## Newton`s Laws of Dynamics

Newton`s laws apply only to a specific set of frames called Newtonian or inertial frames of reference. Some authors interpret the first law as defining what an inertial reference system is; From this point of view, the second law applies only if the observation is made from an inertial reference system, and therefore the first law cannot be proved as a special case of the second. Other authors treat the first law as a consequence of the second. [21] [22] The explicit concept of an inertial system was not developed until long after Newton`s death. The discovery of the laws of dynamics, or laws of motion, was a dramatic moment in the history of science. Before Newton`s time, the motions of things like planets were a mystery, but after Newton, there was a complete understanding. Even slight deviations from Kepler`s laws due to planetary disturbances were predictable. The movements of pendulums, oscillators with springs and weights, etc. could all be fully analyzed after Newton`s laws were pronounced. It is the same with this chapter: before this chapter, we could not calculate how a mass would move on a spring; even less could we calculate the disturbances on the planet Uranus by Jupiter and Saturn. After this chapter, we will be able to calculate not only the motion of the oscillating weight, but also the perturbations on the planet Uranus generated by Jupiter and Saturn! The above analysis is very pleasant for the movement of an oscillating spring, but can we analyze the motion of a planet around the sun? Let`s see if we can approach an ellipse for orbit. We will assume that the sun is infinitely heavy, in the sense that we will not include its motion. Suppose a planet starts at a certain place and moves at a certain speed; it revolves around the sun in a curve, and we will try to analyze what the curve is through Newton`s laws of motion and his law of gravity.

How? At some point, he is in a position in space. If the radial distance of the sun at this position is called $r$, then we know that there is an inner force which, according to the law of gravity, is equal to a constant multiplied by the product of the mass of the sun and the mass of the planet divided by the square of the distance. To analyze this in more detail, we need to know what acceleration is generated by this force. We will need the components of acceleration in two directions, which we call $x$ and $$y. So, if we specify the position of the planet at a given time by specifying $x$ and $ $y (we assume that $ $z is always zero because there is no force in the direction $z $, and if there is no muzzle velocity $v_z$, there is nothing to do $z$ except zero), the force is directed along the line, which connects the planet to the Sun, as shown in Figures 9 to 5. Another example, let`s say we were able to build a device (Fig. 9-3) that applies a force proportional to the distance and directed in the opposite direction – a spring. If we forget gravity, which is naturally compensated by the initial stretching of the spring, and talk only about excessive forces, we see that when we pull the mass down, the spring pulls up, while when we push it up, the spring pulls down. This machine has been carefully designed so that the force is all the greater the more we pull it upwards, exactly proportional to the displacement of the balanced state, and the upward force is also proportional to the distance at which we shoot. If we observe the dynamics of this machine, we see a fairly pleasant movement – up, down, up, down, . The question is whether Newton`s equations will correctly describe this motion. Let`s see if we can calculate exactly how it moves with this periodic oscillation by applying Newton`s law (9.7).

In this case, the equation is begin{equation} label{Eq:I:9:11} -kx=m(dv_x/dt). end{equation} Here we have a situation where the speed changes in the direction $x$ in proportion to $x$. Nothing is gained if we keep many constants, so we will either imagine that the time scale has changed, or that there is an accident in the units so that we have $k/m = $1. Therefore, we will try to solve the equation begin{equation} label{Eq:I:9:12} dv_x/dt=-x. end{equation} To continue, we need to know what $v_x$ is, but of course, we know that speed is the rate of change of position. In their original form, Newton`s laws of motion are not sufficient to characterize the motion of rigid and deformable bodies. In 1750, Leonhard Euler introduced a generalization of Newton`s laws of motion for rigid bodies, called Euler`s laws of motion, which were later applied to deformable fields, which were assumed to be continuums. If a field is represented as a collection of discrete particles, each determined by Newton`s laws of motion, then Euler`s laws can be derived from Newton`s laws. However, Euler`s laws can be thought of as axioms describing the laws of motion for extended bodies independent of any particle structure. [20] Newton`s laws of motion are important because they are the basis of classical mechanics, one of the main branches of physics.

Mechanics is the study of how objects move or do not move when forces act on them. Leaving the domain of statics, the acceleration of an object is associated with the total force F acting on it by the famous law of Newtonian dynamics, F = mg, where m is the mass of the object and g is its acceleration vector. This makes it possible to define the unit of force, the Newton (N), as the force that produces an acceleration of 1 (m/s)/s on a mass of 1kg, which is written 1 N = 1 kg m s-2. To use Newton`s laws, we must have a formula for force; These laws say, watch the forces. When an object accelerates, an agency is at work; Find. Our program for the future of dynamics must be to find the laws of power. Newton himself then gave some examples. In the case of gravity, he gave a specific formula for force. In the case of other forces, he gave some of the information contained in his third law, which we will examine in the next chapter, and which concerns equality of action and reaction. The amplitude of a motion of an elementary mass (considered as point) is defined as the product of mass and velocity, a definition that can be extended to any physical system by adding (vector) the sets of motion of each of its elementary masses.

It is easy to show that the range of motion is equal to the range of motion of the inertial center (barycenter) of the system affected by its total mass. Newton`s law of dynamics then shows that the time derivative of momentum is equal to the sum of the forces acting on the system. From this figure we see that the horizontal component of the force is related to the total force in the same way as the horizontal distance $x$ to the complete hypotenuse $r$, because the two triangles are similar. If $x$ is positive, $F_x$ is negative. That is, $F_x/abs{F}=-x/r$ or $F_x=$ $-abs{F}x/r=$ $-GMmx/r^3$. Now we use the dynamic law to discover that this force component is equal to the mass of the planet multiplied by the rate of change of its velocity in the direction $x$.