Foundations of dynamics

What is dynamics?

dunamis and energeia are terms used by Aristotle, which are generally translated as power and actuality. dunamis is also translated as potential. energeia is associated with ergon, the act, or work.

Potential and actuality are inseparable. Everything that is actual is the actualization of a potential. Conversely, if there were no actuality, there would be no potential, since potential is always the potential to do something.

Dynamics is the science of potential and its actualization.

To exist is always to dance. The actualization of a potential is always a movement. Dynamics is therefore the science of movement and its causes.

What modern physicists call potential energy resembles what Aristotle called dunamis.

Power, according to modern physics, is closely associated with potential energy, because it is the amount of energy that can be supplied per unit of time.

Kinetic energy is the energy of moving masses. It resembles Aristotle’s energeia.

Forces are the causes of variations in the movement of masses. They resemble Aristotle's driving causes.

Dynamics, in the modern sense, is at the same time the science of the movement of masses, the science of energy, or power, in all its forms, and the science of forces.

Kinetic energy is more actual than potential energy because it is more visible, more manifest. But we can also say of potential energy that it is actual. Because to be actual, or simply to be, to exist, is always to be able to have an effect on other beings. A being that cannot have an effect on other beings cannot physically exist. A being's actual being is therefore defined by what it can do, by its potential. It is what it can do. It is its being, even if it hasn't done it yet, and even if it never will.

To be actual is to have potential. Potential is always the potential to have an effect on other actual beings, or on oneself. So potential is always the potential to have an effect on the potential of other beings, or on oneself.

Everything that exists has momentum

Everything that exists physically always has momentum.

Proof: what exists physically always acts on other beings which exist physically. A being that never acted on other physical beings would never have any effect and could never be observed. It would have no physical existence. When a being acts on another, it modifies its movement, therefore its momentum $\mathbf {p} =m\mathbf {v}$ for a body of mass $m$ and speed $\mathbf {v}$. But the total momentum is always conserved. If one body increases the momentum of another, it loses momentum. If one body decreases the momentum of another, it gains momentum. Therefore a body without momentum cannot act on another and cannot exist physically.

Having momentum is a necessary condition for physical existence. Physical quantities are therefore often defined from momentum, and particularly forces and energies:

• A force is the rate of change of a momentum.

• An energy is the work of a force on a path.

What is a force?

If a body is not subjected to any force, it retains its mass and its velocity vector, and therefore its momentum. Its movement is in a straight line at constant speed. It is a uniform rectilinear movement.

Newton's first law:

The movement of a body which is not subject to any force is uniform rectilinear.

It is a law of inertia of movement. The velocity vector does not vary if nothing makes it vary.

From this first law, we deduce that if the momentum of a body is not constant, then there exists a force that causes its momentum to vary.

The fundamental law of dynamics:

A force is the rate of change of a momentum.

$\mathbf {F} ={\frac {d\mathbf {p} }{dt}}$

where $\mathbf {F}$ is the force acting on a body of momentum $\mathbf {p}$.

Newtonian physics assumes that the mass of a body does not depend on its speed, but the theory of relativity does not. For speeds small compared to that of light, the variation in mass is negligible. If we neglect the variations of $m$ as a function of the speed $\mathbf {v}$, we obtain Newton's second law:

$\mathbf {F} =m\mathbf {a}$

where $\mathbf {a} ={\frac {d\mathbf {v} }{dt}}$ is the acceleration of a mass $m$ which experiences a force $\mathbf {F}$.

What is energy?

We define energy from the work of forces: the energy gained or lost by a body is the work of the forces exerted on it. When a body moves against forces, it must give up energy. When a body is pushed by forces, it gains energy.

We can move a heavy object effortlessly on an ice rink, because we don't have to fight against the force of gravity. On the other hand, it takes a lot of effort to lift a heavy object vertically, because we have to oppose the force of gravity. In the first case, the force of gravity does not work, because the movement is horizontal. In the second case, the force of gravity works, because the movement is vertical.

The work W of a force f on a mobile moving in a straight line over a length d is equal to the scalar product of the force vector f and the displacement vector d:

W = f.d = f d cos $\theta$

where $\theta$ is the angle between the force vector f and the displacement vector d. f and d are the lengths of the vectors f and d.

We can consider with Newton that gravity is a force. We then understand that we must provide energy to lift a heavy body, because we must exert a force which works against the force of gravity. The force of gravity is vertical. It does not work for a horizontal movement because it is perpendicular to the movement, cos 90° = 0. The only energy that must be provided to move a heavy body horizontally is the work against the friction forces.

The work of the force of gravity $m\mathbf {g}$ on a mass $m$ which is raised from a height $h$ is

$W=-mgh$

If in W = f.d = f d cos $\theta$, $\theta$ > 90°, then cos $\theta$ < 0 and W < 0. The work of the force has a negative value because it is the energy lost by a body that moves while fighting against the force. This lost energy can be the kinetic energy E = 1/2 mv². The speed v decreases because the body is braked by the force. If $\theta$ < 90°, cos $\theta$ > 0 and W > 0. The work of the force has a positive value because it is the energy acquired by a body that moves by being pushed and accelerated by force.

In the international system of units of measurement (MKSA, meter, kilogram, second, Ampere), the unit of energy is the Joule (J). One Joule is the work required to move a body one meter against a force of one Newton (N).

1 J = 1 N. 1 m = 1 N.m

The force of gravity on the surface of the Earth is approximately equal to 9.8 N, almost 10 N. One Joule is therefore approximately the energy required to lift a 1 kg body ten centimeters.

The fundamental law of dynamics $\mathbf {F} ={\frac {d\mathbf {p} }{dt}}$ and the definition of energy from the work of a force teach where to find energy and how to appropriate it. Energy is where there are forces. To appropriate energy, it is enough to make the forces work. So $\mathbf {F} ={\frac {d\mathbf {p} }{dt}}$ gives us the secret of power. For example, we can find E = m c², the Einstein equation which revealed the power of the atom, from $\mathbf {F} ={\frac {d\mathbf {p} }{dt}}$.

The kinetic energy of a moving mass

Consider two masses $m_{1}$ and $m_{2}$ connected by a spring.

When compressed or stretched, the spring exerts two forces $F_{1}$ and $F_{2}$, one on $m_{1}$, the other on $m_{2}$. If the mass of the spring is negligible compared to the masses it connects then we always have $F_{1}=-F_{2}$.

We assume that each mass is only subjected to the force of the spring and that they are released against each other, at rest, after having extended the spring:

(Animation)

According to the fundamental law of dynamics:

$F_{1}={\frac {dp_{1}}{dt}}$

where $p_{1}=m_{1}v_{1}$ is the momentum of $m_{1}$. $v_{1}$ is its speed.

The forces $F_{1}$ and $F_{2}$ on the masses $m_{1}$ and $m_{2}$ are always in the direction of movement. The work $dW_{1}$ of $F_{1}$ on $m_{1}$ for a small displacement $dx_{1}$ is

$dW_{1}=F_{1}dx_{1}$

The work $W_{1}$ of $F_{1}$ between two instants $t_{1}$ and $t_{2}$ is

$W_{1}=\int _{t_{1}}^{t_{2}}dW_{1}=\int _{t_{1}}^{t_{2}}F_{1}dx_{1}$

The energy gained or lost by the mass $m_{1}$ is therefore

$W_{1}=\int _{t_{1}}^{t_{2}}F_{1}{\frac {dx_{1}}{dt}}dt=\int _{t_{1}}^{t_{2}}F_{1}v_{1}dt$

$W_{1}=\int _{t_{1}}^{t_{2}}m_{1}{\frac {dv_{1}}{dt}}v_{1}dt=\mathbf {[} {\frac {1}{2}}m_{1}v_{1}^{2}\mathbf {]} _{t_{1}}^{t_{2}}$

$E_{c}={\frac {1}{2}}mv^{2}$ is the kinetic energy of a mass $m$ which goes at the speed $v$ .

When a force is exerted on a mass in the direction of its movement, it gives it kinetic energy by increasing its speed.

When a force is exerted on a mass in the opposite direction of its movement, it takes kinetic energy from it by slowing it down.

When a force is exerted on a mass in a direction perpendicular to its movement, the mass retains its kinetic energy, therefore the magnitude $v$ of its velocity vector $\mathbf {v}$ .

Einstein understood that all energy has mass, even kinetic energy. So the mass of a body depends on its speed. The calculation above is not exact. The faster a body goes, the more its mass increases and the more difficult it becomes to accelerate it, because the acceleration given by a force is inversely proportional to the mass of the body on which it is exerted. A mass can never reach the speed of light because it would need infinite energy to do so.

A flywheel is a massive wheel that is rotated around its axis. Mass is put especially at the periphery because that's where it goes the fastest. If friction is low, a flywheel can maintain its rotational speed for a very long time. It thus functions as an energy reservoir, because it retains its kinetic energy of rotation.

The potential energy of a spring

A spring can serve as an energy reserve. For example, 16 mm cameras work without electricity and allow continuous filming for several minutes, simply with a spring, which is wound by the crank.

To extend or compress a spring, the two forces exerted on it are in the same direction as the movement of each of its ends, we must therefore transfer energy to the spring. We have to expend energy, so we have to make an effort to tension or compress a spring.

A spring of stiffness $k$ near its equilibrium position exerts two forces $F_{1}=-F$ and $F_{2}=F$ on each of the bodies, at its left and its right, which compress or extend it:

$F=-kx$

where $x$ measures the variation in length of the spring compared to its equilibrium length $l_{0}$. This is Hooke's law of elasticity. It is true for all solids, provided that they are little deformed by the forces exerted on them. Springs are designed to respect Hooke's law even if they are very deformed.

If $dx_{1}$ and $dx_{2}$ are small displacements of the ends of the spring, the work $dW$ that must be provided to compress or extend the spring is

$dW=-F_{1}dx_{1}-F_{2}dx_{2}=-F(dx_{2}-dx_{1})=-Fdx$

$dx=dx_{2}-dx_{1}$ because $x=x_{2}-x_{1}-l_{0}$.

To extend or compress a spring over a length $x$, it is therefore necessary to provide it with energy

$W=\int _{0}^{x}dW=-\int _{0}^{x}Fdx=\int _{0}^{x}kx$ $dx={\frac {1}{2}}kx^{2}$

$E_{p}={\frac {1}{2}}kx^{2}$ is the elastic potential energy of a spring of stiffness $k$ where $x$ is its variation in length relative to its equilibrium length.

The cohesive forces of matter are electric. Elastic potential energy is electrical potential energy. It is a difference in energy of the electric field produced by the electrons and the nuclei of the spring. When we compress or extend a spring, we increase the electric potential energy conserved in the electric field produced by its electrons and its nuclei.

Conservation of energy

When the spring sets the masses in motion, its elastic potential energy is transformed into the kinetic energy of the masses. When the movement of the masses compresses or extends the spring, their kinetic energy is transformed into elastic potential energy of the spring.

Let $E=E_{c1}+E_{p}+E_{c2}$ be the sum of the kinetic energies of the two masses and the elastic potential energy of the spring.

${\frac {dE}{dt}}={\frac {d}{dt}}[{\frac {1}{2}}m_{1}v_{1}^{2}+{\frac {1}{2}}kx^{2}+{\frac {1}{2}}m_{2}v_{2}^{2}]={\frac {1}{2}}[2m_{1}v_{1}{\frac {dv_{1}}{dt}}+2kx{\frac {dx}{dt}}+2m_{2}v_{2}{\frac {dv_{2}}{dt}}]$

now

${\frac {dx}{dt}}={\frac {dx_{2}}{dt}}-{\frac {dx_{1}}{dt}}=v_{2}-v_{1}$

so

${\frac {dE}{dt}}=-Fv_{1}-F(v_{2}-v_{1})+Fv_{2}=0$

The total energy $E$ shared between the spring and the two masses is therefore conserved.

The law of conservation of energy:

The amount of energy gained or lost by a physical system is always equal to the amount of energy it received or gave up to another physical system.

We can also state it:

When two physical systems transfer energy from one to the other, their total energy does not change.

Conservation of momentum

The momentum $\mathbf {p}$ of a body is always the product of its mass $m$ and its velocity $\mathbf {v}$ :

$\mathbf {p} =m\mathbf {v}$

Let $p$ be the total momentum of the two masses $m_{1}$ and $m_{1}$:

$p=p_{1}+p_{2}$

${\frac {dp}{dt}}={\frac {dp_{1}}{dt}}+{\frac {dp_{2}}{dt}}=F_{1}+F_{2}=0$

So the total momentum of the two masses is conserved.

The force $F_{2}$ exerted by the mass $m_{1}$ on the mass $m_{2}$, via the spring, is equal and opposite to the force $F_{1}$ exerted by the mass $m_{2}$ on the mass $m_{1}$, if we neglect the mass of the spring. The action of $m_{1}$ on $m_{2}$ is equal and opposite to the reaction of $m_{2}$ on $m_{1}$. Conservation of total momentum is therefore equivalent to the equality of action and reaction.

The law of equality of action and reaction

If two bodies A and B interact, the force exerted by A on B is equal and opposite to the force exerted by B on A.

Theorem: we cannot go to the Moon by pulling on our boots.

Proof: the hands exert on the boots a force directed upwards exactly equal and opposite to the force exerted downwards by the boots on the hands. The sum of the two is zero and therefore cannot give an upward acceleration.

The law of equality of action and reaction is Newton's third law. It is strictly exact only if A and B are in contact. The two equal and opposite forces are then exerted at the same point, the point of contact. But if A and B are far apart, they cannot be instantly sensitive to variations in each other's movement, because there is no instantaneous action at a distance. Information and physical beings always move with finite speed, never with infinite speed.

The law of conservation of momentum is better than the law of equality of action and reaction, because it avoids the problem of instantaneous action at a distance:

The law of conservation of momentum:

The momentum gained or lost by a physical system is always equal to the momentum it received or gave up to another physical system.

We can also state it:

When two physical systems transfer momentum from one to the other, their total momentum does not change.

Angular momentum

A body which rotates on itself and which is not subject to any external force maintains its rotational movement. The axis and speed of rotation do not change. This is what happens to a spinning top if it is in free fall. The rotational inertia of their wheels balances bicycles in motion. A bicycle at rest falls, because there is no longer any rotational inertia.

Angular momentum is a rotational momentum. It is to constant rotational movement what momentum is to uniform rectilinear movement. Like momentum, it is always conserved in the absence of an external force. The angular momentum lost or gained by one body is always the angular momentum gained or lost by another body.

The laws of the gift of momentum and energy

A body cannot give more momentum, angular momentum and energy than it has. If it gives all its energy then it is no more, because its mass is energy.

To provide momentum and angular momentum, it must exert forces. To give energy, it must also exert forces, because variation in energy comes with variation in momentum.

When a force is perpendicular to the speed of a mass, it does not work. It varies the momentum vector, but not its length, so it does not increase the kinetic energy of the body on which it is exerted. The body that exerts this force varies the momentum without losing its energy.

The fundamental laws of physics determine the forces that bodies can exert on each other. They are therefore the laws of the gift of momentum, angular momentum and energy.

Energy and momentum of fields

A field is a physical quantity defined at each point in space-time.

There is no instant action at a distance. Two bodies far apart that interact always do so through a force field, where information is propagated at a finite speed, always equal to or less than that of light. For example, two masses linked by a spring interact through the pressure field in the spring. In a pressure field, information is propagated at the speed of sound.

Fundamental forces are the forces between particles. When a body A exerts a force F on a body B, F is the vector sum of all the forces exerted by a particle x on a particle y, for all the particles x of A and all the particles y of B.

The force fields between particles are the electromagnetic field, which exerts electric and magnetic forces on particles, and the nuclear force field, which explains the stability and instability of the nuclei of atoms. Radioactivity is the consequence of nuclear instability: unstable nuclei disintegrate spontaneously, without the slightest force being exerted to break them.

According to Newtonian physics, gravitation is a field of forces between all masses, but the speed of propagation of information is infinite, because the law of universal gravitation imposes instantaneous action at a distance. According to Einstein's theory of general relativity, gravitation is not a force, but a field of space-time distortions, where information can never propagate faster than light.

Like everything else that exists physically, force fields have energy, momentum, and angular momentum. When a force is exerted on a particle, it always varies its momentum, and it can also vary its energy and angular momentum. These variations are caused by transfers of momentum, energy and angular momentum between the field and the particle on which it acts.

The energy or momentum of a physical system is always the sum of the energies or momentum of the particles that constitute it plus the sum of the energies or momentum of the fields that these particles produce.

We often count the energy of the field by assigning potential energy to the particles on which it exerts its forces. For example, we attribute electric potential energy to an electrically charged body. It is the sum of the electrical potential energies of all its electrically charged particles. But this potential energy is not an energy carried by the particles, because it does not vary their mass. It is an energy of the electric field, localized in the space around the particles. Counting the potential energy of particles is only one way of counting the energy of the field they produce.

The energy of mass

A particle is restless if and only if there does not exist an inertial frame of reference where it is motionless.

A particle is with rest if and only if there exists an inertial frame of reference where it is motionless.

Particles with rest have rest mass.

Everything that exists physically has mass, because everything that exists physically has momentum $\mathbf {p} =m\mathbf {v}$. Restless particles have mass, like particles with rest, but they have no rest mass, because they have no rest.

Photons are restless particles. The mass of a photon is

$m_{p}={\frac {p}{c}}={\frac {E}{c^{2}}}$

where $E$ is its energy, $p$ its momentum and $c$ the speed of light.

Even a mass at rest is the energy of a field, the field of force that gave it rest. The rest mass of a particle resembles the work of the force on a path that took it from no rest to rest. For an electrically neutral particle, this work is equal to

$E=m_{0}c^{2}$

where $m_{0}$ is the rest mass of the particle.

For an electrically charged particle of rest mass $m_{0}$, we must add to the work of the force which gave it rest the energy of the electric field that it produces around it to obtain $E=m_{0}c^{2}$.

Let $m_{n}c^{2}$ be the work of the force which gave a charged particle rest. Let $m_{C}$ be the mass of the Coulomb field produced by this charge if it were isolated.

$m_{0}=m_{n}+m_{C}$

In general, we measure $m_{0}$, not $m_{n}$ and $m_{C}$ separately, because we cannot undress a charged particle and ask it to leave its Coulomb field in the locker room before measuring its mass $m_{n}$.

When an electrically charged particle and its antiparticle, an electron and a positron for example, are exactly superimposed, the electric field they produce together is equal to zero everywhere in space. The mass $m$ of the pair, if it were at rest, would therefore be

$m=2m_{n}$

because the energies $m_{C}c^{2}$ of the Coulomb field of a particle and its antiparticle are equal, and their rest masses $m_{0}$ too.

The minimum energy required to create a particle-antiparticle pair is equal to $2m_{0}$ not $2m_{n}$. When a pair is created, the difference $m_{0}-m_{n}$ is the minimum kinetic energy of the particle or its antiparticle, for the pair to be created.

After being created, the electrically charged particle and its antiparticle move away from each other. They thus produce a dipolar electric field. They give energy to this field while losing part of their kinetic energy.

The kinetic energy $E_{c}$ of a particle is always the difference between its mass multiplied by c² and its rest mass multiplied by c² also:

$E_{c}=(m-m_{0})c^{2}$

The kinetic energy of a particle depends on the frame of reference in which it is measured, and therefore its mass too.

The mass of a particle, including its kinetic energy, is always the energy of one or more fields, the energy of the field that put it at rest, if it is a particle with rest, plus the energy fields that it produces around it.

When we count the energy of a physical system, we must add up all the energies of the fields associated with it, including the force field that gives the particles rest. Particles do not exist without the fields that they produce or that produced them, because they are quanta of fields. There are only fields and their quanta, the particles.

The fundamental laws of physics are therefore always the laws of the gift of momentum, energy and angular momentum from one field to another field. When the kinetic energy of a charged particle is transformed into electric potential energy, the field that gives the particles their rest and their kinetic energy gives up part of its energy to the electromagnetic field.

The gift of kinetic energy

The mass of an electrically neutral body with rest is the mass of the field which gives it rest. It is the sum of its rest mass and the mass of its kinetic energy. When one mass gives up kinetic energy to another, it loses part of its mass. Part of the energy of the field which gives it rest is transferred to the field which gives rest to the other mass.

Let two equal masses $m_{1}$ and $m_{2}$, $m_{1}=m_{2}=m$, which bounce on each other with speeds $v_{1}=v$ and $v_{2}=-v$ equal and opposite, measured in a reference frame R.

If the masses are perfectly elastic, they retain their kinetic energy after the rebound:

$v_{1out}=-v_{1}$

so

${\frac {1}{2}}m_{1}{v_{1out}}^{2}={\frac {1}{2}}m_{1}v_{1}^{2}$

The same for $m_{2}$.

In a reference frame R' which goes at speed $v$ with respect to R, $m_{1}$ is initially at rest. After the rebound it gains a kinetic energy equal to ${\frac {1}{2}}m_{1}(2v)^{2}=2mv^{2}$, given up by the mass $m_{2}$.

In a reference frame R" which goes at speed $-v$ with respect to R, $m_{2}$ is initially at rest. After the rebound it gains a kinetic energy equal to $2mv^{2}$, given up by the mass $m_{1}$.

From the point of view of R', $m_{1}$ gives up kinetic energy to $m_{2}$. From the point of view of R", $m_{2}$ gives up kinetic energy to $m_{1}$. From the point of view of R, $m_{1}$ and $m_{2}$ each retain their kinetic energy. How is this possible?

Energy always has mass. Mass is always the mass of the quanta of a field. If particles move from A to B from the point of view of one frame, they move from A to B from the point of view of all frames, because the presence of a particle in A or in B does not depend on the point of view. So it seems that a transfer of kinetic energy cannot depend on a point of view.

Restless particles carry energy at the speed of light. Their energy $E=pc$ depends on the frame of reference, because their momentum $p$ depends on the frame of reference, by Doppler effect. When the two masses bounce off each other, there are two particle flows, one from $m_{1}$ to $m_{2}$, the other from $m_{2}$ to $m_{1}$. From the point of view of R, these energy flows are exactly equal. This is why the two masses each retain their kinetic energy. But from the point of view of R' or of R", these two flows do not transfer the same energy, because of the Doppler effect. The difference is the transfer of energy from one mass to the other.

The principle of least action

We can find the fundamental equations of the dynamics of all physical systems by reasoning with the following three principles:

• Kinetic energy $E_{c}$ is expense per unit of time.

• Interaction energy $E_{i}$ is income per unit of time.

• All paths naturally followed by a physical system are paths of least loss or maximum gain, paths for which the difference between total revenues and total expenses is maximum.

A path AB followed by a physical system is a path of least loss if it is such that all mathematically possible paths from A to B have a higher loss. It is a path of maximum gain if it is such that all mathematically possible paths from A to B have a lower gain.

The difference $E_{c}-E_{i}=L$ is called the Lagrangian of the system. The integral $\int _{A}^{B}L$ $dt=\int _{A}^{B}E_{c}$ $dt-\int _{A}^{B}E_{i}$ $dt=S$ is called the action. It has the dimensions of energy multiplied by time. It is the analogue of a loss: total expenses minus total income. Its opposite $-S$ is the analogue of a gain. The principle of least loss, or maximum gain, is called the principle of least action.

Time is reversible. Nature does not differentiate between the past and the future.

Proof: if AB is a path of maximum gain, the same path BA taken in the opposite direction is also naturally possible, because it is a path of minimum loss.

This theorem is almost always true for the fundamental equations of microscopic motions. The arrow of time, from the past to the future, does not appear in these equations, but only in the equations of macroscopic movements given by statistical physics and thermodynamics.

The principle of least action does not impose that all naturally possible paths go towards increasing gains, since time is reversible, it only imposes that all naturally possible paths maximize gains or minimize losses.

We can find Newton's three fundamental laws from the principle of least action, provided that we choose the appropriate Lagrangian.

Consider a race from A to B between a hare and a tortoise. The tortoise moves forward at a constant speed while the hare stops to take a nap. To arrive at the same time as the tortoise, the hare must make up for lost time. So he arrives all out of breath while the turtle arrives calmly. We can calculate that the tortoise chose a path of least action for the Lagrangian $L=E_{c}={\frac {1}{2}}mv^{2}$, while the hare did not minimize its losses . We thus find Newton's first law: in the absence of forces, movements are always uniform rectilinear. That there is no force is reflected by the absence of interaction energy in the Lagrangian.

Points A and B are points in configuration space. A configuration is defined by the positions of all the bodies that are part of the system. If there is a single moving point, the configuration space is real, three-dimensional space. If there are n moving points, the configuration space has 3n dimensions.

An optimal path is determined by two points A and B and by a delay to reach B starting from A. The path of least action is the optimal path among all the paths which have the same endpoints and the same delay. A is like a starting point at a fixed time and B is a meeting point, at an equally fixed time. The path of least action is the optimal path to arrive at the meeting place at the appointed time.

When we know the path of least action, we can calculate the speeds along the entire path, therefore the speeds at point A and point B. We therefore obtain a function between the final speed at B and the initial speed at A. We can thus calculate the final speed after a certain delay according to the initial speed. By making the delay tend towards zero, we thus find the rate of variation of the speeds of the points of the system, therefore the rate of variation of their quantities of movement, therefore the forces which are exerted on them.