Chapter 01. Classical Mechanics
As pointed out before, every physical law contains space-time relationships in terms of length and time.
Length
Length is measured by means of measuring rods. A fundamental property of measuring rods is that if any two rods have once been found to coincide when superimposed they will remain equal forever and always, that is to say, they will always coincide whenever they are superimposed. Only rigid bodies can be used as measuring rods.
Time interval
Time intervals are measured by clocks; any system in which a repeatable process takes place can be used for this. A fundamental concept of classical mechanics concerning the dimensions of bodies and intervals of time is their absolute nature; a measuring rod, if it is a good one, is always of the same length regardless of whether it is moving with respect to an observer or not; two clocks once set by one another will show the same time, if they have the same rhythm, irrespective of how they may be moving.
Space and time as physical entities
Space and time are physical entities like any other, but their importance by far surpasses that of any other physical entity. To study the properties of space and time one must observe the motion of bodies in them; investigation of the manner in which this motion proceeds offers a clue to the properties of space and time.
Three-dimensional space
The statement of one dimension in space is not enough to locate a point uniquely. From experience we know that three dimensions must be stated to define the position of a point in space relative to other points. The numerical values of these dimensions are called the coordinates of the point. The fundamental property of three-dimensional space is that three coordinates are sufficient to locate precisely a point relative to some rigid body, the frame of reference.
Space homogeneity and isotropy
In further investigating the motions of bodies one finds that the properties of space are identical at different points, and that at each point they are the same in all directions. In other words, space is both homogeneous and isotropic. These properties of space find expression in the laws of conservation of linear momentum (or simply momentum) and angular momentum (or moment of momentum).
Time homogeneity
The same experiments also reveal that different moments of time are equivalent, as far as their physical properties are concerned, i.e., time is homogeneous. An expression of this is the law of conservation of energy.
The whole progress of physics has served to confirm that the energy and momentum conservation laws are fundamental laws of nature. This is seen in the fact that the vast number of known physical laws can be developed from a very few general relationships, which include the conservation laws.
Euclidean space
Another property of space is its flatness. This is known as Euclidean space, because it conforms to Euclidean geometry.(1)
The classical notions of space and time mentioned here were established experimentally and are valid in a vast domain of physical phenomena.
The particle
We shall now examine the properties of motion of the simplest physical entity, a material point or particle. A material point is a body whose dimensions can be neglected in describing its motion. The location of a material point, as mentioned before, is determined by three spatial dimensions, its coordinates.
Frame of reference. Coordinates of a particle
Three coordinates uniquely define the position of a point with respect to a body of reference. If a Cartesian coordinate system is attached to this reference body, the location of the particle in space can be characterized by its radius vector, the components of which parallel to the axes of the coordinate system are equal to the Cartesian coordinates x, y, z of the point.(2)
Classical Motion
According to classical notions of the spatio-temporal properties of motion, a material point occupies a definite place in space at every instant and it possesses definite coordinates. When a material point moves its coordinates change. Accordingly, the radius vector r of a material point can be considered as a function of time: 
Velocity of a particle
The velocity of a material point is  i.e., the derivative of the radius vector with respect to time.
Acceleration
Acceleration is defined as 
Degrees of freedom
Mechanics studies the motion of systems of material points. The number of independent coordinates required to describe the motion of a mechanical system is called the number of degrees of freedom of that system. A material point, evidently, has three degrees of freedom, and a system consisting of n material points has 3n degrees of freedom.
State of a classical particle
If the state of a mechanical system at any given instant is known, its state at any other moment of time can be determined. The state of a system, it is found, is completely defined by stipulating the coordinates and velocities. Acceleration cannot be stated arbitrarily as it is a function of the position coordinates and velocity.
The formulas that correlate acceleration, coordinates and velocity are called the equations of motion of a system.
Classical mechanics
The fundamental problem of classical mechanics (that is, mechanics based on the classical notions of space, time and motion) is the study of the motion of any mechanical system by determining its position coordinates as functions of time from stated initial conditions, i.e., from the coordinates and velocities at some initial time.
Inertial frame of reference
As mentioned before, physical phenomena cannot be investigated without reference to a coordinate system. One is free to choose any of a countless number of reference systems moving in all manners with respect to one another. The manifestations of the laws of nature, however, may differ in different systems. If an arbitrary reference system is chosen one may find that the laws governing the simplest phenomena are involved indeed. The problem thus is to find a frame of reference in which all the laws of nature would appear in as simple a form as possible; such a reference system would evidently be best suited for describing physical phenomena.
In seeking such a frame of reference we proceed from the simplest type of motion, that of a material point so distant from all other bodies that any interaction between it and those bodies can be neglected. Such a material point is said to be in free motion.
Free motion
Let us take an arbitrary frame of reference and study its properties with the aid of a free material point. We assume that at the initial moment the point was at rest in that system. Generally speaking, then, the next instant the point will no longer be at rest, having begun to move in some direction. In this sense, we may state that a frame of reference with arbitrary properties is neither homogeneous nor isotropic.
Two free moving bodies, however, may remain at rest relative to each other indefinitely. Therefore motion can be referred to a coordinate system rigidly attached to some free moving bodies. Such a frame is called an inertial system. In an inertial system all directions are physically equivalent and the properties of different points in space are identical, i.e., it is mechanically homogeneous and isotropic.
The inertia law
These properties of an inertial reference system lead to the conclusion that in it the free motion of a material particle takes place at uniform velocity. This statement is known as the inertia law (Newton’s first law). In particular, as assumed before, if at some initial time the velocity of a material point is zero, it remains at rest indefinitely.
The relativity principle
If we consider a reference frame moving with respect to an inertial system in some arbitrary manner, the former will not, in the most general case, be an inertial system. It does not follow from this, however, that there is only one inertial frame of reference in the world. It will be readily observed that there exist an infinite number of such systems, all of them being in uniform rectilinear motion with respect to each other.
The classical relativity principle
To inertial frames of reference is applicable the relativity principle, which states that their physical properties are equivalent. Coupled with the concept of absolute time, this is known as Galileo’s relativity principle. From the equivalence of all inertial frames of reference it follows that the equations of motion of any mechanical system do not change in passing from one inertial system to another.
The Galilean transformations
Let r be a radius vector defining the position of a material point relative to an inertial reference systemO at some moment of time t. Let the radius vector and time of the event be respectively r’ and t’ in another inertial reference system O’ moving with a velocity  relative to the unprimed system.
According to the classical notions of space and time, the formulas for going over from one set of coordinates and time measurement to another have the form
 (1)
The first equation expresses the absolute quality of spatial dimensions, the second, the absolute quality of time. These relations are known as the Galilean transformations.
Composition of velocities
Differentiating (1) with respect to time, we obtain
 (2)
This simple formula defines the principle of composition of velocities: velocity v relative to the unprimed reference system is compounded of velocity v’ relative to the primed system and the velocity  of system O’ relative to system O.
Acceleration
Differentiating (2) with respect to time, and taking into account that  is constant, we obtain
i.e., the acceleration is the same in all inertial frames of reference.
Relative and absolute
Insofar as there exists no distinguished “absolute” reference system, the concept of absolute rest is devoid of meaning. If a body is at rest in one inertial system, it is moving with some uniform velocity relative to other systems, and there is no reason why one should be given preference before another. Similarly, the notion of absolute velocity is also meaningless; only the relative velocity of bodies with respect to one another has physical meaning. Absolute acceleration, on the other hand, is physically meaningful since, as we have found out, acceleration is the same in different inertial systems, and the difference between inertial and non-inertial systems is of an absolute nature. In future, unless otherwise specified, we shall always be referring to inertial systems.
The principle of action at a distance
It follows from Galileo’s relativity principle that interactions between bodies propagate through space instantaneously, i.e., if the state of one body is, altered, it will immediately affect all other bodies interacting with it, no matter how far away they are. In fact, if interaction propagated with a finite velocity then, from the rule of velocity composition (2), the velocity would be different in different frames of reference and it would be physically possible to distinguish between them, which contradicts the relativity principle. As the velocity of propagation of interactions is assumed to be infinite, classical (Newtonian) mechanics is said to be based on the principle of action at a distance.
Systems of particles
Up till now we have been speaking of the properties of motion of a single free moving material point. Now consider a system of material points, assuming it to be so far away from any other bodies that its interactions with them can be neglected (they do not have to be taken into account in investigating the motion of the system). Such systems are termed closed.
Conservation laws
In investigating the changes that take place with free moving bodies after they have interacted for some time, we find that, irrespective of the nature of the interaction, certain conservation laws hold. In other words, there are certain quantities characterizing the state of bodies, the totality of which over all the interacting bodies is not affected by the interaction.
Conservation of momentum
Consider a closed system of particles. Since, by virtue of spatial homogeneity, all configurations occupied by it as a whole in space are equivalent, we may assert that its properties will not change if it is moved parallel to itself to any distance.
A consequence of this circumstance is the conservation of a certain vector quantity characterizing the system. Namely, there exists a vector characterizing every material point, such that the sum of all the vectors over all the particles of a closed system is independent of time. It is called the vector of linear momentum. Momentum is related to velocity by a direct proportionality. The coefficient of proportionality, which is different for different material particles, is called mass.(3) The law of conservation of the momentum of a system is thus expressed by the formula
 , (3)
where  is the number of particles.
Mass of a particle
The law of conservation of momentum suggests a mode for correlating the masses of material points. Thus, for two colliding particles we can rewrite (3) as:
where  and  represent the change in the velocities of the respective particles. Hence,
Knowing  and  , it is possible to correlate the masses of both particles. Obviously the basic mass cannot be measured and it must be taken for unity. It can be chosen arbitrarily: choice carries no deep physical meaning.
Newton’s first law
The law of conservation of momentum in a closed system can be regarded as a generalization of the inertia law (Newton’s first law). In fact, for a free moving particle the momentum  remains constant. In the case of a system of material points interacting in any way the momentum of each particle is not constant, but the sum of the momenta of all the particles is conserved, and
Evidently, the quantity
 , (4)
Force
which expresses the change in the momentum of a particle in unit time, is a measure of the external action on that particle. This quantity is the force which the particles of a system exert on the particle .
Potential function
The interaction of material points is described in classical mechanics in terms of the potential energy of interaction
 ,
which is a function of the coordinates of the interacting particles. The type of potential function is determined in each case by the nature of the interactions. With distances between material points increasing to infinity the potential energy vanishes to zero.
It is apparent that this method of describing interactions presumes their instantaneous propagation, i.e., it is in accord with Galileo’s principle of relativity. Actually, the force(4)
 (5)
exerted on a particle by other particles depends at any instant, in this mode of description, only on the positions of the particles at that instant. A change in the position of one particle affects the other particles at the same instant.
This remark leads us to the conclusion that forces depend only on the mutual position of material points and not on their motion.
Newton’s second law
It follows from equations (4) and (5) that
 (6)
This formula gives the most general equation of motion of a system of material points. The equation of mechanics in the form (6) are called the Newtonian equations (Newton’s second law).
Newton’s third law
Summing the equations of motion (6) over a system of particles and taking into account that, from the law of conservation of momentum, the left-hand side of the equation becomes zero, we find that the sum of all the forces in a closed system is zero:
In the special case of a system consisting of two material points,
This means that the force exerted on the first particle by the second is equal in magnitude and opposite in direction to the force exerted by the second particle on the first. This is known as the law of action and reaction (Newton’s third law).
Some properties of mass
The concept of momentum can be used to formulate the concepts of rest and velocity of a system as a whole. A system of material points is at rest in that frame of reference in which its momentum is zero. The velocity of a system of material points relative to some reference frame O is defined as the velocity of a frame of reference  in which the system is at rest.
Denoting by  and  the velocities of a material point  relative to reference systems O and  , and by v the velocity of system relative to system O, the Galilean transformations yield
 .
Multiplying through by the mass of the corresponding material points and summing over all the points, we obtain
 ,
The system of material points is at rest relative to the reference frame , hence,  and
 ,
As P is the momentum and v the velocity of the system as a whole, the latter relationship expresses the additivity of mass: the mass of a composite body is equal to the sum of the masses of its component parts. In other words, we have the law of conservation of mass.
Here are some other properties of mass. Making use of the relationship between the momentum and velocity of a particle, equation (6) can be written for one particle in the form
 .
According to Galileo’s relativity principle, this equation does not change in passing from one inertial reference system to another. It follows, in particular, that the mass m of a particle is invariant, i.e., independent of the choice of reference system and therefore is a true characteristic of the particle.
Another property of mass to be noted is that it cannot be negative,(5) i.e., the directions of the momentum and velocity of a particle coincide.
Law of conservation of angular momentum
We have seen that the law of conservation of linear momentum follows from the homogeneity of space with respect to a closed system of particles. Space is also isotropic; and any direction is as good as the other. Hence the properties of a closed system should not change if the whole system is turned through an arbitrary angle about an arbitrary axis. This condition implies the conservation of some vector quantity characterizing the system. Namely, every material point of a system is characterized by a vector, such that the sum of all the vectors over all the particles of a given closed system is independent of time. This is the vector of moment of momentum, or angular momentum as it is more commonly called. It is related in a specific way with the linear momentum vector: the angular momentum of a particle is equal to the product of the radius vector and the linear momentum of the particle. Thus the law of conservation of angular momentum takes the form
 .
So far we have been speaking of closed systems only.
Conservation laws for two interacting systems
Let us now consider the motion of a system interacting with another system whose motion is known. In this case the concept of motion in an external field is introduced.
We know already that interactions of material points are described by means of the potential function
 ,
which is a function of the coordinates of the interacting particles. It describes the interactions of the material points of a system and, if the system is not closed, the interactions between the material points of the system and other bodies.
For a non-closed system in given external conditions the potential function may be explicitly time-dependent:
 .
It represents the potential energy in an external field.
And so, motion in external fields or, more precisely, the conservation laws for open systems.
In the most general case, the angular momentum of a system in an external field is not conserved, though it may be in some special cases. Thus, if a system is in a central-symmetrical field (one in which potential energy depends only on the distance from some fixed point, the center) all directions in space from that center are equivalent, and the angular momentum of a system with respect to that center is conserved. With respect to any other point in space, however, it will naturally change.
If a system is not closed but its interactions with surrounding bodies are such that the external conditions do not change in its displacement in some direction l (an axial-symmetrical field), the projection of the linear momentum of the system on that direction is conserved:
 .
Similarly, in an axial-symmetrical field the component of the angular momentum along the axis of symmetry is conserved.
We shall restrict ourselves to the cited cases as they are the most important.
Law of conservation of energy
Let us now consider the very important case of motion of an open system in a uniform external field. If a system is not in a variable external field its properties are not explicitly time-dependent. This follows from the consideration that, in the absence of an external field (or in a uniform external field), all instants of time are equivalent with respect to a given physical system. A consequence of this condition is the conservation of a scalar quantity E characterizing the system,
 , (7)
called energy. The expression
gives the kinetic energy of the system. The function U was mentioned before.
The relationship (7) expresses the law of conservation of energy of a system. It will be observed that energy comprises two essentially different components. The kinetic energy T is a quadratic function of velocity, the potential energy U is independent of velocity.
Isotropy of time
In developing the law of conservation of energy we spoke of time homogeneity, according to which the physical properties of different moments of time are equivalent. Equation (7) indicates that time is not only homogeneous but isotropic as well, i.e., its properties are the same in both directions: substitution of –t for t leaves equation (7), as well as the equation of motion (6), unchanged. In other words, if a system is capable of some type of motion, a reverse motion (one in which the system goes through the same states in reverse order) is always possible. In this sense, according to the laws of classical mechanics, all motions are reversible.
Non-inertial frames of reference
Up till now we have been considering the motion of a material point or a system of particles relative to inertial frames of reference. Accordingly, the equations of motion of a point had the form
 (8)
Such equations of motion hold for inertial reference frames, which, as we know, move relative to one another with uniform velocity. But if we go over to a non-inertial (accelerated) reference system the equations of motion change.
Consider a frame of reference O’ in translatory motion with some velocity V(t) relative to an inertial system O. Denoting by v the velocity of a material point with respect to system O, and by v’ its velocity with respect to system O’, we get
 .
Substituting this expression for v into equation (8), taking into account that V(t) is a given function of time, and introducing the vector  , which is the acceleration of the primed coordinate system, we obtain the equation of motion of a material point in system O’ :
 (9)
where r’ is the radius vector of the particle in the primed system.
A comparison of equations (8) and (9) shows that in an accelerated system moving in a straight line there appears an additional uniform field of force,(6) the force acting on the material point being equal to the product of its mass and acceleration W(t) and directed opposite the acceleration. This force is commonly called the inertia force.
If, in addition to translatory motion, the frame of reference is also rotating with an angular velocity  (7), the equation of motion of a material point in that reference system can be obtained by the necessary velocity transformations. It will differ from (9) by three additional terms found in the right-hand side:  , the inertia force;  , called the centrifugal inertia force; and  , called the Coriolis inertia force. Note that the latter, unlike the forces considered before, depends on velocity.
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1. The question of the geometry of real space is a purely physical one which can be answered only by experiment.
2. Other types of coordinate systems may also be used. They are usually chose according to their convenience in describing motion.
3. This is its physical definition, on which experiments are based.
4. The derivative of a quantity with respect to a vector is a vector the components of which are equal to the derivatives of that quantity with respect to the components of the vector.
5. This follows from the very general physical principle known as the principle of least action.
6. That is, a field in which the force acting on a particle is the same at all points in space at a given moment.
7. It is defined as  , where  angle of rotation.
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