Laplace–Runge–Lenz vector

The Laplace -Runge -Lenz vector ( in the literature also Runge -Lenz vector, Lenz shear vector, etc., according to Pierre -Simon Laplace, Carl Runge and Wilhelm Lenz) is a conserved quantity of motion in the potential ( Coulomb potential, gravitational potential ).

In classical mechanics, the vector is mainly used to describe the shape and orientation of the orbit of an astronomical body around another, such as the orbit of a planet around its star. For two interacting on the basis of Newtonian physics body of the vector is a constant of the motion, that is, it is at any point of the path is equal to ( conserved quantity ).

Definition

It is defined as

With

• : Pulse of the body
• : Angular momentum of the body
• : Mass of the body
• : Constant of proportionality of the potential ( for Kepler with the gravitational constant, for the Coulomb electric field constants)
• : Radial unit vector
• : Position vector of the body
• : Amount of the position vector

And allows the derivation of the elegant trajectory of a particle (for example, the planet Kepler problem particles scattered from the nucleus ), which acts on the resultant force of such a potential. This is the axis vector: it shows from the focal point of the track ( power center ) to the nearest railway point ( perihelion at Earth ) and thus has a direction parallel to the major rail axis.

Also in the quantum mechanics of the hydrogen atom, the vector plays a role.

Proof of the conservation

In a system with 1/r-Potential isotropy applies. Therefore, conservation of angular momentum applies, with the consequence that the motion takes place in a plane perpendicular to the angular momentum and there is a simple relationship between angular momentum and angular velocity:

The angular velocity determines the time derivative of the second term of, for a unit vector can only be changed by rotation:

The potential V generated by a force

Where the last equality follows from Newton's law. Therefore applies to the first term of

By taking the difference now follows the constancy of the Runge -Lenz vector

Derivation of the trajectory

For this purpose, usually, that is, if one prefers to work with the energy as a conserved quantity, an elaborate integration needed with multiple substitutions. In contrast, it follows from the multiplication of the Runge -Lenz vector by now, stick to the cosine relationship of the scalar product ( arrow loose letters always denote the amounts of the corresponding vector ):

Here, the cyclical nature of the Spatproduktes as well as the angular momentum definition was used. denotes the angle between the Runge- Lenz vector and location.

By rewriting the typical conic equation arises in polar coordinates:

This elliptical parabola) is the numerical eccentricity of the conic section, the said web forming circuit ( ), (), () or a hyperbola (determined.

Properties

• The Runge- Lenz vector lies in the orbital plane because it is perpendicular to the angular momentum vector:

• The Runge -Lenz vector points from the center of power of the web ( one of the two foci ) to the pericenter, ie the center point of the next track. This follows immediately from the above path equation, since the angle between local and Runge -Lenz vector is minimal and is designed for maximum denominator, that is.
• The Runge -Lenz vector has as the amount times the numerical eccentricity of the orbit. This has already been shown in the derivation of the same.

Perihelion for deviations from Kepler potential

The conservation of the Runge -Lenz vector implies that the ellipses of planetary motion in the Kepler potential have a fixed orientation in space.

For small deviations from 1/r-Potential, eg by the presence of other planets in the solar system or as a result of Einstein's theories of relativity, there is a slow rotation of the path axis ( perihelion ). If a deviation is so small that its square may be neglected, then fault the Kepler track is elementary calculated using the Runge -Lenz vector. It is the interference potential, which is added to the Kepler potential. For the Runge -Lenz vector is found ( cf. Proof of Conservation )

The z direction is perpendicular to the orbital plane. Apparently, the movement of the Runge -Lenz vector is not at any time a rotation. A rotation arises but, if infinitesimal changes through a round to be integrated. For this, one can first find

Since quadratic effects should be of negligible, can be used for the unperturbed trajectory. Said radial unit vector is decomposed into components parallel and perpendicular to the rail axis,

In the Kepler ellipse is a function of, therefore, the integral over one period by a factor for each interference potential zero. It remains only

Was being used and the rotation angle is given by the expression:

The failure of a planetary orbit by the presence of other planets, the interference potential is not directly on the form, but receives this form by averaging over many rounds of planets in a common orbital plane.