar enough to show the same time, can exist? As physicist Lee Smolin pointed out in his recent books, when dealing with time, we do not deal with the past, but with the records of past events. In my view, this relates the metaphysical problem of time to some modern concepts of computer science, such as the “P!=NP” problem. The latter is about whether it is easier to deduce new knowledge from scratch or to learn it from another person, which are two fundamental human ways of dealing with knowledge. However, I will leave this topic for some future blog entry.
Newton’s Third law.
The Third law states that the action is equal to the reaction. Again, it is hard to accept it as a falsifiable law of nature. Can you imagine a situation when point A acts upon point B with a force of Fab, while point B acts upon point A with a different force of Fba? This would result in a spontaneous motion of their common center of masses and perhaps would yield a number of logical paradoxes, preventing us from dividing objects into parts and building free force diagrams.
My general point is that mechanics deserves recognition as a fundamental science since it studies fundamental concepts, which are sometimes different from similar concepts in other parts of physics. I will continue with examples in one or two other blog posts.
Other blog entries in this series:
1. A point-mass and a rigid body
2. Are Newton’s laws laws of nature?
3. Does Statics logically precede Dynamics?
4. Can rotational dynamics be deduced from Newton’s laws?