Invariant of Runge Lenz

In the case of the Keplerian Movement, there exists, in more the mechanical energy Eo and of the kinetic Moment, $\ vec {L_o}$ , three other constants of the movement, gathered in the shape of a vector called “vector invariant of Runge-Lenz”, or “vector eccentricity”: This vector, constant, point towards the perigee (thus e.Lo = 0 checks).

According to the Theorem of Noether, any preserved quantity is equivalent to a symmetry of the problem - that implies a study of a theoretical nature on the groups of symmetry SO (4) or SO (3,1).

In addition, the contribution of Hermann Jacob to his discovery (1713) is discussed an enough point of history of sciences.

Symmetry SO (4) and degeneration

That the vector E is constant results directly owing to the fact that its temporal derivative is null (immediate checking!).

To claim to find the vector E , by such or such method is " astucieux" , but actually:

The degeneration and the existence of E are related to this symmetry SO (4), known as dynamics;

The article on the Keplerian Movement watch why Goursat and Bohlin, then others whose Levi-Civita, made the connection between harmonic oscillator (of Hooke) and movement of Kepler via transformation Z - > Z = z^2, on a scale of time close (Newtonian Temps), and there is as well sure a great symmetry of the Hamiltonian of the harmonic oscillator.

In the presence of a constant force Fo , the movement of the harmonic oscillator is only moved: it results from it that in the case of Kepler symmetry is not completely raised, and besides the movement can be integrated via the elliptic functions, but the number of junctions is rather astonishing: it is the Stark effect known as traditional.

In the same way, it still remains of symmetry in the case of two fixed identical centers of force: the movement is still integrable: it is celebrates it problem of Alembert, recently rewritten by Vinti for its utility in space Géodésie .

Finally let us announce that one can still see further: SO (4,2) is also symmetry of the Hamiltonian, that is to say 15 generators of the algebra of corresponding Dregs: outline.

Note history: of Hermann with Laplace

Alain Albouy (IMCCE) recalled the history of discovered of this vector eccentricity, which goes back apparently to Jakob Hermann.

BUT it is not known if the following kinematic problem had been solved:

a boat sails by leaving a headlight F with port side, the course 90° and thus described a circle around the headlight, and its Hodographe is a circle. With an additional uniform current of value U lower at the V° speed of the boat, the hodograph is a excentré circle and THUS the boat describes an ellipse of hearth F: was this the solution of the elliptoïde of Hooke (1679)? It is known that Hooke marvelously drew well (see Micrographia and especially a marvellous drawing of the " trajectory of Hooke") : such a step by step solved problem, point by point, was not inaccessible for him. Unfortunately this point of history will undoubtedly remain a question mark, since, following the estrangement 1679-1684, Newton with the ombrageux character, will burn papers of Hooke, with his death, as soon as he is president of Royal Society. In any event, the masterly demonstration of Newton of 1684 is mathematically rigorous and proves that the trajectory is an ellipse (and not a vague ellipse " numérique"). Arnold (1990) is convincing in its definition of the " preuve" at the 17th century.

At the 18th century, the business is heard, and Laplace uses the vector eccentricity for calculations of Perturbation of the Keplerian movement. But Lagrange knew it already and much others. It is thus necessary to go back to Jakob Hermann (1713) to find the trace first vector eccentricity: obviously, it is about a feat of ingenuity, and he is never question of vectors there! To already include/understand the differential notation of the time is an exploit.

Utility of the vector eccentricity

Its constancy in direction is the confirmation of the quiescence of the perigee. Its constancy in module is the constancy of the eccentricity of the ellipse.

Its introduction by Goldstein takes eight lines of calculation. The introduction in France east often this one:

$D \ vec {V} /dt = - GM \ vec {U} /r^2 = D \ vec {V} /d \ theta \ cdot (D \ theta/dt) = D \ vec {V} /d \ theta \ cdot C/r^2$

The term R ² disappears, and the equation of the excentré circle appears: it is the Hodographe:

From there follows that the trajectory is an ellipse (Danjon Course: cosmography, baccalaureat, 1952) and/or the existence of E .

One can prefer the demonstration of Goldstein-Chester, because it results from a result of very elegant Cinématique vectorial; for any vector, let us say R :

Good Luck for Newton' S law! $\ dowry {\ vec {U}} = \ dowry {\ vec {v}} \ wedge \ vec {L}/\ mu$ . And this gives by integrating the vector eccentricity!

Others " astuces" exist. But the TRUE reason is in the generators of SO (4), more exactly since energy is negative, those of SO (3,1): exactly those of restricted relativity. Then not astonishing to have correspondences between problems of aberration-Doppler and problems of ellipses of Kepler.

Consequently, the utility is clear: the symmetry of the Hamiltonian high risk to transpose itself in quantum mechanics (not relativist). As well as the problems of disturbance: gained! that is true; Stark effect, linear Zeeman effect, then quadratic can be taught differently.

Quantum vector eccentricity

outline

Runge and Lenz

The attribution of this vector with Runge and Lenz is due to the unequalled notoriety of the treaty of mechanics of Goldstein. Nevertheless the new edition Goldstein and Safko (ISBN 0-201-65702-3) corrects: vector of Laplace-Runge-Lenz. Should the list really be lengthened?

External bonds

Demonstration of the laws of Kepler and properties of an ellipse

See too

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