Laws of the movement of Newton

The laws of the movement of Newton are in fact of the principles at the base of the great theory of Newton concerning the movement of the bodies, theory which one names today Newtonian Mécanique or traditional Mécanique. To these general laws of the movement based in particular on the principle of relativity of the movements, Newton added the law of the universal Gravitation making it possible to interpret the fall of the bodies as well as the movement of the the Moon around the Ground.

First law of Newton or principle of inertia

Statement

The original statement of the first law of the movement is the suivant :

Any body perseveres in or the uniform movement at-rest state in straight line in which it is, unless some force does not act on him, and does not force it to change state.

In other words, if there is no force which is exerted on a body (isolated body), or if the sum of the forces being exerted on him is equal to the null vector (pseudo-isolated body), the direction and the Norme of its Speed do not change or, which returns to same, its Accélération is null. This first law cancels the laws of the physics of Aristote, according to which one thought that to maintain the speed of a mobile constant, it was necessary to apply a force to him.

Although Newton did not specify it in its work, this law is valid only in a Référentiel galiléen. The first law of Newton can thus be reformulated in a more modern language:

In a reference frame galiléen, the Flight Path Vector of the Center of inertia of a system is constant if and only if the sum of the vectors forces which are exerted on the system is a null vector.

Problem of the reference frame galiléen

See also: Search for an inertial reference frame

The definition of a reference frame galiléen appears fundamental and is often formulated ainsi :

a reference frame galiléen is a reference frame in which the first law of Newton is checked

Thus the first law of Newton applies only in one reference frame galiléen and a reference frame galiléen is a reference frame where the first law of Newton applies … what seems to be a circular definition. To avoid this problem, one can rewrite the principle of inertia as follows  :

There exists a family of reference frames, called galiléens or inertial, such as, compared to one of these reference frames, any material point isolated (which is subjected to no external action) is either at rest, or animated of a rectilinear motion and uniform.

The determination of a good reference frame galiléen is actually experimental and like often in Physics, only coherence between the theory (here the first law of Newton) and the measurement (uniform rectilinear motion) validate the choice a posteriori .

Second law of Newton or basic principle of the dynamics of translation

Statement

The basic principle of the dynamics of translation (PFDT) (sometimes called fundamental Relation of dynamics or RFD) states itself as follows: That is to say a body of constant mass m ,

the Accélération undergone by a body in a reference frame galiléen is proportional to the resultant of the forces which it undergoes, and inversely proportional to its mass m . .

This is often recapitulated in the equation:

$\backslash \; vec\; \{has\}\; =\; \backslash \; frac\; \{1\}\; \{m\}\; \backslash \; sum\; \{\backslash \; vec\; \{F\_i\}\}$

or

$\backslash \; sum\; \{\backslash \; vec\; \{F\_i\}\}\; =\; m\; \backslash \; vec\; \{has\}$

where $\backslash \; vec\; \{F\_i\}$ indicates the forces exerted on the object, m is its mass, and $\backslash \; vec\; \{has\}$ corresponds to the Accélération of sound Center of inertia G .

Teaching of the second law of Newton

The students have great difficulties in using the laws of Newton such as they are traditionally stated and it is not without reason. Indeed the forces seem to be exerted as if they even existed in them, ex abrupto. Thus let us propose a new formulation of the law by accompanying it by its procedure so that each one can apply it rationally, i.e. while arguing. In a reference mark galiléen the sum of the forces $\backslash \; vec\; \{F\}\; \_\; \{Oext/B\}\; (T)$ that the objects external with the object B exert on B is equal to the product the bulk $m\_B$ of B by acceleration $\backslash \; vec\; \{has\}\; \_B\; (T)$ of b:

$\backslash \; sum\; \backslash \; vec\; \{F\}\; \_\; \{Oext/B\}\; (T)\; =\; m\_B\; \backslash \; vec\; \{has\}\; \_B\; (T)$

The writing reveals that the forces and acceleration vary during time, whereas the mass of the object is regarded as constant in the field of validity considered and that the speeds of the macroscopic objects are low compared to speed of light.

In problems which aims at describing the movement of an object B when it is immutable and representable by a point (what is called the model of the “material point”), the procedures of use are the following ones:

• To select by the thought the object B which one wants to describe the movement.
• To index all the objects which are external for him and which exert a “notable” force on B. to reach that point it is essential to know the orders of magnitude of the various interactions between the material involved objects in order to treat on a hierarchical basis them; only most intense intervene in the modeling of the situation where arises the difficulty to solve.

Theorem of the momentum

A more general, also valid form if the mass changes during time is

the force is equal to the changes of Quantité of movement per unit of time.

This is often recapitulated in the equation:

$\backslash \; sum\; \{\backslash \; vec\; \{F\_i\}\}\; =\; \backslash \; frac\; \{D\; \backslash \; vec\; \{p\}\}\; \{dt\}$

where $\backslash \; vec\; \{F\_i\}$ indicates the forces exerted on the object, $\backslash \; vec\; \{p\}\; =\; m\; \backslash \; vec\; \{v\}$ is the Quantité of movement, equal to the product of its Masse m and of its Speed $\backslash \; vec\; \{v\}$.

This theorem is called theorem of the momentum. For a fixed solid of mass in Newtonian mechanics, it is equivalent to the second law.

Thus, the force necessary to accelerate an object is the product of its mass and sound accélération : the larger the mass of an object is large, is the necessary force to accelerate it at a given speed (in a fixed amount of time). Whatever the mass of an object, any not-null clear force which is applied to him produces an acceleration.

Return on the principle of inertia

For a body subjected to a null resultant of the forces one finds well the first law of Newton, i.e. a uniform rectilinear motion. In first analysis, one can wonder which is the utility of the first law since it seems to be a consequence of the second. Actually, in the statement of Newton, it of it is nothing because the first law is not presented like a particular case of the second but like a sufficient condition to the application of the latter. Indeed, to state the first law, it is to affirm the existence reference frames galiléens. That constitutes an extremely strong postulate which allows, in the modern talks of the traditional Mécanique, to define the reference mark galiléens which are the only reference marks in which the second law is valid. In the absence of the first law, the second law is inapplicable since one cannot define his field of validity. Consequently, the logical order in which the laws are stated is not the fruit of the chance but well that of a coherent intellectual construction.

Reference frames non-galiléens

See also: Inertia

Let us note finally that it is possible to reformulate in a broader way the second law of Newton in a reference frame not galiléen by adding terms in the equation which are homogeneous with forces, and which one often calls " forces of inertie". These terms are not force S in the usual sense but of the corrective terms of geometrical and kinematic origin.

Third law of Newton or principle of the reciprocal actions

Any body has exerting a force on a body B undergoes a force of equal intensity, of the same direction but of opposed direction, exerted by the body B .

has and B being two bodies in interaction, the force $\backslash \; vec\; \{F\}\; \_\; \{A/B\}$ (exerted by has on B ) and forces it $\backslash \; vec\; \{F\}\; \_\; \{B/A\}$ (exerted by B on has ) which describes the interaction are directly opposite:

$\backslash \; vec\; \{F\}\; \_\; \{A/B\}\; =\; -\; \backslash \; vec\; \{F\}\; \_\; \{B/A\}$

In the case of the mechanics of the point, the third law also specifies:

$\backslash \; vec\; \{F\}\; \_\; \{A/B\}\; \backslash \; wedge\; \backslash \; vec\; \{AB\}\; =\; \backslash \; vec\; 0$ : the force of interaction is carried by the line connecting the positions of the Particule S.

These forces have the same line of action, the opposite directions and the same standard. These two forces are always directly opposite, that has and B is motionless or moving.

This law is sometimes called law of action - reaction , a formulation at best vague, in the worst case involving many confusions. In particular, this old formulation conveys the idea that there is always a force which is the " cause" (the action), the other being only one kind of consequence (reaction).
Another difficulty encountered by the students is the lapse of memory that these 2 forces $\backslash \; vec\; \{F\}\; \_\; \{A/B\}$ and $\backslash \; vec\; \{F\}\; \_\; \{B/A\}$ is exerted on 2 different bodies. They thus cannot “  to cancel mutuellement  ”. The effect of cancellation intervenes only when one considers a system made up of various bodies and that one is interested in the resultant of the forces : in this case, the interior Forces are cancelled indeed, and only the sum of the external Forces is to be taken into account (what is happy to study the movement of a solid made up of more than 10²³ elements).

It is advisable to point out here that the law of the reciprocal actions to the disadvantage of supposing the application of the forces as instantaneous (what is abandoned in restricted Relativité). In the case of the remote forces, it is advisable in certain cases to carry out transformations to take account of the delay of propagation.

This correction does not concern relativity. As the electromagnetic forces apply remotely, one had highlighted that these forces are propagated with speed of light and not at infinite speed, and included this nuance in the equations, before the revolution of restricted relativity

Other laws of Newton

Law of gravitational interaction

See also: universal Law of the gravitation

Certain authors (minority) call fourth law of Newton its universal Loi of the gravitation. This denomination is very contestable, but it is mentioned here because of the historical relationship of the lois : if this law does not form part of the principles of mechanics as well as the three others and the principle of relativity, the first success of Newton was to use its mechanical laws plus its law of gravitational interaction to show the empirical laws of Kepler. These are the first successes which established for a long time the domination of the laws of Newton on science.

Let us note that by combining this law and the basic principle of dynamics, one shows the prediction of Galileo according to whom in the vacuum, all the objects fall at the same speed (by admitting implicitly that inertia and gravitational Masse are equal).

" Fifth corollaire" of Newton: principle of relativity

Newton in its Principia highlighted the concept of relativity of the movement in the definitions preceding the book first. However, while introducing into the scholies II and the IV concept of absolute space, it does not release yet the concept of reference frame galiléen such as it is today defined. In addition, Newton does not refer any if a reference frame is not in uniform rectilinear motion compared to what it calls absolute space . Its results are thus implicitly valid in reference frames in uniform rectilinear motion but no invalidation of the validity of its laws in the accelerated reference frames is given in the Principia . It will be necessary to await work of Coriolis and of Foucault at the XIXe century so that the concept of reference frame galiléen such as it is known today releases and so that the formulas of change of reference mark towards (or since) a reference frame not galiléen are established.

The principle of relativity states &thinsp as follows;: “  Two reference frames of space in rectilinear translation uniform one compared to the other are equivalent for the laws of the mécanique.  ”

(for Newton, it would be necessary to be restricted with the reference frames in uniform rectilinear motion compared to absolute space, by remembering that if a reference frame is in uniform rectilinear motion compared to a second itself in uniform rectilinear motion compared to absolute space, then the first reference frame is in uniform rectilinear motion compared to absolute space)

One will be able to check it, by admitting the first three laws, the invariance of time, the mass and the forces (implicit in pre-Einsteinian physics). This is why this principle is called here corollary.

This principle is known as principle of Relativité galiléenne, because one finds of it the trace in famous Dialog of Galileo, though Galileo had supposed that it was the same for a uniform rotation.

A more modern formulation affirms than all the laws of physics are the same ones for two reference frames of space in rectilinear translation uniform one compared to the other. It is this strong formulation which is at the base of the restricted Relativité.

Remark : The heliocentric reference frame is (generally regarded as) galiléen and it is in this reference frame that the movements of planets and the space probes are studied. To regard the geocentric reference frame as galiléen, whereas the center of the Earth is in acceleration around the Sun, amounts neglecting the Forces of tide. To consider the terrestrial reference frame as galiléen amounts neglecting the centrifugal component in the “  Gravity   ”, and the Force of Coriolis if the material point is moving. In a pragmatic way, knowledge to find with which degree of approximation a reference frame can be (regarded as) galiléen is an unceasingly pushed back search.

History and epistemology

Historical context

Isaac Newton stated his laws in the first volume of sound Philosophiae Naturalis Principia Mathematica in 1687 and, using the new mathematical tools that it developed, it proved many results about the movement of the idealized particles.

Certain detractors of Newton say that it took as a starting point the work of Galileo to write its first principle (by taking again the statement of Galileo almost: “  Any body will continue in its movement of straight line AD eternam if it is subjected to no force  ”, by adding however the concept of uniformity of the movement).

It is advisable to moderate: if Newton were informed of work of Galileo, its role was to formalize the ideas of Galileo and to draw from them the conclusions which made it possible to build mechanics. When Newton affirms “  If I further saw the others, it is because I was carried by shoulders of géants.  ”, the informed reader is supposed to understand that work falls under the continuity of that of Galileo. In fact, one could even say that Newton did not specify that the principle of inertia and the principle of relativity, on which it was based to build all mechanics, were enacted by Galileo, quite simply because it estimates that the reader is supposed the savoir !

The first two volumes are mathematical. In the third volume, natural philosophy (old denomination of the physics of the natural phenomena) is expliquée : it showed how its laws of the movement combined with its universal Loi of the gravitation explain the movement of the Planet S and make it possible to derive the Lois from Kepler.

In 1905 the theory of the restricted Relativité of Albert Einstein watch that the concept of absolute Time, is a concept which gives correct results only to the Speed S much smaller than the Speed of light. Another consequence of restricted relativity, any material body cannot exceed a maximum speed called C , which one considers, until today, that it is equal to the celerity of the photon, who was fixed by definition at: C = 299  792  458 m/s, which defines the meter with the same relative precision as the second (about 10^-15).

In the same way in 1915, by generalizing the principle of relativity, Einstein proposes its theory of the gravitation, still in 2005 not tested in a terrestrial laboratory, but checked and not cancelled in astronomy, with an increasing precision. This theory proposes a propagation of the gravitation to speed of light, avoiding the propagation at infinite speed imposed by the equations of Newton. This new vision of gravity stresses the importance of the preliminary result admitted by Newton whereby inertia is equal to the gravitational mass.

Despite everything, this building of the principles remains a monument of the human thought. These simple laws alone make it possible to build all usual mechanics, i.e. to describe all physics except the situations Quantique S or relativists.

The prediction of the movement of planets by the equations of Newton was remarkable. And by taking account of the interactions of planets, the only aberration compared to reality was the small residue of 43 " of arc per century for the advances Mercury perihelion, and one needed general relativity to explain it.

And in the common life low speeds (other thus that architecture “  relativiste  ” of the buildings of the LHC, CERN), one satisfies well these laws of the practical movement of use.

And, as soon as one wants the precision ultimate (for example, a better precision of the systems of total positioning, GPS or Galileo), then it is known that it is necessary to slightly correct Newton by Einstein (“  giants épaulent…   ”  !), or by Heisenberg when the atoms are studied.

Epistemology

The caused laws were formatted and enacted by Newton. But the bases come from work antérieurs : Galileo, Torricelli, Descartes, Huygens, Hooke, … “  I was carried by shoulders of géants.  ” itself Newton recognized.

As Ernst Mach in its work mechanics pointed out it. Exposed historical and critical of its development, the first law is actually a Tautologie (very safe inutile !) definition IV of the Principia , which introduces the concept of force, fundamental in physique :

the printed force (screw impressa) is the action by which the state of the body is changed, either that this state or rest, or the uniform movement in straight line.

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