In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a vector used chiefly to describe the shape and orientation of the orbit of one astronomical body around another, such as a binary star or a planet revolving around a star. For two bodies interacting by Newtonian gravity, the LRL vector is a constant of motion, meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be conserved. More generally, the LRL vector is conserved in all problems in which two bodies interact by a central force that varies as the inverse square of the distance between them; such problems are called Kepler problems.
Thus the hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today.
In classical and quantum mechanics, conserved quantities generally correspond to a symmetry of the system. The conservation of the LRL vector corresponds to an unusual symmetry; the Kepler problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This higher symmetry results from two properties of the Kepler problem: the velocity vector always moves in a perfect circle and, for a given total energy, all such velocity circles intersect each other in the same two points.
The Laplace–Runge–Lenz vector is named after Pierre-Simon de Laplace, Carl Runge and Wilhelm Lenz. It is also known as the Laplace vector, the Runge–Lenz vector and the Lenz vector. Ironically, none of those scientists discovered it. The LRL vector has been re-discovered and re-formulated several times; for example, it is equivalent to the dimensionless eccentricity vector of celestial mechanics. Its first use seems to go back at least to Jakob Hermann.Jakob Hermann: Extrait d’une lettre de M. Herman à M. Bernoulli datée de Padoüe le 12. Juillet 1710. In: Histoire de l’Académie royale des sciences. Année M. DCC. X. Paris 1712, S. 519–521 (online). Various generalisations of the LRL vector have been defined, which incorporate the effects of special relativity, electromagnetic fields and even different types of central forces.
Context A single particle moving under any conservative central force has at least four constants of motion: the total energy and the three Cartesian components of the angular momentum vector with respect to the center of force. The particle's orbit is confined to the plane defined by the particle's initial momentum (or, equivalently, its velocity ) and the vector between the particle and the center of force (see Figure 1). This plane of motion is perpendicular to the constant angular momentum vector ; this may be expressed mathematically by the vector dot product equation . Given its mathematical definition below, the Laplace–Runge–Lenz vector (LRL vector) is always perpendicular to the constant angular momentum vector for all central forces (). Therefore, always lies in the plane of motion. As shown below, points from the center of force to the periapsis of the motion, the point of closest approach, and its length is proportional to the eccentricity of the orbit.
The LRL vector is constant in length and direction, but only for an inverse-square central force. For other central forces, the vector is not constant, but changes in both length and direction. If the central force is approximately an inverse-square law, the vector is approximately constant in length, but slowly rotates its direction. A generalized conserved LRL vector can be defined for all central forces, but this generalized vector is a complicated function of position, and usually not expressible in closed form.
The LRL vector differs from other conserved quantities in the following property. Whereas for typical conserved quantities, there is a corresponding cyclic coordinate in the three-dimensional Lagrangian of the system, there does not exist such a coordinate for the LRL vector. Thus, the conservation of the LRL vector must be derived directly, e.g., by the method of Poisson brackets, as described below. Conserved quantities of this kind are called "dynamic", in contrast to the usual "geometric" conservation laws, e.g., that of the angular momentum.
History of rediscovery The LRL vector is a constant of motion of the Kepler problem, and is useful in describing astronomical orbits, such as the motion of planets and binary stars. Nevertheless, it has never been well known among physicists, possibly because it is less intuitive than momentum and angular momentum. Consequently, it has been rediscovered independently several times over the last three centuries.
Jakob Hermann was the first to show that is conserved for a special case of the inverse-square central force, and worked out its connection to the eccentricity of the orbital ellipse. Hermann's work was generalized to its modern form by Johann Bernoulli in 1710. At the end of the century, Pierre-Simon de Laplace rediscovered the conservation of , deriving it analytically, rather than geometrically. In the middle of the nineteenth century, William Rowan Hamilton derived the equivalent eccentricity vector defined below, using it to show that the momentum vector moves on a circle for motion under an inverse-square central force (Figure 3).
At the beginning of the twentieth century, Josiah Willard Gibbs derived the same vector by vector analysis. Gibbs' derivation was used as an example by Carl Runge in a popular German textbook on vectors, which was referenced by Wilhelm Lenz in his paper on the (old) quantum mechanical treatment of the hydrogen atom. In 1926, Wolfgang Pauli used the LRL vector to derive the energy levels of the hydrogen atom using the matrix mechanics formulation of quantum mechanics, after which it became known mainly as the Runge–Lenz vector.
Definition An inverse-square central force acting on a single particle is described by the equation
The corresponding potential energy is given by . The constant parameter describes the strength of the central force; it is equal to for gravitational and for electrostatic forces. The force is attractive if and repulsive if .
The LRL vector is defined mathematically by the formula
where is the mass of the point particle moving under the central force, is its momentum vector, is its angular momentum vector, is the position vector of the particle (Figure 1), is the corresponding unit vector, i.e., , and is the magnitude of , the distance of the mass from the center of force.
The SI units of the LRL vector are joule-kilogram-meter (J⋅kg⋅m). This follows because the units of and are kg⋅m/s and J⋅s, respectively. This agrees with the units of (kg) and of (N⋅m2).
This definition of the LRL vector pertains to a single point particle of mass moving under the action of a fixed force. However, the same definition may be extended to two-body problems such as the Kepler problem, by taking as the reduced mass of the two bodies and as the vector between the two bodies.
Since the assumed force is conservative, the total energy is a constant of motion,
The assumed force is also a central force. Hence, the angular momentum vector is also conserved and defines the plane in which the particle travels. The LRL vector is perpendicular to the angular momentum vector because both and are perpendicular to . It follows that lies in the plane of motion.
Alternative formulations for the same constant of motion may be defined, typically by scaling the vector with constants, such as the mass , the force parameter or the angular momentum . The most common variant is to divide by , which yields the eccentricity vector, a dimensionless vector along the semi-major axis whose modulus equals the eccentricity of the conic:
An equivalent formulation multiplies this eccentricity vector by the major semiaxis , giving the resulting vector the units of length. Yet another formulation divides by , yielding an equivalent conserved quantity with units of inverse length, a quantity that appears in the solution of the Kepler problem
where is the angle between and the position vector . Further alternative formulations are given below.
Derivation of the Kepler orbits
The shape and orientation of the orbits can be determined from the LRL vector as follows. Taking the dot product of with the position vector gives the equation
where is the angle between and (Figure 2). Permuting the scalar triple product yields
Rearranging yields the solution for the Kepler equation
This corresponds to the formula for a conic section of eccentricity e
where the eccentricity and is a constant.
Taking the dot product of with itself yields an equation involving the total energy ,
which may be rewritten in terms of the eccentricity,
Thus, if the energy is negative (bound orbits), the eccentricity is less than one and the orbit is an ellipse. Conversely, if the energy is positive (unbound orbits, also called "scattered orbits"), the eccentricity is greater than one and the orbit is a hyperbola. Finally, if the energy is exactly zero, the eccentricity is one and the orbit is a parabola. In all cases, the direction of lies along the symmetry axis of the conic section and points from the center of force toward the periapsis, the point of closest approach.
Circular momentum hodographs
The conservation of the LRL vector and angular momentum vector is useful in showing that the momentum vector moves on a circle under an inverse-square central force.
Taking the dot product of
with itself yields
Further choosing along the -axis, and the major semiaxis as the -axis, yields the locus equation for ,
In other words, the momentum vector is confined to a circle of radius centered on .The conserved binormal Hamilton vector on this momentum plane (pink) has a simpler geometrical significance, and may actually supplant it, as , see Patera, R. P. (1981). "Momentum-space derivation of the Runge-Lenz vector", Am. J. Phys 49 593–594. It has length and is discussed in section #Alternative scalings, symbols and formulations. For bounded orbits, the eccentricity corresponds to the cosine of the angle shown in Figure 3. For unbounded orbits, we have and so the circle does not intersect the -axis.
In the degenerate limit of circular orbits, and thus vanishing , the circle centers at the origin . For brevity, it is also useful to introduce the variable .
This circular hodograph is useful in illustrating the symmetry of the Kepler problem.
Constants of motion and superintegrability The seven scalar quantities , and (being vectors, the latter two contribute three conserved quantities each) are related by two equations, and , giving five independent constants of motion. (Since the magnitude of , hence the eccentricity of the orbit, can be determined from the total angular momentum and the energy , only the direction of is conserved independently; moreover, since must be perpendicular to , it contributes only one additional conserved quantity.)
This is consistent with the six initial conditions (the particle's initial position and velocity vectors, each with three components) that specify the orbit of the particle, since the initial time is not determined by a constant of motion. The resulting 1-dimensional orbit in 6-dimensional phase space is thus completely specified.