The law of rotational dynamics states that the moment of inertia times the angular acceleration equals the applied torque. This is analogous to the 2nd Newton’s law. But can the former be logically deduced from the latter?

The rotational law of dynamics is sometimes called the Euler equations or second Euler’s law, and it has the form of dM/dt=T, where M is the angular momentum and T is the torque / moment of force (both M and T are vectors). First Euler’s law is Newton’s Second law, dP/dt=F, where P is the momentum and F is the force vectors.

Physics textbooks usually assume that the Euler equation is deduced from Newton’s Second law given the angular momentum is conserved. The standard derivation is by stating that pairwise internal forces in a rigid body, i.e., force vectors between points A and B of the rigid body, namely F_ab and F_ba are opposite (by Newton’s Third law) and central (which means that they act along the straight line |AB|). Then their combined moment of force is zero: R_A х F_ab = – R_B х F_ba. Using that along with the conservation of angular momentum and applying some vector algebra, the physics textbooks derive the Euler equation, dM/dt=T.

The use of the conservation of angular momentum is not problematic here, because it is a fundamental consequence of the isotropy of the physical space, on the basis of Noether’s theorem. One can raise questions about the constrained motion of points A and B in a rigid body. The definition of the rigid body is that the distance between any two points (for example, points A and B) remains constant. This means a constraint upon the motion of these points. The constraints are equivalent to “virtual forces.” How this virtual forces affect the balance of the angular momentum, may be a matter of a separate analysis.

However, more important is the assumption that the forces act along the straight line. Physicist Frank Wilczek noted:

“When most textbooks come to discuss angular momentum, they introduce a fourth law, that forces between bodies are directed along the line that connects them. It is introduced in order to “prove” the conservation of angular momentum. But this fourth law isn’t true at all for molecular forces.”

Here is what mechanician Clifford Truesdell wrote on this matter:

“It is clear enough that in statics the equilibrium of moments is not assured by the equilibrium of forces, nor vice versa. In dynamics, the principle of moment of momentum developed late, and much of the earlier work concerning it gives the impression that the two principles were somehow hoped to be equivalent, so that there would be but a single law of motion. This illusion is fostered in the teaching of mechanics by physicists today.

… Euler and Daniel Bernoulli in their early work on rigid, linked, or flexible systems frequently invoked the principle of moment of momentum, often disguised by an additional quasi-static assumption. Its importance began to be seen gradually, so that only with difficulty can a particular date be fixed as that in which it rose to the level of a principle of mechanics. I believe the correct date is 1751, the year of publication of EULER’S paper of 1744 which has been mentioned above as the first in which the “Newtonian equations” are recognized as sufficient to give all the mechanical principles determining the motion of a complex system, the case in point being the taut loaded string. The latter part of this same paper obtains the equations of motion for a system of n rigid rods linked together and subject to arbitrary forces at the junctions. Here the “Newtonian equations” do not suffice. In addition, Euler sets up as an independent principle the balance of moment of momentum about the center of mass of each rod.

… These laws, which may be called Euler’s laws of mechanics, imply not only “Newton’s laws” for mass-points but also all the other principles of classical mechanics and are just as convenient for continuous bodies as for discrete systems. The first law is equivalent to Euler’s “first principles” of 1750 and yields the general theorem on the center of mass. The second law follows from it in some cases; for example, if al1 forces are absolutely continuous functions of mass and if all torques are the moments of forces; but in more general systems, such as those in which shear stress is present, the second law is independent of the first.

For a rigid body, Euler was able to derive his former equations of motion directly and easily from the second law. The law of moment of momentum is subtle, often misunderstood even today.

In presentations of mechanics for physicists it is usually derived as a consequence of “Newton’s laws” for elements of mass which are supposed to attract each other with mutual forces which are central and Pairwise equilibrated. Although the formal steps of this derivation are correct, the result is too special for continuum mechanics as well as methodologically wrong for rigid mechanics:

1. Any forces between the particles of a rigid body never manifest themselves, by definition, in any motion. The condition of rigidity applied to Euler’s second law suffices to determine the motion. To hypothecate mutual forces is to luxuriate in superfluous causes, which are to be excised by Occam’s razor.
2. To introduce mutual forces in a rigid body drags ill action at a distance in a case when it is unnecessary to do so.
3. In continuum mechanics the total force acting upon a finite body arises principally from the stress tensor, which represents the contiguous action of material on material. There is no physical reason to assume that forces arise only from action at a distance, and no purpose served by doing so. If rigid-body mechanics is viewed as a special case of continuum mechanics, the stress within the rigid body is indeterminate and need never be mentioned, but a uniform process based upon Euler’s second law remains possible.

No work of Euler, nor of any other savant of the eighteenth century, approaches rigid bodies by hypothecating forces acting at a distance as in modern books. Euler’s method of discovery tacitly assumes there to be no internal forces at all; the right answer comes out, but we are justified in doubting that the argument be general enough. His final treatment does not make any presumption or restriction regarding the presence or nature of internal forces.”

In other words, there are three different cases in mechanics:

1. Continuum mechanics, where the equilibrium of the shear stresses cannot be deduced from the equilibrium of the normal stresses.
2. Statics of the rigid bodies, where the Archimedes law of the lever cannot be deduced from Newton’s Third law, despite the attempts to deduce it using symmetry considerations.
3. Rigid body dynamics, where Euler’s law can be deduced from the Newton’s Second law assuming that there are internal forces in the rigid body and that they are central.

The question of whether one views a rigid body or a point-mass as a fundamental object of mechanics (this is the “Occam’s razor” argument by Truesdell above) is related to the issues which I have discussed in one of my previous blog entries.

The purpose of this series of blog posts, which I called “Why mechanics is a fundamental science” is to show that there are many fundamental logical problems within mechanics and, furthermore that the approach of mechanics may be different from that of physics. Below is a definition by one of my teachers, Prof. Pavel Zhilin (my translation from Russian):

“Mechanics is not a theory of any natural phenomenon but a method of the study of Nature. In the foundation of mechanics, there are no laws which could be falsified experimentally, in principle. Instead, there are logical concepts expressing a balance of certain parameters, which by themselves are insufficient to build self-contained theories and should be complemented by empirical observations.”

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?