Laplace vector

An integral of motion of a point of constant mass $ m $ in the Newton–Coulomb potential field $ V ( r) = \kappa / r $:

$$ \Lambda _ {1} = ( x _ {1} \dot{x} _ {4} - x _ {4} \dot{x} _ {1} ) = \frac{1}{m} ( \dot{x} _ {2} L _ {3} - \dot{x} _ {3} L _ {2} ) + \kappa \frac{x _ {1} }{r} , $$

$$ \Lambda _ {2} = ( x _ {2} \dot{x} _ {4} - x _ {4} \dot{x} _ {2} ) = \ \frac{1}{m} ( \dot{x} _ {3} L _ {1} - \dot{x} _ {1} L _ {3} ) + \kappa \frac{x _ {2} }{r} , $$

$$ \Lambda _ {3} = ( x _ {3} \dot{x} _ {4} - x _ {4} \dot{x} _ {3} ) = \ \frac{1}{m} ( \dot{x} _ {1} L _ {2} - \dot{x} _ {2} L _ {1} ) + \kappa \frac{x _ {3} }{r} , $$

where

$$ r = \sqrt {x _ {1} ^ {2} + x _ {2} ^ {2} + x _ {3} ^ {2} } ; \ \ x = ( x _ {1} , x _ {2} , x _ {3} ) \in \mathbf R ^ {3} ; $$

$$ x _ {4} = \frac{1}{2} \frac{d}{dt} r ^ {2} ; \ m \dot{x} dot _ {4} = \kappa \frac{x _ {4} }{r} ; $$

$ L = ( L _ {1} , L _ {2} , L _ {3} ) $, the angular momentum, determines the plane of the orbit (for $ L \neq 0 $) and, together with the energy integral

$$ E = \frac{m}{2} ( \dot{x} _ {1} ^ {2} + \dot{x} _ {2} ^ {2} + \dot{x} _ {3} ^ {2} ) + \frac \kappa {r} , $$

determines its configuration. The Laplace vector determines the orientation of the Kepler orbit and is proportional to the position vector of its second focus.

An analogue of the Laplace vector also exists for the potential of an isotropic harmonic oscillator, which together with the Newton potential takes an exceptional position among the potentials of a central field.

In a centrally-perturbed Newtonian field the Laplace vector is not an integral, but precesses. For example, in Kepler's problem with relativistic $ 4 $- momentum the angle of rotation of the Laplace vector is

$$ | \Delta \phi | = 2 \pi \left ( \left ( 1 - \frac{\kappa ^ {2} }{c ^ {2} L ^ {2} } \right ) ^ {-} 1/2 - 1 \right ) \approx \ \frac{\pi \kappa ^ {2} }{L ^ {2} c ^ {2} } $$

for the period of $ r $. In quantum theory the existence of the Laplace vector explains the "random degeneration" of the levels of energy of a hydrogen-like atom with respect to the azimuthal quantum number $ l $ in addition to the degeneration with respect to the magnetic quantum number $ m $, which necessary holds for an arbitrary central potential $ V ( r) $. The Schrödinger equation of a Coulomb oscillator corresponds to two identical particles, one of which moves in the field of the Coulomb centre, situated at the first focus of the Kepler ellipse, while the other moves in the field of the second focus. The Hamiltonian of each particle is invariant with respect to the group of orthogonal transformations of its coordinates $ O ( 3) $, and the whole system is invariant with respect to the group $ O ( 3) \times O ( 3) = O ( 4) $ of orthogonal transformations of the $ 4 $- dimensional Euclidean space.

The Laplace vector was introduced by J. Hermann (see [1]) and P. Laplace (see [2]), apparently independently. The Laplace vector is sometimes called the Runge–Lenz vector by physicists.

References

[1]	J. Hermann, Giornale de Letterati d'Italia, Venice , 2 (1710) pp. 447–467
[2]	P.S. Laplace, "Celestial mechanics" , 1 , Chelsea, reprint (1966) (Translated from French)
[3]	O. Volk, "Miscellanea from the history of celestial mechanics II" Celestial Mech. , 14 (1976) pp. 365–382
[4]	G.N. Duboshin, "Celestial mechanics. Fundamental problems and methods" , Moscow (1975) (In Russian)
[5]	V.S. Popov, , Higher energy physics and the theory of elementary particles , Kiev (1967) pp. 702–727 (In Russian)