Some people state that mechanics is not a fundamental science, but rather an applied discipline. I belong to a school of mechanicians, who would strongly disagree with that assertion. For me, mechanics is a part of physics; however, it has a number of its own concepts, which do not always coincide with similar concepts in physics. In this series of blog posts, I intend to bring several examples of fundamental concepts of mechanics, which tend to be less-than-obvious when it comes to their logical definition.
The first one is the concept of a point massor particle. In Russian literature (including English translations of Russian books, such as “Mechanics” by Landau & Lifshits) the term material point is often used. What is a definition of a particle (point mass or mass-point)? Most textbooks, especially those written by physicists, would say that a particle in mechanics is a rigid body, whose size can be ignored but mass cannot be ignored. This sounds like a fair definition.
However, then you need to define what is a rigid body! A non-deformable rigid body is a system of points, the distance between which always remains constant. One defines the concept of a mechanical particle using the concept of a rigid body, while rigid body is defined using the concept of a particle. This sounds like the old “chicken-or-egg” problem, and does not seem a serious practical issue that one should be concerned about. Perhaps this is not even a significant theoretical problem, about which a mathematically-minded mechanician should worry while looking for rigorous logical definitions of mechanical concepts? However, it is!
The reason is that when you keep decreasing the size of a 3D rigid body, not all of its properties irrelevant for a mass-point vanish in the limit of the small size. Take the moment of inertia of a disk rolling on a flat surface with the linear velocity of V. For a disk of radius R, the moment of inertia is J=(3/2)*M*R2, the angular velocity is V/R, and the kinetic energy is K=(J/2)*(V/R)2. In other words, the kinetic energy is K=(3/4)M*V2. It does not depend on the size of the disk! In the limit of R→0 you still get K=(3/4)M*V2. This is not what you expect for the mass-point, K=(1/2)M*V2. Isn’t this amazing and counter-intuitive?
A mass-point has only three translational degrees of freedom, while a rigid body has six degrees of freedom: three translational and three rotational. When the limit of a very small rigid body is considered, R→0, there is no reason that rotational degrees of freedom go away. This results in a certain ambiguity of the definition of a point-mass: does it have rotational degrees of freedom or not?
In a more formal way using Felix Klein’s Erlangen program approach saying that a rigid body is characterized by the group of symmetry SO(3)xR3 (i.e., three rotational and three translational degrees of freedom). A point mass has only three translational degrees of freedom, R3. Thus the definition of the point mass is
SO(3)xR3 → R3.
The definition of the rigid body is that the distances between all points remain the same. The orientation of a rigid body is given by three points (nine degrees of freedom) upon which three constraints (pairwise constant distance) are applied. Each constraint takes away one degree of freedom yielding the mobility of the rigid body 9 – 3 = 6. Therefore, the definition of the rigid body is given by the transformation
R3xR3xR3 → SO(3)xR3.
The ambiguity is caused, most likely, by the fact that our Euclidean space is R3, but it possesses both translational and rotational SO(3)xR3 symmetries. If we accept Leibniz’s view that the space is just a property of bodies’ mutual relationships, we realize that the rigid body has the same dual (R3 and SO(3)xR3) nature.
From this example we find that rigorous definitions of basic mechanical concepts, such as the point mass (particle, material point) and the rigid body, are related to fundamental concepts of motion and symmetry.
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?