Bertrand's Theorem states that among all possible central force laws, only the inverse-square force and Hooke's linear force produce orbits that are always closed, meaning every bounded orbit under either force law retraces itself exactly rather than precessing endlessly. Named for Joseph Bertrand, who proved it in 1873, it explains why closed elliptical orbits are the generic outcome for gravitationally bound bodies, while almost any other force law would instead produce an open, rosette-shaped path.
Facts
StatementAmong central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits. 1 Classification
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Source Bertrand's theorem (Wikipedia)
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Source Bertrand's theorem (Wikipedia)
Sources
1. Bertrand's theorem (Wikipedia)
Statement of the theorem
Among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits.
Naming and publication
The theorem is named after its discoverer, Joseph Bertrand who published it in 1873.
Proved By: Joseph Bertrand, Lead section
The theorem is named after its discoverer, Joseph Bertrand who published it in 1873.
In Group: Classical Mechanics and Dynamics, Lead section
In classical mechanics, Bertrand's theorem states that among central-force
In Branch: Mathematical Physics, Lead section
In classical mechanics, Bertrand's theorem states that among central-force
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