The Least Action Principle and Relativistic Mechanics
(This page, written long ago, is here as a draft to be reworked
and completed later).
The Least Action Principle is the fundamental framework
of classical mechanics. It is most naturally introduced in
complement to the space-time framework of Special Relativity, to
form relativistic mechanics, before reducing it to the approximation
of Galilean space-time, to form classical mechanics. But it can be
used to explain the Einstein field equation of General Relativity as
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
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 x ↦ a + bx + cx2
+ dx3, its variation around x = 0 is
given by the function bx + cx2 + dx3.
Here the first-order of variation is the term bx, since it
is "as small as x" when x is near 0, while the other terms are much
smaller : bx + cx2 + dx3
= x (b + cx + dx2) which
is roughly bx in first approximation.)
It results in conservation laws. Without difficulty we can express a
part of these laws : the conservation of energy and kinetic
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):
In dimension 3 there is the pressure of a gas (except that it is
affected by movements), and in dimension 4 it is the cosmological
The potential energy in equilibrium physics is
translated by the action in relativistic physics;
the differential of the restriction of this potential to the solid
moves (isometries at the first order near identity) of some
external support gives in equilibrium mechanics the torsor of the
force received from this support. The force vector
comes by still restricting this to translations; in the language
of relativistic mechanics it is called energy-momentum
quadrivector ; when this world apparently splits into
time and the 3-dimensional space, its time component is called energy,
and its space component (projection parallel to the time
direction) is called momentum vector.
The relativistic particles are like elactics, with the property
that the tension (norm of the force vector) is constant and called
the mass of the particle. This phenomenon of
constant tension of 1-dimensional objects also exists in larger
dimension: the surface of water also has a constant surface
tension. Such constant tensions (=negative pressures) unaffected
by movement, are associated with a density of energy.
Comment on the conventional sign of action
The action is usually defined as integral over time, of
(kinetic energy minus potential energy).
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.
Another comment I once wrote
The laws of classical mechanics have been explained as derived from the laws of relativistic mechanics, which come as made of 2 principles. One is that time is somehow just another dimension. The other is the least action principle. The least action principle is the space-time equivalent of the space-only law of equilibrium, which says that a state of equilibrium of a physical system (where nothing happens in the time dimension so that the expression of the law of mechanics is reduced to its aspects on the space dimensions here considered) if its potential energy is minimal. We can also consider unstable equilibrium where the potential energy is not minimal but its variations are zero at the first order of approximation with respect to small movements of the system. Consider a wire stretched between two poles. It is subject to gravity, but has to keep a constant length. So it forms a curve which is the one giving it the lowest center of mass among possible shapes with the same length. Each part of the wire (say, each centimeter length that is distinguished by thought and seen as attached to the rest at its two ends) is subject to gravity, that is a force oriented downwards, so that the equilibrium is reached when that part is just a little below the middle between its two attachment points. Each of the ends also exerts a force of pulling in the direction of the wire at the respective points. And these forces are at equilibrium, that is, the variation of the potential energy of the whole with respect to any given possible movement (of this part towards any direction), that is expressed as the sum of variations of potential energies of each part (this part, the left hand part, the right hand part) does not vary at the first order. Now think that this is a wire of electric supply in the countryside on the border of a railway, and you are in a train going along this wire. You look at it by the window, and what do you see ? You see the wire falling (accelerating) upwards. And at each passing pole, you see Bump ! the wire that was falling up is bounced downwards by the pole and now goes down, then is accelerated up again until the next pole. This way we recover the laws of classical mechanics in space-time by applying them in space then letting one space dimension play the role of time, but with some sign changes. Namely, uh, we can say that the pulling force along the wire plays the role of the mass, but a negative one (so that the sign of the acceleration is opposite to that of the force). The stronger the wire is stretched (stronger pulling force), the smaller the acceleration (curvature) for a given force (weight per length).
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