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CLASSIC NORELATIVISTIC MECHANICS

Classic nonrelativistic mechanics is said to be Newtonian mechanics because three laws lying at the heart of it were formulated by I. Newton.

Classic nonrelativistic mechanics consists of three parts kinematics, dynamics and statics.

Statics studies the equilibrium condition of bodies.

Kinematics gives mathematical description of motion of bodies, without regarding for the interaction between bodies causing this motion.

Dynamics treats the influence of the interaction between bodies on their mechanical motion. Mechanical motion can be very complicated, and its description can be too difficult. Therefore simplified model is usually used instead of the real body, such as a material point (pointlike body), perfectly rigid body (rigid body) and so on.

KINEMATICS

Kinematics of material point

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Cartesian coordinates curvilinear motion curvilinear uniform motion straight line motion equation of motion equation of trajectory origin

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Декартові координат криволінійний рух криволінійний рівномірний рух прямолінійний рух рівняння руху рівняння траєкторії початок координат

coordinate material point path position vector reference frame time trajectory displacement

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координата матеріальна точка шлях радіус вектор система відліку час траєкторія переміщення

Space and time. The notion about properties of space and timeis set in the basis of the whole mechanics (as well as physics itself). Newton gave rather original physical determination of space, naming it the «receptacle of all things». However, Newton considered (and afterwards this position became the foundation of all classical physics) that properties of time and space are not connected in any way between each other. Thus, they can be examined separately. As to «empty» space, it is considered that it is homogeneousand isotropic, i.e. all of its points are equal relative to both simple linear transitions (homogeneity) and revolves (isotropy). In addition, it is considered that our space satisfies Evklid’s (flat) metric system of measurement.

The last statement needs some additional explanations. Once upon a time Russian mathematician Lobachevskiy proved the principal possibility of existence of many varieties of the so-called «curve» metric systems of space measurement. But a question appears at once: is our space flat or «curve»? It was the great German mathematician K.Ф.Gauss (1777—1855) who made the first practicable step to finding out the right answer to that question; however, he thought as the real physicist. In fact he did a proper experiment and proved (within measuring accuracy possible by that time) that our space had Evklid’smetric system of measurement indeed. For this purpose the scientist used the well-known theorems of elementary geometry. Within the framework of Evklid’s geometry, the corners sum of any triangle, as it is known, must equal 180°. And in case of the «curve» metric system of measurement this sum must be less. Therefore, Gauss measured the corners sum of a triangle with sides of about 100 km (mountains tops in Alps) and made sure, that geometry of our space is flat within the limits of measuring accuracy, that is Evklid’s. Afterwards, such experiments were conducted many times, but each time for much more greater scales. At first, this result was received in the scale of the Planetary system, then in Galaxy (~1020), and then in Metagalaxy (~1026). However, it should be pointed out that there is no complete confidence in the rightness of such conclusion at the border of accessible for supervision Universe even by means of available experimental facilities.

In physics, space quantitative description is expressed by means of the reference frames. The reference frame is the system of coordinates, related to some body of reference. More detailed concept of the reference frame follows below.

If the interpretation of notion «space» is more or less obvious, it can’t be said in such a way about concept «time» — it is one of the most difficult in physics. In the simplest (quantitative) sense, “time” is the readings ona clock. More precisely speaking, it is an interval between any two measurements on a clock. In physics, any material object, which performs periodic oscillations, is considered the clock. For example: a mechanical pendulum (period); the Earth which rotates around its own axis (a day), the Earth which rotates around the Sun (a year); the Sun which rotates around the center of Galaxy (a galactic year) and so on.

It is considered that time flows uniformly, and thus, a «clock» goes uniformly, that is, its period is stable and does not change from one oscillation to another. Let’s note that both supposition on homogeneity and isotropy of space and supposition on uniformity of time flow are in general sense hypotheses. But, as it was shown by the further development of physics, these hypotheses were confirmed later by numerous indirect experiments. In particular, the scientists utilized the fact that the energy conservation law resulted from the supposition on homogeneity of time, the momentum conservation law resulted from the supposition on homogeneity of space, and the angular momentum conservation law — from the property of space isotropy. As all these laws are fundamental and according to this are reliably proved with experiments, the just formulated hypotheses are true indeed.

To describe the motion of a body certain characteristics are to be introduced.

A body is said to be a material point if the size of the body is negligibly small in comparison with the distance covered by the body.

Розглянемо по порядку такі кінематичні характеристики: система відліку, радіус-вектор, рівняння руху, траєкторія, рівняння траєкторії, швидкість, прискорення.

Frame of reference. Any motion is relative. Therefore it is necessary to point out a body with respect to which the motion of the body being studied. The set of chosen reference body (point), clock and system of reference represents frame of reference.

Fig. 1.1

Position vector. We can state the position of a material point in the space by a position vector starting at the reference point and ending at the point where the object is found at a given moment of time. Fig. 1.1 shows the position vector of a point at different instants of time. In the unit-vector notation can be written

where , and are the vector components of and the coefficients x, y, and z are their scalar components. The coefficients x, y, and z give the object's location along the axes and relative to the origin.

Equation of motion. The first step in the study of motion is to describe the position of a moving object. We have to specify its position relative to some reference point. The choice of the reference point is arbitrary so long as it is clearly stated and understood by everyone, and mathematical description of the motion relative to this reference point is complete but not complicated. Assume is the position vector of an object at an instant and is the position vector of this moving object at an instant . Thus we can describe the position of a material point at any time by its position vector,

and we can describe the change of the position of a moving object as time progresses by means of the displacement vector . This is the vector method of describing motion.

Any formula, which gives unique relationship between position-vector
and time, is called equation of motion

Fig. 1.2

It is often convenient to deal not with , but with Descartes (or orthogonal) coordinates designated as (Fig. 1.2). In this case the last vector equation resolves into three scalar equations:

Here , and . This method of describing motion is known as coordinate method.

Trajectory (or path) is a line a material point travels along. If this line is straight, the trajectory is a straight-line one and the motion of the material point is called straight-line motion. In case the line is curve, we deal with a curve-line path of the material point and its curvilinear motion.

Distance and displacement. Distance the material point travels is the length of its path. There is an example of a class of quantities called scalars which have only one characteristic - numerical value. Obviously, the numerical value of the distance may be only positive, and the distance always increases never mind constant or not the direction of the material point motion is. The distance is usually designated by and measured in meters (in SI).

Displacement is a vector, it possesses two characteristics — both magnitude and direction. The magnitude of any vector is only positive number and usually we speak about 'the modulus of the vector'. The vector of displacement starts at the initial point of the trajectory and points at its final point. It may therefore be represented to scale by a line drawn in a particular direction. The displacement is usually designated by and measured in meters (in SI).

Vector of displacement is equal to change of position vector or it increment per time interval D t:

.

Using the unit-vector notation, we can rewrite this displacement as

Or

where coordinates (x_{1 ,} y_{1 ,} z₁) correspond to position vector , and coordinates (x₂, y₂, z₂) correspond to position vector . We can also rewrite the displacement by substituting x for (x₂ – x₁), y for (y₂ – y₁), and z for (z₂ – z₁).

Fig. 1.3

Equation of trajectory is the equation of line inspace, which connects the coordinates of point during motion.It can be written as:

Equation of trajectorycan be obtained from equation of motion, excluding the time.

To illustrate the coordinate method, let us analyse an example: motion of a ball projected horizontally at an initial velocity (Fig. 1.3). We write the kinematical equation of the motion and make the equation of the trajectory.

Since is horizontal, it has no component in a vertical direction. Similarly, since acceleration acts vertically, it has no component in a horizontal direction. Thus we can treat vertical and horizontal motion independently. The motions:

1) the uniform motion along the axis its equation is , and

2) uniformly accelerated motion along the axis its equation is .

These two equations together form the kinematical equations of the motion of the ball.

To get the equation of the trajectory (its general form is , this is the equation of a line) we find time from the first equation: , and substitute this expression for in the second equation: Þ - the trajectory of the object thrown forward at a horizontal velocity is a parabola.

The reverse problem of getting the kinematical equation of motion from the equation of the trajectory is the problem without a unique solution.

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