Laplace–Runge–Lenz vector

In classical mechanics, the Laplace–Runge–Lenz vector (LRL vector) is a special tool. It helps describe the shape and direction of the path an astronomical body takes as it moves around another object. Examples include a planet moving around a star or stars moving around each other in a binary star system.

For two objects that pull each other by Newtonian gravity, the LRL vector stays the same no matter where you measure it. This makes it a constant of motion, which means it is conserved. This property also works for any system where two objects interact through a force that gets weaker with the square of the distance between them, called an inverse square force. These are known as Kepler problems.

The LRL vector was important in early studies of quantum mechanics. It helped explain the spectrum of the hydrogen atom before the Schrödinger equation was developed. Even though it is not used as much today, the LRL vector shows a special balance, or symmetry, in these systems. It connects to a four-dimensional space.

The vector is named after Pierre-Simon de Laplace, Carl Runge, and Wilhelm Lenz. Scientists have also created versions of the LRL vector that work with special relativity and other kinds of forces.

Context

A single particle moving under a central force has several constants of motion, like total energy and angular momentum. The Laplace–Runge–Lenz vector (LRL vector) is another special constant. It helps describe the shape and direction of the orbit. It always points toward the closest point in the orbit and its length relates to how stretched the orbit is.

This special vector stays the same only when the force between two objects follows an inverse-square law, like gravity. For other forces, the vector changes. The LRL vector is unique because it doesn’t come from a simple coordinate in the system’s equations.

History of rediscovery

The LRL vector is a special tool that helps us understand how planets and stars move in their orbits. Even though it’s very useful, many scientists haven’t heard much about it because it’s harder to understand than other ideas like momentum.

Over the past few hundred years, many smart people discovered this idea again and again without knowing others had found it before. The first person known to find it was Jakob Hermann. Later, others like Pierre-Simon de Laplace and William Rowan Hamilton also worked on it. In the 1900s, Wolfgang Pauli used this idea to help explain how atoms work. Because of this history, it’s sometimes called the Runge–Lenz vector.

Jakob Hermann Johann Bernoulli Pierre-Simon de Laplace William Rowan Hamilton JosiahWillard Gibbs Carl Runge German Wilhelm Lenz Wolfgang Pauli matrix mechanics ellipse vector analysis

Definition

The Laplace–Runge–Lenz vector is a special tool used in physics. It helps us understand the shape and direction of an orbit, like the path of a planet around the sun.

This vector stays the same no matter where you measure it along the orbit. This makes it useful for studying movements in space. It works for situations where two objects pull each other with a force that follows an inverse-square law. This means the force gets weaker with the square of the distance between them. This is common in gravitational systems, like planets around a star.

Derivation of the Kepler orbits

The Laplace–Runge–Lenz vector helps us learn about the shape and direction of orbits, like the paths of planets around the sun. Using this vector, we can see that orbits are shaped like conic sections — which include circles, ellipses, parabolas, and hyperbolas.

The vector points toward the closest point in the orbit, called the periapsis. Whether an orbit is a closed ellipse or an open hyperbola depends on the energy of the moving body. If the energy is negative, the orbit is an ellipse; if it is positive, the orbit is a hyperbola; and if the energy is zero, the orbit is a parabola. This shows how the Laplace–Runge–Lenz vector explains the paths of objects in space.

Main article: Conic section
Main article: Eccentricity
Main article: Hyperbola
Main article: Parabola

Circular momentum hodographs

The Laplace–Runge–Lenz vector and angular momentum help us learn about how the momentum vector changes in some orbits. With certain forces, this momentum vector moves along a circle. This idea shows us patterns in how objects like planets travel around each other.

Constants of motion and superintegrability

The Laplace–Runge–Lenz vector is one of a group of special numbers in physics that do not change, no matter where you look at in an orbit. These special numbers help us understand the paths of objects, like planets moving around the Sun.

When a system has more of these special, unchanging numbers than normal, it is called "superintegrable." The motion of planets around the Sun is a good example of this. It follows special rules that make its path easy to describe with different math methods.

Evolution under perturbed potentials

The Laplace–Runge–Lenz vector works best when the force between two objects follows the exact inverse-square law, like in perfect planetary motion. But in real life, extra forces can slightly change this pattern, causing the orbit to slowly rotate. This rotation is called apsidal precession.

Scientists use this rotation to learn about the extra forces. For example, Einstein's theory of general relativity adds a tiny change to the normal gravity between objects. This change helps explain why the orbit of Mercury and some binary pulsars doesn't match exactly what we’d expect from simple gravity alone.

Poisson brackets

The Poisson brackets help us understand how different parts of the Laplace–Runge–Lenz vector and angular momentum relate to each other in physics. For the angular momentum vector L, the Poisson brackets show how its parts change in relation to each other.

Similarly, the Laplace–Runge–Lenz vector A has special relationships with L.

When we look at the parts of A with each other, these relationships depend on the system’s energy. For systems with negative energy (like planets orbiting a star), the relationships between the parts of a changed version of A, called D, are simpler and form a pattern like rotations in four dimensions. For systems with positive energy, the relationships are different, forming a pattern related to rotations in a space with one negative dimension.

These math tools help explain why some orbits repeat their shapes and why energy levels in atoms can be predicted.

Quantum mechanics of the hydrogen atom

In 1926, Wolfgang Pauli used a special math method to find the energy levels of hydrogen-like atoms. He did this before the Schrödinger equation was created. This helped explain how these atoms give off light.

Scientists also made special math tools, called ladder operators. These tools connect different energy levels. They show that energy levels depend only on one main number, n. This matches what we see in experiments and helps us understand how atoms are built.

Conservation and symmetry

The Laplace–Runge–Lenz vector is linked to a special balance in how objects move around each other because of gravity. In physics, some balances, called symmetries, mean that even if you move an object to a new spot in its path, its energy does not change. For example, turning an object keeps its angular momentum the same.

For objects pulled together by gravity, like planets around a star, there is an even more special balance. This balance helps keep both the angular momentum and the Laplace–Runge–Lenz vector steady. When we study these movements, this balance means that energy levels stay the same no matter how the object spins. This balance is special because it needs thinking about space in more than three directions.

Rotational symmetry in four dimensions

The Laplace–Runge–Lenz vector helps us understand how objects, like planets, move around stars. It connects this motion to a special kind of balance in a space with four dimensions instead of three.

By adding this extra dimension, we can see that orbits with the same energy are linked by a hidden kind of turning motion. This motion is not easy to notice in our usual three-dimensional space. This hidden balance is a big reason why the Laplace–Runge–Lenz vector is important for studying how things move under gravity.

Main article: Laplace–Runge–Lenz vector

Apollonian circles

great circles

action-angle variables

Jacobi's elliptic functions

Generalizations to other potentials and relativity

The Laplace–Runge–Lenz vector can help us learn about special properties in other situations. When there is an electric field, we can change the vector to find another important value that stays the same even as things move.

We can also change this idea to work with more complex forces and even situations where space and time bend, called special relativity. This helps scientists understand how objects move in different conditions, like when they move back and forth regularly.

Proofs that the Laplace–Runge–Lenz vector is conserved in Kepler problems

The Laplace–Runge–Lenz vector helps us understand the shape and direction of an orbit, like a planet going around the sun. In problems where two objects interact through gravity (or any force that follows an inverse-square law), this vector stays the same no matter where you measure it in the orbit. This means the vector is "conserved."

There are a few ways to show this conservation. One method uses basic physics principles to follow how the vector changes over time and shows it doesn’t change. Another method uses a special set of coordinates to simplify the problem. Finally, there’s a mathematical approach using symmetry principles that also confirms the vector’s conservation. All these methods help explain why the orbits of planets and stars follow predictable paths.

Alternative scalings, symbols and formulations

Unlike other vectors that describe motion, the Laplace–Runge–Lenz vector can look different depending on how scientists write about it. Sometimes they make it smaller or use different letters, but it always does the same job.

One common way to change how it looks is by dividing it by a number. This gives a new version that points in the same direction and shows how stretched or squashed the orbit is. Scientists can change its size in other ways too, and sometimes they flip the direction. Even with these changes, the vector always behaves the same way: it never changes while the object is moving.