# Laplace–Runge–Lenz vector

The **Laplace–Runge–Lenz (LRL) vector** is a conserved vector of the [Kepler problem](https://www.edgepedia.org/kepler-problem), the motion of a body under an attractive inverse-square central force, defined by \( \vec{A} = \vec{p} \times \vec{L} - km\,\hat{r} \), where \( \vec{p} \) is the linear momentum, \( \vec{L} = \vec{r} \times \vec{p} \) the angular momentum, \( m \) the mass, and \( k \) the force constant (for gravity, \( k = GM \)).<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup> Energy and angular momentum are conserved for any central force, but for the inverse-square law alone a third quantity is also conserved, and its conservation is what forces bound orbits to be closed ellipses that never precess.<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup><sup> • </sup><sup>[2](https://www.physics2u.com/vault/kepler-problem-closed-orbits.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \vec{A} = \vec{p} \times \vec{L} - km\,\hat{r} \), constant throughout the motion for an inverse-square force \( f(r) = k/r^{2} \)<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup> |
| Geometry | Lies in the plane of the orbit (perpendicular to \( \vec{L} \)) and points along the major axis toward the periapsis<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup><sup> • </sup><sup>[3](https://www.mdpi.com/2073-8994/15/12/2114)</sup> |
| Magnitude | \( e = A/km \), so the vector's magnitude is proportional to the orbital eccentricity<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup><sup> • </sup><sup>[4](https://www.mdpi.com/2218-2004/11/7/105)</sup> |
| Dynamical symmetry | With rotations, the normalized vector closes under Poisson brackets into \( \mathfrak{so}(4) \) for negative energy and \( \mathfrak{so}(3,1) \) for positive energy<sup>[3](https://www.mdpi.com/2073-8994/15/12/2114)</sup><sup> • </sup><sup>[5](https://www.sciencedirect.com/science/article/pii/S0375960126006018)</sup> |
| Quantum role | The hermitian vector operator let Pauli derive the hydrogen spectrum algebraically and explains the \( n^{2} \)-fold Coulomb degeneracy of the energy levels<sup>[6](https://sedici.unlp.edu.ar/bitstream/handle/10915/131597/Documento_completo.pdf?sequence=1)</sup><sup> • </sup><sup>[7](https://www.sciencedirect.com/science/article/abs/pii/S0009261498009610)</sup> |
| History | First found in 1710 by Jakob Hermann and Johann Bernoulli; later used by Laplace, rediscovered by Hamilton, and named after Runge and Lenz, who came last<sup>[4](https://www.mdpi.com/2218-2004/11/7/105)</sup><sup> • </sup><sup>[8](https://www.researchgate.net/publication/307830504_LAPLACE-RUNGE-LENZ_VECTOR_IN_QUANTUM_MECHANICS_IN_NONCOMMUTATIVE_SPACE)</sup> |

## Definition and orbital geometry

The vector follows from Hamilton's vectorial treatment of the equation of motion. Applying the operator \( \vec{L} \times \) to \( m\ddot{\vec{r}} = -f(r)\,\hat{r} \) gives \( \vec{L} \times m\ddot{\vec{r}} = -mr^{2}f(r)\, d\hat{r}/dt \), which is useless for a general force law but integrates immediately when \( f(r) = k/r^{2} \), yielding the constant \( \vec{A} = \vec{p} \times \vec{L} - km\,\hat{r} \).<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup>

Two dot products fix the orbit. Since \( \vec{A} \cdot \vec{L} = 0 \), the vector lies in the orbital plane. Dotting \( \vec{A} \) with \( \vec{r} \) gives \( L^{2} = kmr + \vec{A} \cdot \vec{r} \), or

\[ \frac{\ell}{r} = 1 + e\cos\theta, \qquad \ell = \frac{L^{2}}{km}, \quad e = \frac{A}{km}, \]

the standard polar equation of a conic section with the focus at the origin, where \( \theta \) is measured from \( \vec{A} \).<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup> The eccentricity classifies the orbit: \( e = 0 \) is a circle, \( e < 1 \) an ellipse, \( e = 1 \) a parabola, and \( e > 1 \) a hyperbola.<sup>[9](https://math.mit.edu/~dunkel/Teach/18.354_2015S/slides/L03_HamKepler.pdf)</sup> Because \( \vec{A} \) is fixed, the direction of the major axis remains the same and the ellipse repeats indefinitely; the conservation law is what enforces closure, not an accident of solving the differential equation.<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup><sup> • </sup><sup>[2](https://www.physics2u.com/vault/kepler-problem-closed-orbits.html)</sup>

**Sensitivity to perturbations.** If the force law is changed slightly from inverse-square, \( \vec{A} \) is no longer conserved and the whole ellipse rotates around the focus: the orbit precesses.<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup> The historically famous case is Mercury, whose leftover perihelion precession, unexplained by planetary perturbations, was finally accounted for by Einstein's general relativity.<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup> The LRL method has also been applied perturbatively to compute perihelion precession in the equatorial plane of a Kerr black hole.<sup>[10](https://arxiv.org/abs/2512.23871)</sup>

## Hidden symmetry

By [Noether's theorem](https://www.edgepedia.org/noethers-theorem) a conserved quantity corresponds to a symmetry, and the LRL vector corresponds to a symmetry that is not manifest in three-dimensional space.<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup> The three components of \( \vec{L} \) and the three components of the rescaled vector \( \vec{A} = c\,(m/2|E|)^{1/2}\,\vec{\epsilon} \) close under the Poisson brackets,

\[ \{L_{i}, A_{j}\} = \epsilon_{ijk}A_{k}, \qquad \{A_{i}, A_{j}\} = \epsilon_{ijk}L_{k}, \]

forming a six-generator [Lie algebra](https://www.edgepedia.org/lie-algebra).<sup>[3](https://www.mdpi.com/2073-8994/15/12/2114)</sup> When the energy is negative this is \( \mathfrak{so}(4) \), the algebra of rotations in four dimensions, and when the energy is positive it is \( \mathfrak{so}(3,1) \); these are respectively the infinitesimal isometry algebras of a 3-sphere \( S^{3} \) in \( \mathbb{R}^{4} \) and a 3-hyperboloid \( H^{3} \) in [Minkowski space](https://www.edgepedia.org/minkowski-space).<sup>[5](https://www.sciencedirect.com/science/article/pii/S0375960126006018)</sup> The \( O(4) \) symmetry is characteristic of the central \( 1/r \) potential and is therefore called a dynamical symmetry, unlike the ordinary \( O(3) \) rotational symmetry present for any central force.<sup>[3](https://www.mdpi.com/2073-8994/15/12/2114)</sup>

The four-dimensional picture has a geometric origin in Moser's observation that, for negative energy, the Kepler problem is equivalent to non-affine geodesic motion on \( S^{3} \) under stereographic projection, with the auxiliary-space rotations forming the \( SO(4) \) group that combines physical rotations and LRL transformations.<sup>[5](https://www.sciencedirect.com/science/article/pii/S0375960126006018)</sup> Bertrand's theorem supplies the converse context: the inverse-square and harmonic-oscillator potentials are the only central forces for which all bounded orbits are closed, and the LRL vector is the Kepler case's distinguishing extra integral of motion.<sup>[11](https://pmc.ncbi.nlm.nih.gov/articles/PMC5014100/)</sup>

A 2026 analysis in *Physics Letters A* sharpened the picture by separating the vector's magnitude from its direction. Since the magnitude is only a function of energy and angular momentum, the part of the LRL vector carrying new information is its direction angle in the plane of motion; the infinitesimal dynamical symmetry generated by this direction part, written in the physical time, position, and velocity variables, forms the semi-direct product of \( SO(3) \) and \( \mathbb{R}^{3} \), in contrast to the \( SO(4) \) group obtained when LRL symmetries are combined with rotations in the usual auxiliary four-dimensional space.<sup>[5](https://www.sciencedirect.com/science/article/pii/S0375960126006018)</sup>

## Quantum mechanics and the hydrogen atom

In quantum mechanics the LRL vector becomes a hermitian operator,

\[ \vec{C} = \frac{1}{2m}\left(\vec{p} \times \vec{L} - \vec{L} \times \vec{p}\right) - \frac{e^{2}}{r}\,\vec{r}, \]

for the hydrogen atom, and the rescaled operator \( \vec{K} = \sqrt{-m/2H}\,\vec{C} \), well defined on bound states of negative energy, together with \( \vec{L} \) generates the hidden \( SO(4) \) symmetry algebra.<sup>[6](https://sedici.unlp.edu.ar/bitstream/handle/10915/131597/Documento_completo.pdf?sequence=1)</sup> Using this conserved vector operator, [Wolfgang Pauli](https://www.edgepedia.org/wolfgang-pauli) calculated the spectrum of the hydrogen atom by purely algebraic means, expressing the Hamiltonian in terms of the Casimir operator \( K^{2} + L^{2} \) and obtaining the bound-state energies by group theory.<sup>[6](https://sedici.unlp.edu.ar/bitstream/handle/10915/131597/Documento_completo.pdf?sequence=1)</sup>

The same symmetry explains the [accidental degeneracy](https://www.edgepedia.org/accidental-degeneracy) of the hydrogen spectrum: all energy levels with the same principal quantum number \( n \) are \( n^{2} \)-fold degenerate, the so-called Coulomb degeneracy, because the LRL operator connects states of the same energy but different total angular momentum.<sup>[7](https://www.sciencedirect.com/science/article/abs/pii/S0009261498009610)</sup><sup> • </sup><sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup> The construction generalizes: in \( d \) dimensions the LRL vector together with the \( SO(d) \) rotations generates a dynamical \( SO(d+1) \) symmetry that allows a purely algebraic solution of the hydrogen atom in arbitrary dimensions and in supersymmetric extensions.<sup>[6](https://sedici.unlp.edu.ar/bitstream/handle/10915/131597/Documento_completo.pdf?sequence=1)</sup>

## History and naming

The vector's history is longer than its name suggests. Jakob Hermann discovered the invariant in 1710, and [Johann Bernoulli](https://www.edgepedia.org/johann-bernoulli) developed it in their correspondence of that year; the vector was then used intensively by Laplace, rediscovered by Hamilton, described in a didactic textbook by [Carl Runge](https://www.edgepedia.org/carl-runge), and finally used by [Wilhelm Lenz](https://www.edgepedia.org/wilhelm-lenz) and Pauli in quantum mechanics for the hydrogen atom.<sup>[4](https://www.mdpi.com/2218-2004/11/7/105)</sup><sup> • </sup><sup>[8](https://www.researchgate.net/publication/307830504_LAPLACE-RUNGE-LENZ_VECTOR_IN_QUANTUM_MECHANICS_IN_NONCOMMUTATIVE_SPACE)</sup> For this reason some authors argue that a more accurate name would be the Hermann–Bernoulli–Laplace vector, and the alternative names Hermann–Bernoulli vector and Ermanno–Bernoulli vector appear in the literature.<sup>[6](https://sedici.unlp.edu.ar/bitstream/handle/10915/131597/Documento_completo.pdf?sequence=1)</sup><sup> • </sup><sup>[8](https://www.researchgate.net/publication/307830504_LAPLACE-RUNGE-LENZ_VECTOR_IN_QUANTUM_MECHANICS_IN_NONCOMMUTATIVE_SPACE)</sup> One classical-mechanics text puts it bluntly: Runge and Lenz "don't really deserve the fame" since they rehashed Gibbs' work in a textbook.<sup>[1](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)</sup>

## References

1. [15.5: A Vectorial Approach – Hamilton's Equation and the Runge Lenz Vector, Physics LibreTexts (Michael Fowler, University of Virginia)](https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Graduate_Classical_Mechanics_%28Fowler%29/15%3A_Keplerian_Orbits/15.05%3A_A_Vectorial_Approach-__Hamiltons_Equation_and_the_Runge_Lenz_Vector)
2. [Kepler Orbits and the Laplace-Runge-Lenz Vector, physics2u](https://www.physics2u.com/vault/kepler-problem-closed-orbits.html)
3. [Orbital Regularity of Exoplanets and the Symmetries of the Kepler Problem, Symmetry 15(12), 2114 (2023), MDPI](https://www.mdpi.com/2073-8994/15/12/2114)
4. [On Invariant Vectors in the Presence of Electric and Magnetic Fields, Entropy 11(7), 105 (2025), MDPI](https://www.mdpi.com/2218-2004/11/7/105)
5. [Symmetry transformation group arising from the Laplace–Runge–Lenz vector, Physics Letters A 591, 131926 (2026)](https://www.sciencedirect.com/science/article/pii/S0375960126006018)
6. [Kirchberg, Lange, Pisani, Wipf (2003). The Laplace–Runge–Lenz vector in supersymmetric quantum mechanics, Annals of Physics 303, 359–388](https://sedici.unlp.edu.ar/bitstream/handle/10915/131597/Documento_completo.pdf?sequence=1)
7. [The Lenz vector in the confined hydrogen atom problem, Chemical Physics Letters (1998)](https://www.sciencedirect.com/science/article/abs/pii/S0009261498009610)
8. [Laplace-Runge-Lenz Vector in Quantum Mechanics in Noncommutative Space, ResearchGate](https://www.researchgate.net/publication/307830504_LAPLACE-RUNGE-LENZ_VECTOR_IN_QUANTUM_MECHANICS_IN_NONCOMMUTATIVE_SPACE)
9. [L03: Hamiltonian Dynamics and the Kepler Problem, MIT 18.354 course notes](https://math.mit.edu/~dunkel/Teach/18.354_2015S/slides/L03_HamKepler.pdf)
10. [Kerr Perihelion Precession via the Laplace-Runge-Lenz Vector Method, arXiv:2512.23871](https://arxiv.org/abs/2512.23871)
11. [N-body dynamics on closed surfaces: the axioms of mechanics, Proceedings of the Royal Society A (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC5014100/)

---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Motion, forces, and dynamics › Newtonian dynamics of particles › Newton's laws of motion*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license