The Lagrangian one of the system can be written: $\backslash \; L\; =\; \backslash \; frac\; \{\backslash \; driven\; \backslash \; V^2\}\; \{2\}\; +\; \backslash \; L\_\; \{int\}$

and energy $\backslash \; E\; =\; \backslash \; frac\; \{\backslash \; driven\; \backslash \; V^2\}\; \{2\}\; +\; \backslash \; E\_\; \{int\}$

The reference frame to privilege to facilitate calculations is the reference frame of the center of mass whose impulse and the kinetic moment are those seen above.

Change of gauge

More precisely: let us suppose that $\backslash \; U\; (a.q\_1,\; a.q\_2,\dots .,\; a.q\_n)\; =\; a^k.U\; (q\_1,\; q\_2,\dots ,\; q\_n)$

A total change of gauge is a change of measurements: $\backslash \; Q\; \backslash \; to\; a.q$ and $\backslash \; T\; \backslash \; to\; b.t$

What means that: $\backslash \; frac\; \{Q\}\; \{q\text{'}\}\; =\; has$ and $\backslash \; frac\; \{T\}\; \{you\}\; =\; B$ where $\backslash \; Q$, $\backslash \; q\text{'}$ on the one hand, and $\backslash \; t$ and $\backslash \; t\text{'}$ of the other, is measurements different from the same physical or temporal distances.

Then $v\; =\; \backslash \; frac\; \{dq\}\; \{dt\}$ undergoes the change of measurement $\backslash \; v\; \backslash \; to\; \backslash \; frac\; \{has\}\; \{B\}\; .v$, therefore the kinetic energy undergoes a change of measurements of factor $\backslash \; frac\; \{a^2\}\; \{b^2\}$

So that the Lagrangian one is only multiplied by a constant number, it is necessary that: $\backslash \; frac\; \{a^2\}\; \{b^2\}\; =\; a^k$, i.e.: $\backslash \; B\; =\; a^\; \{1\; \backslash \; frac\; \{K\}\; \{2\}\}$

• For example with the Newtonian gravitational potential , one a: $\backslash \; K\; =\; -1$, therefore $\backslash \; B\; =\; a^\; \{\backslash \; frac\; \{3\}\; \{2\}\}$, i.e.: $\backslash \; q^3$ proportional to $\backslash \; t^2$, which corresponds to the 3 {{E}} law of Kepler.

In restricted Relativity

With or without quadri-writing

• In restricted relativity, the bodies evolve/move in the space time of Minkowski where each reference frame galiléen has its coordinates of space $\backslash \; x\_1;\; x\_2;\; x\_3$ and its coordinate of time $\backslash \; x\_0\; =\; c.t$, undergoing a whole a modification in the event of change of reference frame galiléen. Thus more absolute time ago, however the time of an unspecified reference frame, galiléen or not , always makes it possible to parameterize the evolution of a physical system.

While choosing to locate the system in an unspecified reference frame galiléen, therefore with the coordinates $\backslash \; (x\_0;\; x\_1;\; x\_2;\; x\_3)$, one can choose a time $t\_0$ of another unspecified reference frame, galiléen or not , to parameterize its evolution.

Lagrangian the $\backslash \; L\; =\; L\; (x\_0;\; x\_1;\; x\_2;\; x\_3;\; V\_0;\; V\_1;\; V\_2;\; V\_3)\; =\; L\; (x\_i;\; V\_i)$ expressed using the coordinates and speed can thus be written $\backslash \; L\; (x\_i;\; V\_i)\; =\; L\; (x\_i\; (t\_0);\; V\_i\; (t\_0))$, with $V\_i\; =\; \backslash \; frac\; \{dx\_i\}\; \{dt\_0\}$.

• If one chooses $t\_0\; =\; \backslash \; frac\; \{x\_0\}\; \{C\}\; ~~$ the time of the reference frame of the coordinates, the Lagrangian one and the equations which of it are drawn give, with the approximation at the low speeds in front of $c~$, the Lagrangian one and the properties of traditional mechanics. One will then say to work without the quadri-writing bus only the space coordinates appear in general.

• If one chooses $t\_0\; =~~$ the clean time, with $dt\_0\; =\; \backslash \; frac\; \{ds\}\; \{C\}\; ~~$, one obtains equivalent results but whose writing is considered to be more elegant and approaches that of general relativity. It will be noticed that a clean reference mark is galiléen only if the body is free. With clean time, one will say to work in quadri-writing bus the four coordinates of the reference frame appear in calculations.
• If one chooses $t\_0$ an unspecified different time, one can work more easily with the derivative partial than by using two other previous times; but the results, although equivalents, have a less handy and less elegant writing. In this case, one will also say to work in quadri-writing, for the same reason.

; With the quadri-writing

By convenience, we will adopt the Convention of summation of Einstein in the space of Minkowski: for two quadri-vectors $\backslash \; (V\_0;\; V\_1;\; V\_2;\; V\_3)\; \backslash$ and $\backslash \; (U\_0;\; U\_1;\; U\_2;\; U\_3)$, one defines the scalar product $\backslash \; V^iU\_i$ by $V^iU\_i\; =\; \backslash \; Sigma\_\; \{i=0\}\; ^3V^i.U\_i\; =\; V\_0.U\_0-V\_1.U\_1-V\_2.U\_2-V\_3.U\_3\; =\; V\_iU^i$, with $\backslash \; V^0\; =\; V\_0$ and for i=1; 2; 3 $\backslash \; V^i\; =\; -\; V\_i$

One has then: $\backslash \; (V\_0)\; ^2\; -\; (V\_1)\; ^2\; -\; (V\_2)\; ^2\; -\; (V\_3)\; ^2\; =\; V^iV\_i$

It is shown that: $\backslash \; frac\; \{\backslash \; partial\; V\_jV^j\}\; \{\backslash \; partial\; V\_i\}\; =\; 2V^i$

In a similar way, one will write: $\backslash \; partial^i\; =\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x\_i\}$ and $\backslash \; partial\_i\; =\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; x^i\}$

By using an unspecified unspecified time $\backslash \; t\_0$, the action $\backslash \; S\; =\; \backslash \; int\_\; \{t\_\; \{0i\}\}\; ^\; \{t\_\; \{0f\}\}\; L\_0\; dt\_0$ makes it possible to obtain the equations of Euler-Lagrange, relativists but obtained same manner as in the traditional case, with coordinates moreover:

$\backslash \; frac\; \{d~~\}\; \{dt\_0\}\; \backslash \; frac\; \{\backslash \; partial\; L\_0\}\; \{\backslash \; partial\; V\_j\}\; \backslash \; -\; \backslash \; \backslash \; frac\; \{\backslash \; partial\; partial\; L\_0\}\; \{\backslash \; x\_j\}\; \backslash \; =\; \backslash \; 0~~$ for j=0; 1; 2; 3

It is important to notice that as in the traditional case, the action and the Lagrangian one are not defined in a single way: the action is defined in the addition close to a function of the ends of the way and time, and the Lagrangian one is defined in the addition close to derived from a function from the time (which once integrated gives a function of the ends and time).

Case of a free body

Lagrangian of a free body

In any reference frame galiléen quadri-speed is not null because the body advances at least in temporal dimension.

The Lagrangian relativist of a free body must, at the low speeds and at first approximation, being equal (perhaps to an additive constant near: the addition of a constant does not change the equations of Euler-Lagrange) to Lagrangian traditional.

In the space time of Minkowski, the action determines the trajectory, and this one does not depend on the reference frame from where it is observed. Thus the action does not depend on the coordinates, and, for a free body, depends only on speed and is invariant by the transformations of Lorentz:

$\backslash \; S\; =\; \backslash \; int\_\; \{t\_i\}\; ^\; \{t\_f\}\; L\; dt$ is invariant by the transformations of Lorentz

In the clean reference frame of the body, $\backslash \; dt\; =\; dt\_0$ is the variation of the clean time of the body; $\backslash \; L\; =\; L\_0$ and the space speed of the body is null. In a reference frame galiléen, and with the assumption that the body is free, quadri-speed is constant in time (and is never null) thus the Lagrangian one also because it depends on the only speed.

Thus, in the clean reference frame of the Lagrangian body characteristic, $\backslash \; L\_0$, is a constant in time.

Seen since another reference frame galiléen, moving compared to the reference frame suitable for space speed $\backslash \; v$ constant, one a: $\backslash \; dt\_0\; =\; \backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\; .dt$

Thus: $\backslash \; S\; =\; \backslash \; int\_\; \{t\_i\}\; ^\; \{t\_f\}\; L\_0.\; \backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\; dt$

where $\backslash \; v\; =$ relative space speed enters the reference frame and the clean reference frame of the body = space speed of the body the reference frame.

Thus: $\backslash \; L\; =\; L\_0.\; \backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\; \backslash \; approx\; \backslash \; L\_0.\; (1\; -\; \backslash \; frac\; \{v^2\}\; \{2c^2\})\; =\; L\_0\; -\; \backslash \; frac\; \{L\_0\}\; \{2c^2\}\; v^2$ by the approximation at the low speeds in front of $\backslash \; c$.

While comparing with Lagrangian traditional the $\backslash \; L\; =\; \backslash \; frac\; \{1\}\; \{2\}\; mv^2$ (which is not really modified by the addition of the constant $L\_0$), one obtains: $-\; \backslash \; frac\; \{L\_0\}\; \{2c^2\}\; =\; \backslash \; frac\; \{1\}\; \{2\}\; m$, from where $\backslash \; L\_0\; =\; -\; mc^2$

Conclusion: in an unspecified reference frame galiléen, the Lagrangian one is where $\backslash \; v$ is the space speed of the body in this reference frame.

Impulse and energy

• By definition of the impulse $\backslash \; vec\; \{p\}$, one a: $\backslash \; vec\; \{p\}\; =\; \backslash \; frac\; \{\backslash \; partial\; L\}\; \{\backslash \; partial\; \backslash \; vec\; \{v\}\}\; =\; \backslash \; frac\; \{m\; \backslash \; vec\; \{v\}\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}$

energy is defined by: $\backslash \; E\; =\; \backslash \; vec\; \{p\}.\; \backslash \; vec\; \{v\}\; -\; L$

One obtains: $\backslash \; E\; =\; \backslash \; frac\; \{mc^2\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}$

In particular, for $v=0$, the energy at rest is $\backslash \; E\; =\; mc^2$

• By expressing energy according to the impulse, one obtains: $\backslash \; E^2\; =\; p^2.c^2\; +\; m^2.c^4$ or $\backslash \; m^2.c^4\; =\; E^2\; -\; p^2.c^2$

• One will notice that although having the dimension of an energy, the Lagrangian relativist is not the kinetic energy : the latter is worth

$\backslash \; E\_c\; =\; \backslash \; frac\; \{mc^2\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}\; -\; mc^2\; =\; mc^2.\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}\; -\; 1\; \backslash \; right)$

One has well $\backslash \; E\_c\; \backslash$ ≈ $\backslash \; frac\; \{1\}\; \{2\}\; m\; \backslash \; v^2\; \backslash$ with the approximation at the low speeds in front of $\backslash \; c$

With the quadri-writing

• One notes that $\backslash \; L.dt\; =\; -\; mc^2.\; \backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\; .dt\; =\; -\; mc.\; \backslash \; sqrt\; \{(c.dt)\; ^2-\; (dx\_1)\; ^2-\; (dx\_2)\; ^2-\; (dx\_3)\; ^2\}\; =\; -\; mc\; \backslash \; sqrt\; \{dx\_idx^i\}\; =\; -\; mc.ds$, by using the equality $v\_i\; =\; \backslash \; frac\; \{dx\_i\}\; \{dt\}$ and the adequate definition of $\backslash \; ds$, called “clean time” of the body.

By using the fact that $\backslash \; ds\; =\; \backslash \; sqrt\; \{dx\_idx^i\}$ is the clean time of the body, the action minimized between two points of the space time $\backslash \; S\; =\; -\; mc\; \backslash \; int\_A^Bds$ watch that the way followed by the particle to go from the point has at the point B is that which minimizes clean time .

While factorizing by $\backslash \; dt\_0$, an unspecified time parameterizing the system (and is thus not obligatorily clean time), one obtains: $\backslash \; L.dt\; =\; -\; mc.\; \backslash \; sqrt\; \{V\_iV^i\}\; .dt\_0\; =\; L\_0.dt\_0$

, by using quadri-speed not obligatorily clean $\backslash \; V\; =\; (V\_0;\; V\_1;\; V\_2;\; V\_3)$ defined by: $\backslash \; V\_i\; =\; \backslash \; frac\; \{dx\_i\}\; \{dt\_0\}$, with $\backslash \; x\_0\; =\; c.t$

the Lagrangian relativist of a free particle, parameterized by unspecified time $\backslash \; t\_0$, is thus expressed:

• One remembers that the quadri-impulse, as the impulse, is defined by $P^i\; =\; \backslash \; frac\; \{\backslash \; partial\; L\_0\}\; \{\backslash \; partial\; V\_i\}$

From where: $P^i\; =\; -\; mc\; \backslash \; frac\; \{V^i\}\; \{\backslash \; sqrt\; \{V^jV\_j\}\}$

For $i=0$, one obtains: $P^0\; =\; -\; mc.\; \backslash \; frac\; \{(\backslash \; frac\; \{c.dt\}\; \{dt\_0\})\}\{(\backslash \; frac\; \{ds\}\; \{dt\_0\})\}\; =\; -\; mc.\; \backslash \; frac\; \{c.dt\}\; \{ds\}\; =\; -\; \backslash \; frac\; \{mc\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}\; =\; -\; \backslash \; frac\; \{E\}\; \{C\}$

For $i=1;\; 2;\; 3$, in a way similar to the case i=0, one obtains: $(P^1,\; P^2;\; P^3)\; =\; \backslash \; frac\; \{m\; \backslash \; vec\; \{v\}\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}\; =\; \backslash \; vec\; \{p\}$.

the square of the " norme" quadri-impulse is $\backslash \; P^jP\_j\; =\; (P\_0)\; ^2-\; (P\_1)\; ^2-\; (P\_2)\; ^2-\; (P\_3)\; ^2\; =\; \backslash \; frac\; \{(-\; E)\; ^2\}\; \{c^2\}\; -\; (\backslash \; vec\; \{p\})\; ^2$, and also $\backslash \; P^iP\_i\; =\; m^2.c^2.\; \backslash \; frac\; \{V^iV\_i\}\; \{(\backslash \; sqrt\; \{V^jV\_j\}\; \backslash )\; ^2\}\; =\; m^2.c^2$

From where the formula already seen: $\backslash \; m^2.c^4\; =\; E^2\; -\; p^2.c^2$

• the constancy of the quadri-impulse, shown starting from the equations of Euler-Lagrange, makes it possible to show that energy E and the space impulse are constant compared to time $\backslash \; t\_0~$.

• the constant $\backslash \; P^iV\_i-L$ compared to time $\backslash \; t\_0~$ is in fact constant 0; a small handling makes it possible to deduce the equality already seen from it $\backslash \; m^2.c^4\; =\; E^2\; -\; p^2.c^2$

• One shows easily that whatever the time $\backslash \; t\_0~$ chosen, if it are that of a reference mark galiléen, speed $\backslash \; V\_i$ and the " pseudo-norme" $\backslash \; sqrt\; \{V^jV\_j\}$ is constant compared to time $\backslash \; t\_0~$: it is a direct consequence of the definition of the reference marks galiléens, and owing to the fact that the body is free.

• In the particular case where $\backslash \; dt\_0$ is clean time $\backslash \; frac\; \{ds\}\; \{C\}$, then quadri-speed and quadri-impulse are clean, and one with the particular equality $\backslash \; sqrt\; \{V^jV\_j\}\; =\; \backslash \; frac\; \{ds\}\; \{dt\_0\}\; =\; C.\; \backslash \; frac\; \{ds\}\; \{ds\}\; =\; c$, which can be embarrassing for the use of the derivative partial in the work above, and which gives $\backslash \; P^0\; =\; -\; m.V^0\; =\; -\; \backslash \; frac\; \{E\}\; \{C\}$ and for i=1; 2; 3 $\backslash \; P^i\; =\; m.V^i\; =\; p\_i$.

Case of a body in a electromagnetic Field

Without the quadri-writing

$L\; =\; L\; (\backslash \; vec\; \{Q\},\; \backslash \; vec\; \{v\},\; T)\; =\; \backslash \; -\; m.c^2.\; \backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\; -\; E\; \backslash \; phi\; (\backslash \; vec\; \{Q\},\; T)\; +\; E.\; \backslash \; vec\; \{v\}\; \backslash \; cdot\; \backslash \; vec\; \{has\}\; (\backslash \; vec\; \{Q\},\; T)$

By still taking $\backslash \; vec\; \{p\}\; =\; \backslash \; frac\; \{m\; \backslash \; vec\; \{v\}\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}$, the equation of Euler-Lagrange gives:

The impulse is defined by: $\backslash \; vec\; \{P\}\; =\; \backslash \; frac\; \{\backslash \; partial\; L\}\; \{\backslash \; partial\; \backslash \; vec\; \{v\}\}\; =\; \backslash \; frac\; \{m\; \backslash \; vec\; \{v\}\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}\; +\; E.\; \backslash \; vec\; \{has\}\; =\; \backslash \; vec\; \{p\}\; +\; E.\; \backslash \; vec\; \{has\}$

One will thus take care to distinguish $\backslash \; vec\; \{p\}$ and $\backslash \; vec\; \{P\}$

Energy is defined by: $\backslash \; E\; =\; \backslash \; vec\; \{P\}.\; \backslash \; vec\; \{v\}\; -\; L$

One obtains: $\backslash \; E\; =\; \backslash \; frac\; \{mc^2\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}\; +\; E.\; \backslash \; phi$

And after some calculations to express energy according to the impulse: $\backslash \; (E-e\; \backslash \; phi)\; ^2\; =\; \backslash \; left\; (\backslash \; vec\; \{P\}\; -\; E.\; \backslash \; vec\; \{has\}\; \backslash \; right)\; ^2.c^2\; +\; m^2.c^4$

All the approximations at the low speeds in front of $c$ give again the traditional results.

Starting from the electromagnetic potential, the first group of the Maxwell's equations is shown without difficulty: the equation of Maxwell-Faraday and the conservation equation of the magnetic flux.

With the quadri-writing

The definition of the infinitesimal relativistic action of an electromagnetic field is thus $\backslash \; L.dt\; =\; \backslash \; -\; mc.\; \backslash \; sqrt\; \{dx\_j.dx^j\}\; -\; e.A^j.dx\_j$.

One poses $F^\; \{ij\}\; =\; \backslash \; partial^i\; A^j\; -\; \backslash \; partial^j\; tensor\; A^i$ electromagnetic field.

By taking $\backslash \; t\_0$ the clean time of the particle, the equations of Euler-Lagrange give the equations of the movement of the particle:

That one can also write:

While taking: electric field = $\backslash \; vec\; \{E\}\; =\; C.\; \backslash \; left\; (F^\; \{01\},\; F^\; \{02\},\; F^\; \{03\}\; \backslash \; right)$ and magnetic field = $\backslash \; vec\; \{B\}\; =\; \backslash \; left\; (F^\; \{23\},\; F^\; \{31\},\; F^\; \{12\}\; \backslash \; right)$, one finds the Force of Lorentz under its usual writing.

• One remembers that the quadri-impulse, as the impulse, is defined by $P^i\; =\; \backslash \; frac\; \{\backslash \; partial\; L\_0\}\; \{\backslash \; partial\; V\_i\}$

From where: $P^i\; =\; -\; mc\; \backslash \; frac\; \{V^i\}\; \{\backslash \; sqrt\; \{V^jV\_j\}\}\; -\; e.A^i$

For $i=0$, one obtains: $P^0\; =\; -\; mc.\; \backslash \; frac\; \{(\backslash \; frac\; \{c.dt\}\; \{dt\_0\})\}\{(\backslash \; frac\; \{ds\}\; \{dt\_0\})\}\; -\; e.A^0\; =\; -\; mc.\; \backslash \; frac\; \{c.dt\}\; \{ds\}\; -\; E.\; \backslash \; frac\; \{\backslash \; phi\}\; \{C\}\; =\; -\; \backslash \; frac\; \{mc\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}\; -\; \backslash \; frac\; \{E\}\; \{C\}.\; \backslash \; phi\; =\; -\; \backslash \; frac\; \{E\}\; \{C\}$

For $i=1;\; 2;\; 3$, in a way similar to the case i=0, one obtains: $(P^1,\; P^2;\; P^3)\; =\; \backslash \; frac\; \{m\; \backslash \; vec\; \{v\}\}\; \{\backslash \; sqrt\; \{1\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}\; +\; E.\; \backslash \; vec\; \{has\}\; =\; \backslash \; vec\; \{p\}\; +\; E.\; \backslash \; vec\; \{has\}\; =\; \backslash \; vec\; \{P\}$.

In a way similar to the case of a free body, the constant $\backslash \; P^iV\_i-L$ compared to time $\backslash \; t\_0~$ is in fact constant 0, and a small handling makes it possible to deduce the equality already seen from it $\backslash \; m^2.c^4\; =\; (E\; -\; E.\; \backslash \; phi)\; ^2\; -\; \backslash \; left\; (\backslash \; vec\; \{P\}\; -\; E\; \backslash \; vec\; \{has\}\; \backslash \; right)\; ^2.c^2$

The invariance of gauge of the potential and the tensor electromagnetic

By applying the variational method which varies the way by keeping the fixed ends, the term $\backslash \; \backslash \; phi\; (x\_i\; (t\_f))\; -\; \backslash \; phi\; (x\_i\; (t\_i))$ is eliminated. Thus the two potentials $\backslash \; A^j$ and $\backslash \; A\text{'}^j\; =\; A^j\; +\; \backslash \; partial^j\; \backslash \; phi$ give the same equations of the movement: one calls that the “invariance of gauge”.

It is noted besides that in the equations of the movement, the electromagnetic tensor, term representing the influence of the electromagnetic field, is well invariant of gauge: $F\text{'}^\; \{ij\}\; =\; \backslash \; partial^i\; A\text{'}^j\; -\; \backslash \; partial^j\; A\text{'}^i\; =\; \backslash \; partial^i\; A^j\; -\; \backslash \; partial^j\; A^i\; +\; \backslash \; partial^i\; \backslash \; partial^j\; \backslash \; phi\; \backslash \; partial^j\; \backslash \; partial^i\; \backslash \; phi\; =\; \backslash \; partial^i\; A^j\; -\; \backslash \; partial^j\; A^i\; =\; F^\; \{ij\}$

by the theorem of Schwarz: $\backslash \; partial^i\; \backslash \; partial^j\; =\; \backslash \; partial^j\; \backslash \; partial^i$.

Case of a field “of force”

Thus, between the influential body and the infuencé body, it ballade something in space, in general with the speed of light, which répend in the space and of which the effect is a change of trajectory of the influenced body.

One can answer these questions using the principle of less action.

Lagrangian density and associated equations of Euler-Lagrange

• a field is characterized by an important extent in space, one cannot thus locate it by the coordinates $(T,\; \backslash \; vec\; \{Q\},\; \backslash \; vec\; \{v\})$, but one can locate it (or rather quantify it) by his projections $\backslash \; (A\_0,\; A\_1,\; A\_2,\; A\_3)$ on the axes $(x\_0,\; x\_1,\; x\_2,\; x\_3)$ and by the variations $\backslash \; frac\; \{\backslash \; partial\; partial\; A\_i\}\; \{\backslash \; x\_j\}\; =\; \backslash \; partial^j\; A\_i$ of his projections (by supposing by advance that we will be able to deduce the derivative from it second, like in the case of a body localized).

We will thus use $\backslash \; (A\_0,\; A\_1,\; A\_2,\; A\_3)$ and $\backslash \; frac\; \{\backslash \; partial\; partial\; A\_i\}\; \{\backslash \; x\_j\}\; =\; \backslash \; partial^j\; A\_i$, with I, J ∈ {0,1,2,3} for a field in the same way as the quadri-coordinates and quadri-speed for a body located, the coordinates $(x\_0,\; x\_1,\; x\_2,\; x\_3)$ playing the part of parameters, as only time made it front.

• the action of a field is thus of the form: $S\; =\; \backslash \; int\_V\; \backslash \; Lambda\; (A\_i;\; \backslash \; partial^jA\_i)\; D\; \backslash \; Omega$

Where V is the quadri-volume in which one will apply the variational method, $\backslash \; \backslash \; Lambda$ is called the “Lagrangian density” and $\backslash \; D\; \backslash \; Omega\; =\; dx\_0.dx\_1.dx\_2.dx\_3\; =\; c.dt.dx\_1.dx\_2.dx\_3$

• By a demonstration similar to that already seen in the case of a body localisable, and by using the convention of summation of Einstein, one obtains the equations of Euler-Lagrange for the Lagrangian density:

Tensor impulse-energy of a field

while posing

tensor “impulse-energy”, one a: what expresses its conservation.

While posing: $\backslash \; W\; =\; T^0\_\; \{~0\}\; =\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; partial\; Lambda\}\; \{\backslash \; (\backslash \; partial\_0A\_j)\}.\; \backslash \; partial\_0A\_j\; -\; \backslash \; Lambda$ density of energy and $P^k\; =\; c.T^k\_\; \{~0\}\; =\; partial\; C.\; \backslash \; frac\; \{\backslash \; \backslash \; partial\; Lambda\}\; \{\backslash \; (\backslash \; partial\_kA\_j)\}.\; \backslash \; partial\_0A\_j$ for k∈ {1; 2; 3} components of the vector $\backslash \; vec\; P$.

We have the two equivalent equations then

who is the “conservation equation of energy”: locally, the variation in the time of the density of energy $\backslash \; w$ is equal to opposite of the variation of density of impulse by the space components $\backslash \; P\_i$.

Lagrangian density of a free electromagnetic field

Equations of the electromagnetic field

The Lagrangian density to use is:

The equations of Euler-Lagrange give:

For K = 0, one obtain $\backslash \; mathrm\; \{div\}\; \backslash \; \backslash \; overrightarrow\; \{E\}\; \backslash \; =\; \backslash \; \backslash \; frac\; \{\backslash \; rho\}\; \{\backslash \; varepsilon\_0\}$ the equation of Maxwell-Gauss or conservation equation of the load.

For k∈ {1; 2; 3} one obtains $\backslash \; overrightarrow\; \{\backslash \; mathrm\; \{belch\}\}\; \backslash \; \backslash \; overrightarrow\; \{B\}\; \backslash \; =\; \backslash \; \backslash \; mu\_0\; \backslash \; overrightarrow\; \{J\}\; \backslash \; +\; \backslash \; \backslash \; varepsilon\_0\; \backslash \; mu\_0\; \backslash \; \backslash \; frac\; \{\backslash \; partial\; \backslash \; overrightarrow\; \{E\}\}\; \{\backslash \; partial\; T\}$ the equation of Maxwell-Amp.

Moreover, starting from $\backslash \; partial^i\; F\_\; \{ik\}\; =\; \backslash \; mu\_0.J\_k$, and by using the anti-symmetry of $\backslash \; F\_\; \{ik\}$ and the theorem of Schwarz ($~~\; \backslash \; partial^\; \{ik\}\; =\; \backslash \; partial\; ^\; \{ki\}\; ~~$), one obtains: $\backslash \; mu\_0.\; \backslash \; partial^k\; J\_k\; =\; -\; \backslash \; mu\_0.\; \backslash partial^i\; J\_i$

From where two presentations of “the conservation equation of the load”:

Tensor impulse-energy of the electromagnetic field

In General relativity

One owes with David Hilbert, in 1916, the first use of the principle of less action to obtain the equations of general relativity, in particular the equations of the gravitational field.

For general relativity also, the equations can be obtained without calling upon the principle of less action: the Principe of equivalence, expressed in the form “one can always find a reference frame cancelling the field of gravitation locally”, makes it possible to directly find the equations of the movement of a particle; and the unicity of the shape of the geometrical tensor which is cancelled by the derivative covariante, unicity proven by Élie Cartan, makes it possible to find the equations of the field of gravitation, which was the original method of Einstein (although unicity in question was not proven yet at the time).

If the equations of general relativity are given, one can deduce the action from it allowing to apply the principle. In particular, with the equations of geodetic one can find metric the $ds^2\; \backslash ,$ associated.

Particle

Particle in a field of Gravitation

Within the framework of restricted relativity, by taking a reference frame accelerated (coordinated $\backslash \; (x\text{'}\_0;\; x\text{'}\_1;\; x\text{'}\_2;\; x\text{'}\_3)$), local perception is thus a field of gravitation, and the change of reference frame compared to an inertial reference frame (coordinates $\backslash \; (x\_0;\; x\_1;\; x\_2;\; x\_3)$) imposes a Métrique on the noncommonplace coefficients: $\backslash \; ds^2\; =\; (x\_0)\; ^2-\; (x\_1)\; ^2-\; (x\_2)\; ^2-\; (x\_3)\; ^2\; =\; g^\; \{ij\}\; (x\text{'})\; x\text{'}\_ix\text{'}\_j$. It is enough to determine the equations of the movement in this reference frame because of the principle of less action in restricted relativity.

The principle of equivalence makes it possible to say that a real gravitational field (nondue to the choice of the reference frame) is also determined by metric the $\backslash \; ds^2$ (and the metric one is determined by the field of gravitation); although the use of metric a $\backslash \; ds^2\; =\; g^\; \{ij\}\; (X)\; x\_ix\_j\; =\; g^\; \{ij\}\; x\_ix\_j$ which is not caused, and thus not compensable beyond a local field of the space time, by a change of reference frame implies that the space time is not Euclidean (see the experiment by the thought of the disc in rotation, described in General relativity), and that one leaves then the framework of relativity restricted to build a new theory: the General relativity.

One can thus remain in the continuity of restricted relativity, and affirm that the infinitesimal action of a specific particle, influenced by the only gravitation, in general relativity is:

where it is supposed that $\backslash \; g^\; \{ij\}\; =\; g^\; \{ji\}$ without nothing to remove with the general information.

By using the fact that $\backslash \; ds\; =\; \backslash \; sqrt\; \{g^\; \{ij\}\; dx\_idx\_j\}$ is the clean time of the particle, the action minimized between two points of the space time $\backslash \; S\; =\; -\; mc\; \backslash \; int\_A^Bds$ watch that it is clean time to go from the point has at the point B which is minimized (locally) by the principle. the geodetic ones are the ways which minimize (locally) the clean time of the particle .

To keep physical coherence, one needs to suppose that the $\backslash \; g^\; \{ij\}$ are continuous; to be able to work with known tools, i.e. derivations, but to as suppose as the gravitational field is continuous, one must suppose that they are differentiable. Thereafter, for the equations of Einstein, it will be essential to suppose that they are C.

By considering an unspecified time $t\_0$:

One always uses the equations of Euler-Lagrange $\backslash \; \backslash \; frac\; \{d~~\}\; \{dt\_0\}\; \backslash \; frac\; \{\backslash \; partial\; L\_0\}\; \{\backslash \; partial\; V\_k\}\; \backslash \; -\; \backslash \; \backslash \; frac\; \{\backslash \; partial\; partial\; L\_0\}\; \{\backslash \; x\_k\}\; \backslash \; =\; \backslash \; 0~~$ after having divided by the coefficient $\backslash \; -\; mc$ here useless.

The equation is obtained:

that one can also write:

or:

with the “derivative covariante”: $DV\_k\; =\; dV\_k\; +\; \backslash \; Gamma\_k^\; \{ij\}\; V\_idV\_j$ and $DV^k\; =\; dV^k\; +\; \backslash \; Gamma^k\_\; \{ij\}\; V^idV^j$, where $\backslash \; V\_k\; =\; \backslash \; frac\; \{dx\_k\}\; \{dt\_0\}$ for $\backslash \; t\_0\; =$ clean time.

The symbol of Christoffel $\backslash \; Gamma\_k^\; \{ij\}$ is essential like the manifestation of the gravitation in the equations of the movement.

The equations of the movement do not depend on the mass of the particle (named thus because we neglected its space extent and its influence on its environment): all the particles follow the same trajectories (in identical initial conditions), it is the equation of geodetic in general relativity, in the presence of the only gravitation.

However, these equations of the movement are not valid for a particle of null mass because in this case, one has upon the departure $~dS\; =\; 0~~$, which prohibits all calculations carried out above; there is also $~ds\; =\; c.dt\_0\; =\; 0~~$ because clean time does not pass for a particle of null mass (see restricted Relativité), the term $\backslash \; dowry\; \{V\}\; \_m$ cannot in no case to have direction. It is necessary to consider the wave associated with the particle to have an equation having a direction, moreover the Lumière was included/understood like a wave (electromagnetic) and a particle (the Photon, of null mass) when general relativity was written.

Particle in a electromagnetic Field

By perfectly similar calculations, one draws the equations from them from the movement:

that one can write:

or:

Field of Gravitation

In order to determine the Lagrangian density of it, then the equations, it is necessary to develop a little certain considerations approached above, and even some news.

Lagrangian density in curved space

By noting $\backslash \; \backslash \; Lambda$ the scalar of the field, invariant compared to the changes of reference frames, the Lagrangian density will be: $\backslash \; L\; =\; \backslash \; Lambda.\; |G|^\; \{\backslash \; frac\; \{1\}\; \{2\}\}$

Definitions of the Tensor S of Riemann, Ricci, and the Curve

In mathematical terms, the four-dimensional space defined by the considerations above is a variety C² where quadri-speeds are vectors belonging to the tangent vector space at the point where one derived, this vector space being provided with metric the $\backslash \; g^\; \{ij\}$.

Let us recall that the coordinates $(x\_0;\; x\_1;\; x\_2;\; x\_3)$ is the coordinates of the points of the variety, provided with an unspecified frame of reference, representing the arbitrary choice of the physical reference frame of the observer.

The measurement of the gravitation, which influences the geodetic ones, can be done through the difference in orientation between two vectors resulting from transport from only one vector from origin by two different geodetic ways towards the same final point.

• the equation of geodetic the $\backslash \; dowry\; \{V\}\; \_m\; +\; \backslash \; Gamma\_m^\; \{ij\}\; V\_iV\_j\; =\; 0$ is equivalent to $\backslash \; frac\; \{dV\_k\}\; \{dt\_0\}\; =\; -\; \backslash \; Gamma\_k^\; \{ij\}\; V\_i\; V\_j$.

Owing to the fact that $V\_j\; =\; \backslash \; frac\; \{dx\_j\}\; \{dt\_0\}$, one deduces: $dV\_k\; =\; -\; \backslash \; Gamma\_k^\; \{ij\}\; V\_i\; dx\_j$; knowing that one has $\backslash \; Gamma\_k^\; \{ij\}\; =\; \backslash \; Gamma\_k^\; \{ji\}$ as one sees it starting from his definition, one could as well write $dV\_k\; =\; -\; \backslash \; Gamma\_k^\; \{ij\}\; dx\_i\; V\_j$.

In a similar way, one obtains $dV^k\; =\; -\; \backslash \; Gamma^k\_\; \{ij\}\; V^i\; dx^j$

• a vector $\backslash \; left\; (A\_i\; \backslash \; right)$ is known as in parallel transported along geodetic if the variations of its coordinates check $dA\_k\; =\; -\; \backslash \; Gamma\_k^\; \{ij\}\; A\_i\; dx\_j$ when it is moved of $\backslash \; (dx\_j)\; \_\; \{j=0;\; 1;\; 2;\; 3\}$ along the geodetic one.

• One defines the tensor Riemann by:

• the tensor of Ricci is a contraction of the tensor of Riemann: $R^\; \{ij\}\; =R\_k^\; \{I,\; kj\}$

• the Courbure riemannienne is the number obtained by contraction of the tensor of Ricci: $\backslash \; R=g\_\{ij\}R^\{ij\}$

• All equalities used in “ details of the method of Élie Cartan ” being independent of the reference frame chosen, and it is also the case for the definitions of the tensors of Riemann and Ricci (it is besides why one allows oneself to name them Tenseur ). It is also the case of the curve $\backslash \; R$ which is thus candidate to be $\backslash \; \backslash \; Lambda$ the scalar invariant of the field of gravitation.

• Élie Cartan showed that the scalars invariants by change of reference frame are form $\backslash \; \backslash \; alpha\; R\; +\; \backslash \; beta~$.

Analytical tools

An application of the Principle of inertia in curved space

• the tangent vector spaces (of dimension 4) are provided with their “natural” base {$\backslash \; \backslash \; \{\backslash \; vec\; \{E\}\; ^\; \{~0\};\; \backslash \; vec\; \{E\}\; ^\; \{~1\};\; \backslash \; vec\; \{E\}\; ^\; \{~2\};\; \backslash \; vec\; \{E\}\; ^\; \{~3\}\}$}: if $\backslash \; M\; (x\_0;\; x\_1;\; x\_2;\; x\_3)$ is the point where tangent space is considered, one poses $\backslash \; vec\; \{E\}\; ^\; \{~i\}\; =\; \backslash \; left\; (~\; \backslash \; frac\; \{\backslash \; partial\; x\_j\}\; \{\backslash \; partial\; x\_i\}\; ~\; \backslash \; right)\; \_\; \{j=0,1,2,3\}$; what one often writes $\backslash \; vec\; \{E\}\; ^\; \{~i\}\; =\; \backslash \; frac\; \{\backslash \; partial\; ~\}\; \{\backslash \; partial\; x\_i\}$.

the equations of geodetic are properties concerning the coordinates $\backslash \; frac\; \{dx\_i\}\; \{dt\_o\}$ or $\backslash \; frac\; \{dx\_i\}\; \{ds\}$ quadri-speed along this trajectory, they do not give an indication for the variation (derivation) of a quadri-vector $\backslash \; vec\; \{E\}\; ^\; \{~i\}$ of a point to another of space, nor even for the derivation of the quadri-vector speed $\backslash \; vec\; V\; =\; V\_i\; \backslash \; vec\; e^\; \{~i\}$.

For that, we can use a physical principle rewritten to measure for general relativity:

• Principle of inertia: along geodetic, and in the absence of external intervention, it (quadri-) Flight Path Vector of a particle is constant.

By analyzing the equations of geodetic or by taking account of the fact that the “axes” of the coordinates are not obligatorily the geodetic ones, one cannot affirm that the coordinates of the quadri-vector speed are constant.

The derivative covariante

There a: $d\; \backslash \; vec\; \{is\}\; (X)\; =\; (dA\_i)\; \backslash \; vec\; e^\; \{~i\}\; +\; A\_i\; D\; (\backslash \; vec\; e^\; \{~i\})\; =\; (\backslash \; partial^j\; A\_i\; +\; A\_k\; \backslash \; Gamma\_i^\; \{jk\})\; \backslash \; vec\; e^\; \{~i\}\; dx\_j\; =\; D^j\; A\_i.\; \backslash vec\; e^\{~i\}dx\_j$

By defining the derived covariante by:

Property:

And so on with all the indices of a tensor, according to their positions.

Where one finds the tensors of Riemann, etc

One thus obtains the concepts already introduced “with the manner of Élie Cartan”.

Useful equalities and properties

• Theorem of Ricci: $\backslash \; D\_kg^\; \{ij\}\; =\; 0~\; \backslash \; quad\; ~$ and $~\; \backslash \; quad\; D\_kg\_\; \{ij\}\; =\; 0~$

• By posing $\backslash \; G\; =\; \backslash \; det\; (g^\; \{ij\})\; \backslash \; qquad$, one a: $|G|\; =\; -\; G\; \backslash \; qquad\; g^\; \{ij\}\; .g\_\; \{ij\}\; =\; \backslash \; delta^i\_i\; =\; 4\; \backslash \; qquad\; \backslash \; dg\; =\; g~g\_\; \{ij\}\; ~dg^\; \{ij\}$

• Theorem of Ostrogradski: $\backslash \; int\_V\; \backslash \; sqrt\; \{-\; G\}\; ~\; D\_iA^i~d\; \backslash \; Omega\; =\; \backslash \; oint\_\; \{\backslash \; share\; V\}\; \backslash \; sqrt\; \{-\; G\}\; A^i~dS\_i$, when $\backslash \; A^i$ is a tensor.

• the sum, the difference and the summation of Einstein of Tenseur S defined in same the tangent Espace give a tensor; on the other hand if they are tensors defined in different tangent spaces, it is not sure that gives a tensor.

• a tensorial equality shown in an unspecified point, but by using a particular reference frame, is a true equality in this point and for all the reference frames: it is the principal interest there to use tensors.

Equations of Einstein of the field of gravitation in the external case

The first case of the equations of the fields is the case where there is no matter (locally): one speaks about the “case external”, under heard “with the matter”.

In this case, the only component of the action is the component of the gravitational field $\backslash \; S\_g\; =\; K.\; \backslash \; int\; \backslash \; sqrt\; \{-\; G\}.\; R.d\; \backslash \; Omega$, where $\backslash \; K$ is a constant related to the choice of unitées: for units MKSA, one takes $\backslash \; K\; =\; -\; \backslash \; frac\; \{c^3\}\; \{4\; \backslash \; pi\; G\}$, the sign $\backslash \; -$ being due to the principle of minimization of the action.

To find the equations of the field of gravitation in the shape of tensors of density of energy which are symmetrical, it is simpler to transform the Lagrangian one under the integral of the action than to use the equations of Euler-Lagrange. The variational principle is applied while varying the terms of metric the $\backslash \; g^\; \{ij\}$, which is the Lagrangian demonstration of the gravitation, according to the principle of equivalence as higher applied.

The deduced equations are:

By making the “contraction” $\backslash \; g^\; \{ij\}\; R\_\; \{ij\}\; -\; \backslash \; frac\; \{1\}\; \{2\}\; g^\; \{ij\}\; .g\_\; \{ij\}\; R\; =\; 0$, one obtains $\backslash \; R\; =\; 0$, which does not mean that space is flat, but rather than it is about a minimal Surface with four dimensions, tended between the various masses which evolve/move there.

The equations of Einstein in the external case are thus:

Equations of Einstein of the field of gravitation in the interior case

The second case of the equations of the fields is the case where there is matter (locally): one speaks about the “interior case”, i.e. “in the matter”.

In this case, the action is made up of the action of the gravitational field $\backslash \; S\_g\; =\; K.\; \backslash \; int\; \backslash \; sqrt\; \{-\; G\}.\; R.d\; \backslash \; Omega$ and of the action of the matter, by including there the electromagnetic field, which one writes $\backslash \; S\_m\; =\; \backslash \; frac\; \{1\}\; \{C\}\; \backslash \; int\; \backslash \; sqrt\; \{-\; G\}.\; \backslash \; Lambda\_m\; D\; \backslash \; Omega$.

The deduced equations are:

With the contraction similar to the external case , knowing that $\backslash \; g\_\; \{ij\}\; g^\; \{ij\}\; =\; 4$ and by posing $\backslash \; T\; =\; g^\; \{ij\}\; T\_\; \{ij\}$, one has $\backslash \; R\; =\; -\; \backslash \; chi\; T$. The principal Courbure is thus proportional to the density of total energy $\backslash \; T\; =\; g^\; \{ij\}\; T\_\; \{ij\}$ (or trace of the tensor $\backslash \; T\_\; \{ij\}$ ).

See too

Related articles

• Analytical mechanics
• Lagrangian
• Mechanical Hamiltonian
• Space of Minkowski
• restricted Relativity
• General relativity
• Principle of Fermat, interesting parallel with optics

External bonds

Cours on line

• Site of a student in astrophysics
• C. Cohen-Tannoudji, '' Lagrangian Forme of quantum mechanics '', course of 1966 to the ENS
• memory of magistère on the principle of less action in quantum mechanics