Laplace-Runge-Lenz vector

The Laplace-Runge-Lenz vector (in the literature also Runge-Lenz vector , Lenzscher vector etc., after Pierre-Simon Laplace , Carl Runge and Wilhelm Lenz ) is a conservation quantity of the movement in a 1 / r - potential ( Coulomb -Potential , gravitational potential ), d. i.e. it is the same at every point on the orbit (conservation magnitude). It points from the focal point of the orbit (center of force) to the nearest orbit point (perihelion in the earth's orbit) and thus has a direction parallel to the major orbit axis. Its amount is with eccentricitylinked to the railway. The Laplace-Runge-Lenz vector therefore enables the elegant derivation of the trajectory of a particle (e.g. planet in the Kepler problem, particles scattered on the atomic nucleus) in this force field without having to solve a single equation of motion . ${\displaystyle r(\varphi )}$ ${\displaystyle \alpha }$

In classical mechanics , the vector is mainly used to describe the shape and orientation of the orbit of one astronomical body around another, for example the orbit of a planet around its star.

The vector also plays a role in the quantum mechanics of the hydrogen atom as a Laplace-Runge-Lenz or Laplace-Runge-Lenz-Pauli operator .

definition

Illustration of the Laplace-Runge-Lenz vector on an elliptical trajectory for four different points

Be

${\displaystyle V=-{\frac {k}{r}}}$

a radially symmetrical attractive potential , which is inversely proportional to the relative distance between two objects with a proportionality constant . Then the Laplace-Runge-Lenz vector is defined as ${\displaystyle k}$ ${\displaystyle r}$ ${\displaystyle {\vec {A}}}$

${\displaystyle {\vec {A}}={\vec {p}}\times {\vec {L}}-mk{\vec {e}}_{r}}$,

in which

• ${\displaystyle {\vec {p}}}$the impulse of the body
• ${\displaystyle {\vec {L}}}$its angular momentum ,
• ${\displaystyle m}$its mass and
• ${\displaystyle {\vec {e}}_{r}}$ the radial unit vector

designated.

Proof of conservation

Direct invoice

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

${\displaystyle {\vec {L}}={\vec {r}}\times {\vec {p}}=mr^{2}{\vec {\omega }}=\mathrm {const} }$

The angular velocity determines the time derivative of the second term of , because a unit vector can only change through rotation: ${\displaystyle {\vec {A}}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\,mk\,{\vec {e}}_{r}=mk\,{\vec {\omega }}\times {\vec {e}}_{r}}$

According to Newton's law, the force is the rate of change of the momentum:

${\displaystyle {\vec {F}}=-{\frac {k}{r^{2}}}\,{\vec {e}}_{r}={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}}$

The following applies to the first term of ${\displaystyle {\vec {A}}}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {p}}\times {\vec {L}}=\left(-{\frac {k}{r^{2}}}{\vec {e}}_{r}\right)\times \left(mr^{2}{\vec {\omega }}\right)=mk\,{\vec {\omega }}\times {\vec {e}}_{r}.}$

The constancy of the Runge-Lenz vector now follows by subtracting:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\vec {A}}={\vec {0}}}$

Conclusion from the Noether theorem

Although it is sometimes argued in the literature that there is no associated symmetry transformation of the Lagrangian for the Laplace-Runge-Lenz vector, this can easily be stated. The Lagrangian of an attractive potential is: ${\displaystyle 1/r}$

${\displaystyle {\mathcal {L}}={\frac {1}{2}}m{\vec {v}}^{2}+{\frac {k}{r}}}$

The symmetry underlying the conservation of the Laplace-Rung-Lenz vector is shown under the variable transformation

${\displaystyle r_{i}\to r'_{i}=r_{i}+m\varepsilon _{k}\left(v_{i}r_{k}-{\frac {1}{2}}r_{i}v_{k}-{\frac {1}{2}}{\vec {r}}\cdot {\vec {v}}\delta _{ik}\right)}$

with three infinitesimal parameters . With the help of the equations of motion, the corresponding transformation of the velocities as ${\displaystyle \varepsilon _{k}}$

${\displaystyle v_{i}\to v'_{i}=v_{i}+{\frac {1}{2}}m\varepsilon _{k}\left(v_{i}v_{k}-{\vec {v}}^{2}\delta _{ik}-{\frac {k}{m}}{\frac {r_{i}r_{k}}{r^{3}}}+{\frac {k}{m}}{\frac {\delta _{ik}}{r}}\right)}$

be identified. Inserting the Lagrangian and Taylor expansion up to the order shows that this is like ${\displaystyle {\mathcal {O}}(\varepsilon _{k})}$

${\displaystyle {\mathcal {L}}\to {\mathcal {L}}'={\mathcal {L}}+mk\varepsilon _{k}\left({\frac {v_{k}}{r}}-{\frac {{\vec {r}}\cdot {\vec {v}}}{r^{3}}}r_{k}\right)={\mathcal {L}}+mk\varepsilon _{k}{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {r_{k}}{r}}}$

behaves, where the additional term is a total time derivative and therefore leaves the action of the system invariant. From Noether's theorem it follows that the three components of the vector

${\displaystyle A_{k}=m\left(v_{i}r_{k}-{\frac {1}{2}}r_{i}v_{k}-{\frac {1}{2}}{\vec {r}}\cdot {\vec {v}}\delta _{ik}\right){\frac {\partial {\mathcal {L}}}{\partial v_{i}}}-mk{\frac {r_{k}}{r}}=m^{2}({\vec {v}}\times ({\vec {r}}\times {\vec {v}}))_{k}-mk{\frac {r_{k}}{r}}}$

are preserved.

Conservation in the Hamilton formalism

With the Hamilton function of the system

${\displaystyle {\mathcal {H}}={\frac {{\vec {p}}^{2}}{2m}}-{\frac {k}{r}}}$

follows for the partial derivatives of the Hamilton function and the Laplace-Runge-Lenz vector according to the coordinates and momenta

${\displaystyle {\begin{aligned}&{\frac {\partial {\mathcal {H}}}{\partial r_{i}}}=k{\frac {r_{i}}{r^{3}}}&\qquad &{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}={\frac {p_{i}}{m}}&\\&{\frac {\partial {\vec {A}}}{\partial r_{i}}}=p^{2}{\vec {e}}_{i}-p_{i}{\vec {p}}-mk{\frac {{\vec {e}}_{i}}{r}}+mk{\frac {r_{i}}{r^{3}}}{\vec {r}}&\qquad &{\frac {\partial {\vec {A}}}{\partial p_{i}}}=2p_{i}{\vec {r}}-({\vec {r}}\cdot {\vec {p}}){\vec {e}}_{i}-r_{i}{\vec {p}}&\end{aligned}}}$

and according to Hamilton's equations of motion

${\displaystyle {\frac {\mathrm {d} {\vec {A}}}{\mathrm {d} t}}={\frac {\partial {\vec {A}}}{\partial t}}+\{{\vec {A}},{\mathcal {H}}\}={\frac {\partial {\vec {A}}}{\partial t}}+\left({\frac {\partial {\vec {A}}}{\partial r_{i}}}{\frac {\partial {\mathcal {H}}}{\partial p_{i}}}-{\frac {\partial {\vec {A}}}{\partial p_{i}}}{\frac {\partial {\mathcal {H}}}{\partial r_{i}}}\right)=0}$

Derivation of the trajectory

This is usually, i. That is, if one prefers to work with energy as the conservation quantity, a complex integration with several substitutions is necessary. On the other hand, the multiplication of the Runge-Lenz vector with now simply follows the cosine relation of the scalar product (arrow-free letters always indicate the amounts of the associated vector): ${\displaystyle {\vec {r}}}$

${\displaystyle {\vec {A}}\cdot {\vec {r}}=Ar\cos \varphi ={\vec {r}}\cdot \left({\vec {p}}\times {\vec {L}}\right)-mkr={\vec {L}}\cdot \left({\vec {r}}\times {\vec {p}}\right)-mkr=L^{2}-mkr}$

The cyclicity of the late product and the definition of angular momentum were used. denotes the angle between the Runge-Lenz vector and the position vector. ${\displaystyle \varphi }$

By rewriting a conic section equation arises in polar coordinates:

${\displaystyle r={\frac {L^{2}/(mk)}{1+{\frac {A}{mk}}\cos \varphi }}={\frac {L^{2}/(mk)}{1+\varepsilon _{k}\cos \varphi }}}$,

where the term can be identified as the numerical eccentricity of the conic section , which determines the orbit shape circle ( ), ellipse ( ), parabola ( ) or hyperbola ( ). ${\displaystyle A/mk}$ ${\displaystyle \varepsilon _{k}}$ ${\displaystyle \varepsilon _{k}=0}$ ${\displaystyle 0<\varepsilon _{k}<1}$ ${\displaystyle \varepsilon _{k}=1}$ ${\displaystyle \varepsilon _{k}>1}$

Hodograph of the Kepler Railway; points 1–4 correspond to those in the figure above

Furthermore, it is also possible to derive the hodograph of the Keplerbahn using the Laplace-Runge-Lenz vector. Since the angular momentum vector is perpendicular to the plane of motion , it follows ${\displaystyle {\vec {A}}\cdot {\vec {p}}=0}$

${\displaystyle {\frac {mk}{r}}{\vec {r}}={\vec {p}}\times {\vec {L}}-{\vec {A}}}$

with the Lagrange identity and a cyclic permutation of the late product

${\displaystyle m^{2}k^{2}=p^{2}L^{2}-2({\vec {A}}\times {\vec {p}})\cdot {\vec {L}}+A^{2}}$.

If the coordinate system is chosen so that the angular momentum points in the direction , and the orthogonal Laplace-Runge-Lenz vector in the direction,, follows: ${\displaystyle z}$ ${\displaystyle {\vec {L}}=L{\vec {e}}_{z}}$ ${\displaystyle x}$ ${\displaystyle {\vec {A}}=A{\vec {e}}_{x}}$

${\displaystyle p_{x}^{2}+\left(p_{y}-{\frac {A}{L}}\right)^{2}=\left({\frac {mk}{L}}\right)^{2}}$

The hodograph is thus a circle with a radius that is shifted from the center of force. The following applies to the intersections of the hodograph with the axis : ${\displaystyle mk/L}$ ${\displaystyle A/L}$ ${\displaystyle x}$ ${\displaystyle p_{0}}$

${\displaystyle p_{0}=\pm {\sqrt {{\frac {m^{2}k^{2}}{L^{2}}}-{\frac {A^{2}}{L^{2}}}}}=\pm {\sqrt {2m|E|}}}$

They are therefore independent of the angular momentum and the Laplace-Runge-Lenz vector.

properties

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

${\displaystyle {\vec {L}}\cdot {\vec {A}}={\vec {L}}\cdot \left({\vec {p}}\times {\vec {L}}\right)-mk{\frac {{\vec {L}}\cdot {\vec {r}}}{r}}={\vec {p}}\cdot \left({\vec {L}}\times {\vec {L}}\right)-mk{\frac {\left({\vec {r}}\times {\vec {p}}\right)\cdot {\vec {r}}}{r}}=0}$

• The Runge-Lenz vector points from the center of force of the orbit (one of the two focal points) to the pericenter, i.e. H. closest to the center of the railway. This follows immediately from the above equation, since it represents the angle between the Ort and Runge-Lenz vectors and is minimal for the maximum denominator, i.e. H. . ${\displaystyle \varphi }$ ${\displaystyle r}$ ${\displaystyle \cos \varphi =1\Rightarrow \varphi =0}$
• The amount of the Runge-Lenz vector is times the numerical eccentricity of the trajectory. This has already been shown in the derivation of the same. ${\displaystyle mk}$
• All three components of the Laplace-Runge-Lenz vector are conserved quantities. Since its magnitude is already given by the other conservation quantities angular momentum and energy and its position by the orthogonality to the angular momentum vector, the Laplace-Runge-Lenz vector only provides an independent conservation quantity. The Kepler problem therefore has five independent conserved quantities (energy, 3 components of the angular momentum vector, orientation of the Laplace-Runge-Lenz vector) for six initial conditions; it is therefore a maximally superintegrable system .

Perihelion rotation in the event of deviations from the Kepler potential

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

In the case of small deviations from the 1 / r potential, e.g. B. Due to 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 orbit axis ( perihelion ). If a deviation is so small that its square can be neglected, the disturbance of the Kepler orbit can be calculated elementarily with the help of the Runge-Lenz vector. Let it be the interference potential that is added to the Kepler potential. For the Runge-Lenz vector one finds (see proof of conservation) ${\displaystyle \Phi (r)}$

${\displaystyle {\frac {\mathrm {d} {\vec {A}}}{\mathrm {d} t}}=-\partial _{r}\Phi (r){\vec {e}}_{r}\times mr^{2}{\vec {\omega }}=mr^{2}\partial _{r}\Phi (r)~{\vec {e}}_{z}\times {\vec {e}}_{r}~{\frac {\mathrm {d} \varphi }{\mathrm {d} t}}.}$

The z-direction is perpendicular to the plane of the path. Obviously, the movement of the Runge-Lenz vector is not always a rotation. A rotation occurs, however, when infinitesimal changes are integrated over a cycle. For that one finds first

${\displaystyle \left(\Delta {\vec {A}}\right)_{\mathrm {1\,Umlauf} }=\int _{0}^{T}{\frac {\mathrm {d} {\vec {A}}}{\mathrm {d} t}}\mathrm {d} t=m{\vec {e}}_{z}\times \int _{0}^{2\pi }r^{2}\partial _{r}\Phi (r)~{\vec {e}}_{r}~\mathrm {d} \varphi \qquad \qquad r=r(\varphi ).}$

Since quadratic effects should be negligible, can be used for the undisturbed trajectory. The radial unit vector, broken down into components parallel and perpendicular to the orbit axis, is ${\displaystyle \Phi }$ ${\displaystyle r(\varphi )}$

${\displaystyle {\vec {e}}_{r}(\varphi )={\vec {e}}_{A}\,\cos \varphi +{\vec {e}}_{\perp }\,\sin \varphi .}$

For the Kepler ellipse is a function of , so the integral over a period with the factor for each interference potential is zero. It just remains ${\displaystyle r(\varphi )}$ ${\displaystyle \cos \varphi }$ ${\displaystyle \sin \varphi }$ ${\displaystyle \Phi (r)}$

${\displaystyle \left(\Delta {\vec {A}}\right)_{\mathrm {1\,Umlauf} }={\vec {e}}_{z}\Delta \varphi \times {\vec {A}},}$

where was inserted and the angle of rotation is given by the following expression: ${\displaystyle {\vec {A}}=mk\varepsilon {\vec {e}}_{A}}$ ${\displaystyle \Delta \varphi }$

${\displaystyle \Delta \varphi ={\frac {1}{k\varepsilon }}\int _{0}^{2\pi }r(\varphi )^{2}\partial _{r}\Phi (r(\varphi ))\cos \varphi ~\mathrm {d} \varphi }$

When a planetary orbit is disturbed by the presence of other planets, the disturbance potential is not directly of the shape , but is given this shape by averaging over many orbits of planets in a common plane. ${\displaystyle \Phi (r)}$

Quantum mechanics

In quantum mechanics , the Hermitian operator can be used in the hydrogen problem as an analogue to the Laplace-Runge-Lenz vector

${\displaystyle {\hat {\vec {A}}}={\frac {1}{2m}}\left({\hat {\vec {p}}}\times {\hat {\vec {L}}}-{\hat {\vec {L}}}\times {\hat {\vec {p}}}\right)+Z\alpha \hbar c{\frac {\hat {\vec {x}}}{|{\hat {\vec {x}}}|}}}$

can be defined, where

• ${\displaystyle {\hat {\vec {p}}}}$the momentum operator ,
• ${\displaystyle {\hat {\vec {L}}}}$the angular momentum operator and
• ${\displaystyle {\hat {\vec {x}}}}$are the position operator , as well as
• ${\displaystyle Z}$the atomic number ,
• ${\displaystyle \alpha }$the fine structure constant ,
• ${\displaystyle \hbar }$the reduced Planck quantum of action ,
• ${\displaystyle c}$the speed of light and
• ${\displaystyle m}$are the mass of the electron .

In particular in quantum mechanics , the commutator between the momentum and angular momentum operator does not disappear. The Hamilton operator of the Coulomb problem is ${\displaystyle \textstyle {\hat {\vec {L}}}\times {\hat {\vec {p}}}\neq -{\hat {\vec {p}}}\times {\hat {\vec {L}}}}$ ${\displaystyle {\hat {H}}}$

${\displaystyle {\hat {H}}={\frac {{\hat {\vec {p}}}^{2}}{2m}}-Z\alpha \hbar c{\frac {1}{|{\hat {\vec {x}}}|}}}$

and the commutator relation follows from the definition of the angular momentum operator

${\displaystyle [{\hat {H}},{\hat {A}}_{i}]=0}$

for all components of the Laplace-Runge-Lenz operator. Since this itself is not time-dependent, it follows from Heisenberg's equations of motion for quantum mechanical operators

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\vec {A}}}={\frac {\partial }{\partial t}}{\hat {\vec {A}}}+{\frac {\mathrm {i} }{\hbar }}[{\hat {H}},{\hat {\vec {A}}}]=0}$.

From the interchangeability of the Hamilton operator and the Laplace-Runge-Lenz operator it follows that both have a set of common eigenstates and in particular also the Hamilton operator and the square of the Laplace-Runge-Lenz operator.

The commutator relations for the individual components of the Laplace-Runge-Lenz operator are

${\displaystyle [{\hat {A}}_{i},{\hat {A}}_{j}]=-2\mathrm {i} \hbar {\hat {H}}\varepsilon _{ijk}{\frac {{\hat {L}}_{k}}{m}}}$

and for the commutator of the components of the Laplace-Runge-Lenz operator and the angular momentum operator

${\displaystyle [{\hat {A}}_{i},{\hat {L}}_{j}]=\mathrm {i} \hbar c\varepsilon _{ijk}{\hat {L}}_{k}}$

with the Levi Civita symbol . In particular are ${\displaystyle \varepsilon }$

${\displaystyle [{\hat {\vec {A}}}^{2},A_{j}]=[{\hat {\vec {A}}}^{2},L_{j}]=[{\hat {\vec {A}}}^{2},{\hat {\vec {L}}}^{2}]=[{\hat {\vec {L}}}^{2},{\hat {A}}_{j}]=[{\hat {\vec {L}}}^{2},{\hat {L}}_{j}]=0}$,

thus there exists a set of common eigen-states for both sets of the operators and . ${\displaystyle {\hat {H}},{\hat {\vec {L}}}^{2},{\hat {L}}_{3},{\hat {\vec {A}}}^{2}}$ ${\displaystyle {\hat {H}},{\hat {\vec {L}}}^{2},{\hat {\vec {A}}}^{2},{\hat {A}}_{3}}$

Individual evidence

1. ^ Friedhelm Kuypers: Classical mechanics . 10th edition. Wiley-VCH, 2016, ISBN 978-3-527-33960-0 , pp. 98 .
2. ^ Jean-Marc Lévy-Leblond: Conservation Laws for Gauge-Variant Lagrangians in Classical Mechanics . In: American Journal of Physcis . tape 39 , no. 502 , 1971, pp. 505 , doi : 10.1119 / 1.1986202 (English).
3. ^ Herbert Goldstein: More on the prehistory of the Laplace or Runge-Lenz vector . In: American Journal of Physics . tape 44 , no. 11 , 1976, p. 1123-1124 , doi : 10.1119 / 1.10202 (English).
4. ↑ W. Lenz: About the course of movement and the quantum states of the disturbed Kepler movement. Zeitschrift für Physik A 24 (1924), 197-207.