Newton’s laws of motion

The laws of motion (also called Newton’s laws of motion) are the physical principles on which Newtonian dynamics is based, and describe the relationships between the motion of a body and the entities that modify it.

They are valid in inertial reference systems and accurately describe the behavior of bodies moving at speeds much slower than the speed of light, a condition in which they can be approximated by the more general principles of special relativity.

The three laws were presented together by Newton in 1687 in the work Philosophiae Naturalis Principia Mathematica (The Mathematical Principles of Natural Philosophy). Newton himself called his principles Axiomata, sive leges motus (Axioms or Laws of Motion), noting that they were the fundamental basis of mechanics, just as Euclid’s axioms are for geometry, the validity of which can only be tested by experiment and from which all other laws about the motions of bodies can be derived.

The first law, called the principle of inertia, is traditionally attributed to Galileo’s studies of the orbits of the heavenly bodies and the motion of bodies in free fall. The principle of inertia is opposed to the physical theory of Aristotle, who held that the natural state of all bodies was one of rest and that an external agent was necessary to induce motion. Galileo devised a series of experiments, including mental experiments, to demonstrate the fallacy of this assumption. Descartes came to similar conclusions in his writings on physics.

The second law of dynamics, called the principle of proportionality, is due to Newton and introduces the concept of force as the origin and cause of the change in the state of motion of bodies. Over the centuries, there have been numerous discussions about how and what exactly Newton meant by “force” and “change of state of motion”, especially in relation to today’s formulation of the second law of dynamics.

The third law, called the principle of action and reaction, expresses an important property of forces and was used by Newton to prove the conservation of momentum. According to Nobel Laureate Richard Feynman, the third principle has an important role in the development of mechanics:

[Newton] discovered one rule, one general property of forces, which is expressed in his Third Law, and that is the total knowledge that Newton had about the nature of forces–the law of gravitation and this principle, but no other details.

Richard Feynman

Limits of applicability

The laws of dynamics do not apply to non-inertial reference systems. In fact, in order to study them, it is necessary to introduce apparent interactions, that is, forces and moments due to the accelerations of the reference system. Apparent forces, such as centrifugal and Coriolis forces, have no corresponding reaction, in other words, the third principle of dynamics ceases to apply in noninertial reference systems.

Classical mechanics can be seen as the low-speed approximation to the speed of light of special relativity. For example, the second principle of dynamics is no longer able to correctly describe the events that occur when the velocities of bodies are instead close to that of light, since it always allows the speed of a body to be increased indefinitely by the action of a force. Furthermore, the third principle of dynamics requires that action and reaction are always opposite, creating an instantaneous constraint between distant points outside their respective light cones.

Extending the principles of dynamics to non-inertial systems

In order to extend the validity of the principles of dynamics by extending them to non-inertial and extended systems[unclear], the concept of “action” is restricted to forces and moments; in rational mechanics we speak of generalized real forces to which this principle applies, i.e. involving reaction. Finally, because of the symmetry between the two concepts resulting from this principle, it is now preferred to speak of interaction: “The interaction between bodies is reciprocal and the only source of real force and real mechanical momentum.

A generalized force acting on a body i is “real” when it is due to the influence of another body j, and only then manifests itself on j with an antiparallel orientation”.

Remembering that an inertial system is defined by this principle precisely as a reference system in which only interactions between bodies, i.e., real interactions, are manifested, and apparent interactions are precisely those which, not coming from the bodies in that they are not reciprocal, are imputed to the reference system, and are not real only in the sense that they are not “absolute,” and not in the sense of “irrelevant” on the bodies when present.

Newton’s first law of motion (principle of inertia)

Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus à viribus impressis cogitur statum illum mutare.

Isaac Newton, “Philosophiae Naturalis Principia Mathematica”, Axiomata sive Leges Motus

This principle, also known as the principle of inertia or Galileo’s principle, states that a body will continue to move in a uniform straight line or remain stationary if it is not subjected to external forces. Thus, if the resultant of the forces acting on a body is zero, it will maintain its state of motion. In everyday reality, it is observed that a body in motion tends to slow down slowly until it comes to a stop. However, this does not contradict the first principle, since the force of friction, for example with the air or the ground, acts on the body by changing its state of motion. If it were possible to perform an experiment in which all friction and interactions were eliminated, for example in empty space far from galaxies, it would be observed that the body would continue to move at a constant speed along a straight line indefinitely.

The examples given by Newton regarding the spinning circle and the motion of the planets are actually examples of the conservation of angular momentum and represent the integration of the principle of inertia into the principle of conservation of momentum.

The principle of inertia represents a break with Aristotelian physics in that the absence of forces is related not only to stillness but also to uniform rectilinear motion. Since the peculiarity of uniform rectilinear motion is that velocity is a vector constant, i.e., constant in modulus, direction, and magnitude, it follows that the presence of forces is related to changes in velocity. This leads to the second principle of dynamics.

Newton’s second law of motion (principle of proportionality)

The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.

Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.

Isaac Newton, Philosophiae Naturalis Principia Mathematica, Axiomata sive Leges Motus

Therefore, the second principle, also called the Principle of Proportionality or the Principle of Conservation, states that

\[\vec{F} =m\vec{a}\]

Both force and acceleration are vectors and are shown in bold in the formula. In the text, Newton goes on to say:

Si vis aliqua motum quemvis generet; dupla duplum, tripla triplum generabit, sive simul et semel, sive gradatim et successivè impressa fuerit. Et hic motus (quoniam in eandem semper plagam cum vi generatrice determinatur) si corpus anteamovebatur, motui ejus vel conspiranti additur, vel contrario subducitur, vel obliquo obliquè adjicitur, et cum eo secundùm utrusque determinationem componitur.

Isaac Newton, Philosophiae Naturalis Principia Mathematica, Axiomata sive Leges Motus

Translated: Assuming that any force produces any motion, a double force will produce a double motion, and a triple force will produce a triple motion, whether immediately or gradually and at successive times. And this motion (since it is always determined along the same plane as the generating force), if it is concordant, and if the body has already been moved, is added to the motion of that one; subtracted if contrary, or only obliquely added if oblique, and compounded with it according to the determination of both.

The net force, or resultant force, acting on a body is the vectorial sum of all the forces acting on it. Therefore, the acceleration caused by the forces will have the effect of changing the velocity vector over time. This change can manifest itself as a change in the direction of the velocity, or as an increase or decrease in its modulus.

The mass that appears in the second principle of dynamics is called the inertial mass, that is, it quantitatively measures the resistance of a body to acceleration. In fact, the same force applied to a body with a small mass, such as a push on a table, produces a much greater acceleration than on a body with a large mass, such as a car, which would change its speed only slightly with the same push.

If the inertial mass of the body is not constant, then the second law of dynamics can be generalized by introducing momentum. That is, a material point, that is, a body of negligible size with respect to the reference system under consideration and at the same time endowed with mass, to which a force is applied, varies its momentum proportionally to the force and along the direction of the force. In other words, according to a formulation analogous to Euler’s, the rate of increase of the momentum is equal and parallel to the force applied.

The second principle of dynamics explains the fact that all bodies fall with a velocity that is independent of their mass. A similar result, according to Newton, was obtained by Galileo Galilei with the study of the inclined plane and the experiment of falling bodies. However, any student of Galilei’s works knows that Galileo never came to distinguish the concept of mass from that of weight. On the other hand, this is understandable if one considers the Galilean aversion to any reference to an action “at a distance” between bodies, such as that theorized, for example, by Kepler.

Newton’s third law of motion (action and reaction)

To every action, there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Actioni contrariam semper et equalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi.

Isaac Newton, Philosophiae Naturalis Principia Mathematica, Axiomata sive Leges Motus

For each force, or momentum, that a body A exerts on another body B, there is instantaneously another one equal in modulus and direction, but opposite in direction, caused by body B acting on body A. Mathematically, the third principle can be summarized as


But note that for a full understanding of the principle, it is important to point out that the two forces obviously do not cancel out, since they have different points of application.

As the text continues, Newton brings the following examples:

Quicquid premit vel trahit alterum, tantundem ab eo premitur vel trahitur. Si quis lapidem funi alligatum trahit, retrahetur etiam er equus (ut ita dicam) aequaliter in lapidem: nam funis utrinque distentus eodem relaxandi se conatu urgebit equum versus lapidem, ac lapidem versus equum; tantumque impediet progressum unius quantum promovet progressum alterius. Si corpus aliquod in corpus aliu impigens, motum eius vi sua quomodocunque mutaverit, idem quoque vicissim in motu proprio eandem mutationem in partem contrariam vi alterius (ob aequalitem pressionin mutuae) subibit. His actionibus aequales fiunt mutationes, non velocitatum, se motuum; scilicet in corporibus non aliunde impeditis. Mutationes enim velocitatum, in contrarias itidem partes factae quia motus aequaliter mutantur, sunt corporibus reciprocè proportionales.

Isaac Newton, Philosophiae Naturalis Principia Mathematica, Axiomata sive Leges Motus

Translated: For every action there is an equal and opposite reaction. If a man pushes a stone with his finger, his finger will be pushed by the stone. If a horse pulls a stone that is tied to a rope, the horse will also be pulled toward the stone; for the rope that is stretched between the two parties will, by the same attempt to loosen itself, push the horse toward the stone and the stone toward the horse; and it will hinder the progress of the one as much as it will help the progress of the other. If one body, by colliding with another, has in any way altered the motion of the other by its force, it will in turn, by the opposing force, undergo an equal alteration in its own motion in the opposite direction. To these actions correspond equal changes, not of velocity, but of motion. The changes of velocity, in fact, which are made in the same way in opposite directions, because the motions are changed in the same way, are inversely proportional to the bodies.

The third principle of dynamics implies, in modern terms, that all forces result from the interaction of several bodies. According to the third principle, if there were only a single body in space, that body could not experience any force because there would be no body to which the corresponding reaction could be exerted.

A clear example is the application to the Earth-Moon system, of which the Earth and the Moon are subsystems. The total force exerted by the Earth on the Moon must be equal to the total force exerted by the Moon on the Earth, but in the opposite direction, according to the law of universal gravitation.

A typical example that can be given of a counterintuitive application of this principle is that of simple walking: in this situation, we apply a force backwards to the ground with our foot, and the ground responds with an equal and opposite force, which then propels us forward. The contradiction is resolved by considering that the inertial mass of the Earth is enormous compared to that of the individual, and therefore the force results in an acceleration so small as to be unobservable.

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