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
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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.
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