(This page, written long ago, is here as a draft to be reworked
and completed later).

As Special Relativity explains space-time in the form of a geometrical space where the special role of time is forgotten, thus where things appear "fixed", the laws of mechanics there are also best described as identical to what they usually are when things are fixed (along time).

What does it mean, that things are fixed along time ? Simply that nothing happens in the given direction of time.

Still, the physical configuration in other dimensions (forming an Euclidean space) keeps obeying the same laws of mechanics.

Thus the general laws of mechanics in space-time, are the same as those describing fixed objects in the Euclidean space.

Thus our familiarity with the latter (the mechanics of equilibrium) gives us the key to intuitively understand the former (as the change of underlying geometrical space, from the usual 3-dimensional Euclidean space to the 4-dimensional space-time of Special Relativity, essentially preserves the concepts).

Here is the law of equilibrium :

On a "space of all configurations" of a physical
system, a function of "potential energy" is defined (as the sum of
the potential energies contained in the different components and
interactions).

Then a configuration (a point of this space) is said to be at equilibrium when the variations of the potential energy around this configuration, cancels at the first order (with respect to any movement (distortion) of the system).

Then a configuration (a point of this space) is said to be at equilibrium when the variations of the potential energy around this configuration, cancels at the first order (with respect to any movement (distortion) of the system).

Here, to "cancel at the first order" means that the potential function has a zero derivative at this configuration, in any smooth movement of distortion that passes by it.

(For those who don't know what is a derivative : for a polynomial function

It results in conservation laws. Without difficulty we can express a part of these laws : the conservation of energy and kinetic momentum.

Its two-dimensional version can be easily (and "miraculously") expressed, but its general expression needs (in my opinion) the tool of tensor calculus. if we want to express not only the force vector but also momentums (do you seriously think that the usual presentation of screw theory, without tensor calculus, is nice ?).

Which is the dictionary translating the language of equilibrium into the one of relativistic mechanics ?

And how do the particles look like in terms of equilibrium physics ?

Here is the dictionary (but my English is not perfect so I can
make some mistakes in names):

The ** potential energy** in equilibrium physics is
translated by the

The relativistic particles are like elactics, with the property that the tension (norm of the force vector) is constant and called the

At first glance it seems good since so defined it is minimized by the movements of particles. A posteriori this choice of sign is unfortunate since action is the quadridimensional analog of potential energy, so it would be better to give it the same sign.

Energy measures the variation of action when stretching the space-time region in the temporal direction. Thus, it agrees with the contribution of the potential energy in the expression of action, because this contribution is prolonged during this stretch, proportionally to the duration (the same function is integrated over a longer duration). On the other hand, the link with kinetic energy takes the opposite sign because the temporal stretching of the space-time region does not prolong the motion but dilutes it, giving it a contribution to the action inversely proportional to the duration.

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