Bertrand's theorem

Bertrand's theorem is a fundamental result in classical mechanics stating that, among all central force laws, only the inverse-square law and the linear (Hookean) force law produce closed and stable bounded orbits for all initial conditions in a two-body system.^[1] Named after the French mathematician Joseph Louis François Bertrand, who proved the theorem in 1873, for power-law central forces of the form F(r) \propto r^n, the theorem shows that stability and closure of orbits occur exclusively for n = -2 (inverse-square, as in gravity) and n = 1 (harmonic oscillator), and these are the only such forces in general.^[2]^[3] The theorem's proof involves analyzing the radial orbit equation and perturbation stability around circular orbits, ensuring that small deviations result in periodic oscillations that return to the original path, rather than precessing or filling a rosette pattern.^[1] This result provides a theoretical foundation for the observed near-closure of planetary orbits under Newtonian gravity, with small deviations like Mercury's perihelion precession (approximately 43 arcseconds per century) attributable to general relativistic corrections rather than a violation of the inverse-square law.^[1] Bertrand's theorem has broader implications in dynamical systems, highlighting the exceptional symmetry and integrability of these two potentials, and it has been generalized to higher dimensions and fractional mechanics while retaining its core insight into orbit closure.^[3]^[4]

Introduction

Statement of the theorem

Bertrand's theorem addresses central forces in classical mechanics, which are conservative forces directed toward a fixed center and depending solely on the radial distance r from that center, typically expressed as \mathbf{F} = f(r) \hat{r}.^[5] A stable closed orbit is defined as a bounded trajectory that returns exactly to its initial position and velocity after a finite number of revolutions, remaining unchanged under small perturbations to the initial conditions.^[5]^[6] The theorem asserts that, among all such central force laws where the attraction vanishes at infinite distance, only two produce stable closed orbits for every bound particle with initial speed below a certain limit: the inverse-square law, F \propto -1/r^2, and the isotropic harmonic oscillator, F \propto -r.^[5]^[6] This result, discovered by Joseph Bertrand in 1873, holds fundamental importance in celestial mechanics by identifying the force laws that yield periodic orbits without precession for all bound motions.^[5]

Historical context

Joseph Bertrand published his theorem in 1873 in the Comptes Rendus hebdomadaires des séances de l'Académie des Sciences, presenting it on October 20 under the presidency of Armand de Quatrefages de Bréau.^[5] This work built upon foundational contributions in classical mechanics, particularly Isaac Newton's Philosophiæ Naturalis Principia Mathematica (1687), which established the inverse-square law of universal gravitation and demonstrated that planetary orbits under this force are ellipses, in accordance with Johannes Kepler's laws of planetary motion formulated in 1609 and 1619.^[7] Bertrand's analysis extended these ideas to the broader question of orbital stability in central force fields. These inquiries highlighted uncertainties regarding which central forces would produce bounded orbits that remain closed without precession, beyond the well-understood cases of gravitational and harmonic potentials. Bertrand addressed this gap by rigorously proving that only two specific central force potentials—the inverse-square law and the isotropic harmonic oscillator potential—guarantee closed, bounded orbits for all bound initial conditions, thereby resolving longstanding debates on the exclusivity of such force laws.^[5] The theorem's immediate reception underscored its significance in late 19th-century celestial mechanics, where it influenced studies of orbital dynamics and emphasized the limitations of alternative potentials, such as the inverse-cube law, which fails to produce closed orbits and leads to unstable or precessing trajectories.^[7] Overall, Bertrand's contribution reinforced the symmetry and stability principles central to Newtonian mechanics, shaping subsequent theoretical developments in the field.

Mathematical formulation

Precise conditions

Bertrand's theorem addresses the conditions under which all bounded orbits in a central force field are closed and stable. In the standard setup, the two-body problem with masses m_1 and m_2 interacting via a central potential V(r), where r is the interparticle distance, reduces to an equivalent one-body problem. Here, a fictitious particle of reduced mass \mu = m_1 m_2 / (m_1 + m_2) moves under the same potential V(r) around a fixed center.^[7] The radial motion is governed by the effective potential V_\mathrm{eff}(r) = V(r) + L^2 / (2 \mu r^2), where L is the conserved angular momentum of the system.^[7] This formulation assumes the force derives from a scalar potential depending only on r, ensuring conservation of energy and angular momentum.^[8] For orbits to be stable and closed, bounded trajectories must return exactly to their starting point after a finite time, without precession or dense filling of an annular region. This requires that the apsidal angle \Delta \theta, defined as the angle swept during one half radial period (from periapsis to apoapsis), must be a rational multiple of \pi (\Delta \theta = p \pi / q for integers p, q), independent of the total energy E and L, ensuring closure for arbitrary initial conditions yielding bound motion.^[9] Stability under infinitesimal perturbations further demands that small deviations from circular orbits do not lead to secular changes, but instead result in periodic motion with the same frequency relation.^[8] The theorem specifies that such closure for all bound orbits holds only if the potential satisfies V(r) \propto -1/r (corresponding to an inverse-square force F(r) \propto -1/r^2) or V(r) \propto r^2 (corresponding to a linear force F(r) \propto -r). These forms ensure the required rational periodicity across the entire range of energies yielding bound motion (E < 0 for the inverse-square law; E > minimum of V_\mathrm{eff} for the harmonic oscillator) and for all admissible L > 0.^[5] Potentials yielding closed orbits solely for isolated values of E or L, or only for specific orbit types like circles, are explicitly excluded from the theorem's scope.^[7] The analysis relies on several key assumptions: the force is purely central and isotropic, deriving from a smooth, conservative potential with no dissipative effects; the mechanics is classical and non-relativistic; and the system admits stable circular orbits for the parameters considered, excluding rectilinear or colliding trajectories.^[8] These conditions frame the theorem within the planar motion of non-zero angular momentum orbits in Euclidean space.^[5]

Scope and assumptions

Bertrand's theorem applies to the motion of a single particle in a classical, non-relativistic central force field, where the force is conservative and derives from a scalar potential that depends only on the radial distance from a fixed center.^[10] This setup ensures the force is rotationally symmetric and time-independent.^[7] A key assumption is the conservation of angular momentum, which arises from the central symmetry and restricts all orbits to a fixed plane, reducing the problem to two-dimensional motion.^[10] The theorem specifically concerns bound orbits—those confined to finite regions of phase space (E < 0 for inverse-square; finite E for harmonic)—and requires that all such orbits, for arbitrary initial conditions within the bound regime (i.e., all relevant energies and angular momenta), are closed and stable against small perturbations.^[7] The theorem's scope excludes non-central forces, which violate angular momentum conservation and prevent planar orbits, thereby invalidating the conditions for closed bounded trajectories.^[7] It is also limited to non-relativistic classical mechanics and does not extend to relativistic settings; for example, in special relativity, the dynamics in a Coulomb field do not permit the universal closure of bound orbits as required by the theorem.^[11] Similarly, multi-body systems fall outside its purview, as the net force on any particle is typically non-central due to interactions with multiple bodies.^[10] Potentials of the form V(r) \propto r^{-\alpha} with \alpha \neq 1, 2 are excluded, as their bound orbits generally fail to close, though isolated closed orbits may arise under special initial conditions (e.g., particular energies or angular momenta in an inverse-cube force law).^[12] The theorem demands universality across all bound cases, precluding such exceptions.^[10] This result in classical mechanics is distinct from Bertrand's postulate in number theory, which guarantees a prime number between n and $2n for any integer n > 1.^[13]

Proof

Setup and preliminaries

Bertrand's theorem concerns the motion of a particle subject to a central force, which depends only on the radial distance r from a fixed center. Such problems are naturally formulated in polar coordinates (r, \theta), where r is the distance from the origin and \theta is the polar angle. For a two-body system, the problem reduces to an effective one-body problem with reduced mass m = m_1 m_2 / (m_1 + m_2), assuming the force center is at the origin. The Lagrangian for this system is

\mathcal{L} = \frac{1}{2} m \left( \dot{r}^2 + r^2 \dot{\theta}^2 \right) - V(r),

where V(r) is the central potential energy, independent of \theta.^[14] The cyclic nature of the \theta coordinate implies conservation of angular momentum, given by

L = m r^2 \dot{\theta} = \text{constant}.

This conservation arises from the rotational invariance of the central force. Additionally, for a conservative force derived from V(r), the total energy

E = \frac{1}{2} m \left( \dot{r}^2 + r^2 \dot{\theta}^2 \right) + V(r)

is conserved. Substituting the expression for angular momentum yields r^2 \dot{\theta}^2 = L^2 / (m^2 r^2), so the energy can be rewritten as

E = \frac{1}{2} m \dot{r}^2 + V_{\text{eff}}(r),

where the effective potential is

V_{\text{eff}}(r) = V(r) + \frac{L^2}{2 m r^2}.

The term L^2 / (2 m r^2) represents the centrifugal potential. Bounded orbits occur for energies E such that V_{\text{eff}}(r) has a minimum, with motion confined between turning points r_{\min} and r_{\max} where E = V_{\text{eff}}(r).^[6]^[14] To analyze the orbit shape, the radial equation of motion is transformed into an equation in terms of the angle \theta. Define u = 1/r, so r = 1/u. The radial acceleration follows from the Euler-Lagrange equation or Newton's laws as m \ddot{r} = F(r) + L^2 / (m r^3), where F(r) = -\mathrm{d}V/\mathrm{d}r is the central force (negative for attractive forces). Expressing time derivatives in terms of \theta using \mathrm{d}t = (m r^2 / L) \mathrm{d}\theta (from angular momentum conservation), and substituting u yields the Binet orbit equation:

\frac{\mathrm{d}^2 u}{\mathrm{d} \theta^2} + u = -\frac{m}{L^2 u^2} F\left( \frac{1}{u} \right).

This second-order differential equation describes the orbit r(\theta) or u(\theta).^[6]^[14]^[5] For bounded orbits to be closed, the radial motion must be periodic with a period that commensurates with the angular motion. The radial period T_r is the time for one full oscillation between turning points, while the angular period T_\theta corresponds to the time for \theta to advance by $2\pi. The orbit closes if the ratio T_r / T_\theta is a rational number, ensuring the particle returns to the same angular position after an integer number of radial cycles. In the absence of precession, this ratio is ideally such that \omega_r / \omega_\theta is an integer, where \omega_r = 2\pi / T_r and \omega_\theta = 2\pi / T_\theta.^[6]^[5]

Derivation steps

The derivation of Bertrand's theorem begins with an analysis of small perturbations around stable circular orbits, which reveals the necessary frequency condition for closure. Consider a nearly circular orbit in a central potential V(r), where small deviations in the radial direction lead to oscillations. The effective potential is V_{\rm eff}(r) = V(r) + \frac{L^2}{2m r^2}, with L the angular momentum and m the mass. For a circular orbit at radius r_0, the minimum of V_{\rm eff} satisfies V_{\rm eff}'(r_0) = 0, so V'(r_0) = L^2 / (m r_0^3). Perturbing around r = r_0 + \delta r, the equation of motion approximates a harmonic oscillator:

m \ddot{\delta r} + V_{\rm eff}''(r_0) \delta r = 0,

yielding the radial angular frequency \omega_r = \sqrt{V_{\rm eff}''(r_0)/m}. The angular frequency is \omega_\theta = L / (m r_0^2). For the perturbed orbit to close after one full revolution without precession, the radial motion must complete an integer number q of oscillations per angular period, requiring \omega_r = q \omega_\theta with q integer. The values q=1 (inverse-square law) and q=2 (harmonic oscillator) are the cases that ensure closure for all bound orbits.^[15] Computing V_{\rm eff}''(r_0) explicitly gives

V_{\rm eff}''(r_0) = V''(r_0) + 3 \frac{L^2}{m r_0^4} = V''(r_0) + \frac{3 V'(r_0)}{r_0},

using the circular condition. The frequency condition \omega_r = q \omega_\theta yields V_{\rm eff}''(r_0) = q^2 \frac{L^2}{m r_0^4} = q^2 \frac{V'(r_0)}{r_0}. Thus,

V''(r_0) + \frac{3 V'(r_0)}{r_0} = q^2 \frac{V'(r_0)}{r_0},

V''(r_0) + (3 - q^2) \frac{V'(r_0)}{r_0} = 0.

This differential equation must hold for all r_0 (all energies). For q=1,

V'' + 2 \frac{V'}{r} = 0 \implies \frac{\mathrm{d}V'}{V'} = -2 \frac{\mathrm{d}r}{r} \implies V' \propto r^{-2} \implies V(r) \propto -\frac{1}{r},

the inverse-square law. For q=2,

V'' - \frac{V'}{r} = 0 \implies \frac{\mathrm{d}V'}{V'} = \frac{\mathrm{d}r}{r} \implies V' \propto r \implies V(r) \propto r^2,

the harmonic oscillator. These are the only power-law solutions ensuring constant integer q independent of energy.^[9] To confirm these are the only solutions ensuring closure for all bound orbits, consider the general orbit equation in terms of u = 1/r:

\frac{d^2 u}{d\theta^2} + u = -\frac{m}{L^2 u^2} F\left( \frac{1}{u} \right).

For power-law V(r) = k r^\beta / \beta (\beta \neq 0, -2), F(r) = -k r^{\beta-1}, so

\frac{d^2 u}{d\theta^2} + u = \frac{m k}{L^2} u^{-\beta -1}.

The general solution is periodic with period $2\pi in \theta, independent of energy (amplitude), only for \beta = -1 (Kepler problem: RHS constant, solution u = A + B \cos(\theta - \phi), elliptical orbits) and \beta = 2 (harmonic: RHS \propto u^{-3}, solvable for rotated elliptical orbits). For other \beta, the frequency ratio \omega_r / \omega_\theta = \sqrt{\beta + 2} is not a constant integer, resulting in precessing rosette orbits that do not close.^[15]

Examples and applications

Inverse-square law

The inverse-square law represents one of the two central force potentials that, according to Bertrand's theorem, yield closed bounded orbits for all physically reasonable initial conditions.^[5] This potential is expressed as V(r) = -\frac{k}{r}, with k > 0 for attractive cases, which derives a central force of magnitude F(r) = -\frac{k}{r^2} directed toward the force center.^[7]/11:_Conservative_two-body_Central_Forces/11.10:_Closed-orbit_Stability) In this force field, particle trajectories form conic sections: bound orbits (negative total energy) trace closed ellipses with the force center at one focus and no precession of the periapsis; marginal orbits (zero energy) follow parabolas; and scattering orbits (positive energy) produce hyperbolas.^[5]^[7] Physically, the inverse-square law governs Newton's universal gravitation, where k = G M m for two masses M and m under gravitational constant G, explaining stable planetary motion around stars.^[5]^[7] It also underlies Coulomb's law of electrostatics, with k = \frac{|q_1 q_2|}{4 \pi \epsilon_0} for opposite charges q_1 and q_2 in vacuum permittivity \epsilon_0, modeling classical trajectories such as those in hydrogen-like atoms.^[7]/11:_Conservative_two-body_Central_Forces/11.10:_Closed-orbit_Stability) The theorem's prediction of closed orbits manifests here through Kepler's first law, which asserts that bound trajectories are ellipses with the central body at a focus, a direct consequence of the $1/r^2 force dependence.^[5]^[7] Closure holds universally due to the equality of the radial oscillation frequency \omega_r and the azimuthal frequency \omega_\theta, guaranteeing return to the initial position after one period irrespective of orbital eccentricity or energy.^[7]/11:_Conservative_two-body_Central_Forces/11.10:_Closed-orbit_Stability)

Harmonic oscillator potential

The harmonic oscillator potential is given by V(r) = \frac{1}{2} k r^2, where k > 0 is the force constant and r is the radial distance from the force center.^[16] This potential yields a central force \mathbf{F}(r) = -k \mathbf{r}, which is a linear restoring force proportional to the displacement from the origin.^[16] Bertrand's theorem identifies this potential, alongside the inverse-square law, as one of only two central force laws producing closed bounded orbits for all initial conditions.^[17] Under this potential, bound orbits are closed ellipses centered at the origin, the location of the force center.^[18] This contrasts with the elliptical orbits under the inverse-square law, which are centered at a focus rather than the force center itself. The elliptical shape arises from the superposition of two independent simple harmonic motions in perpendicular directions within the orbital plane, both at the same frequency.^[16] Physically, the isotropic three-dimensional harmonic oscillator potential models systems such as quantum mechanical particles in symmetric traps or classical approximations of molecular vibrations treated as central forces.^[19] In quantum mechanics, it describes the energy levels and wavefunctions of particles confined by quadratic potentials, exhibiting accidental degeneracies due to the separability in multiple coordinate systems.^[19] Classical realizations include small oscillations in molecular bonds approximated as linear restoring forces toward a central equilibrium point.^[19] The closure of orbits is verified by the equality of the radial frequency \omega_r and the angular frequency \omega_\theta, both given by \sqrt{k / \mu}, where \mu is the reduced mass of the system.^[16] This frequency matching ensures that the radial motion completes an integer number of oscillations per angular revolution, independent of the orbit's amplitude or energy, guaranteeing closure for all bound trajectories.^[16]

Generalizations

Extensions to non-central forces have been considered through perturbations of the classical central potentials. For small non-central perturbations to the inverse-square or harmonic oscillator forces, bounded orbits remain stable over exponentially long times, with deviations from exact closure being small, leading to approximate closure for nearly central fields. This stability arises from the quasiconvexity of the perturbed Hamiltonian, excluding the exceptional Keplerian and harmonic cases, as analyzed using Nekhoroshev-type theorems.^[20] In higher spatial dimensions d > 3, the Bertrand theorem generalizes, but the conditions for closed bound orbits become more restrictive. For the Newtonian gravitational potential, which follows a power-law force F \propto -1/r^{d-1} (or V \propto -1/r^{d-2}), no stable bound orbits exist because the effective potential lacks a minimum, preventing structures like planetary systems. Isotropy is crucial; while isotropic harmonic potentials V \propto r^2 still yield closed orbits, anisotropic variants do not unless frequency ratios are rationally related, leading to non-closed trajectories in general.^[21] For general power-law potentials V \propto r^\beta in d dimensions, closed orbits for all bound particles occur only for specific \beta values: \beta = 2 (harmonic) and \beta = 2 - d (generalized inverse power). Other \beta \neq -1, 2 (in 3D) yield conditional closure for particular energies or angular momenta, but not universally, extending the theorem's uniqueness while highlighting dimensionality dependence. This analysis underscores why stable planetary motion is confined to 3D space.^[22] Recent extensions (as of 2024) have generalized Bertrand's theorem to relativistic field theories using the double copy procedure, which maps gravity to Yang-Mills theories while preserving the so(4) symmetry and closed orbit properties in relativistic Kepler-like systems.^[23] In relativistic settings, generalizations deviate from classical closure. For special relativistic central forces, only the inverse-square potential permits stable circular orbits, with harmonic-like potentials failing due to relativistic corrections inducing precession; the apsidal angle shift is \delta\theta = 2\pi (1 - \zeta)/\zeta, where \zeta < 1 is determined by angular momentum. In general relativity, the Schwarzschild metric for a spherically symmetric mass causes perihelion precession, as observed in Mercury's orbit (43 arcseconds per century), breaking exact closure even for nearly Newtonian regimes. Bound orbits exist only in 4D spacetime, analogous to the classical 3D restriction.^[24]^[21]

Quantum analogs

In quantum mechanics, an analog of Bertrand's theorem identifies the Coulomb and isotropic harmonic oscillator potentials as the unique central potentials that admit an infinite number of exactly solvable bound states under natural conditions on the radial Schrödinger equation.^[25] This result parallels the classical theorem by specifying potentials where quantum bound states exhibit properties akin to closed classical orbits, such as high degeneracy or regular spacing in energy levels. For the hydrogen atom, the Coulomb potential V(r) = -\frac{k}{r} (with k > 0) yields energy eigenvalues E_n = -\frac{m k^2}{2 \hbar^2 n^2}, depending solely on the principal quantum number n = 1, 2, \dots, independent of the orbital angular momentum quantum number l. This accidental degeneracy, with each level n having degeneracy n^2, stems from an underlying SO(4) dynamical symmetry that enlarges the SO(3) rotational symmetry, mirroring the closure of all classical Keplerian orbits into ellipses. The exact solution of the Schrödinger equation for this potential, first obtained by Schrödinger in 1926, highlights how the quantum spectrum encapsulates the classical integrability. In contrast, the three-dimensional isotropic harmonic oscillator potential V(r) = \frac{1}{2} m \omega^2 r^2 produces equally spaced energy levels E_n = \hbar \omega \left( n + \frac{3}{2} \right), where n = 0, 1, 2, \dots is the total quantum number, and n = 2 n_r + l combines the radial n_r and angular l quantum numbers. The degeneracy of each level, given by \frac{(n+1)(n+2)}{2}, arises from an SU(3) symmetry and corresponds to the elliptical closed paths in the classical phase space for all bound motions. This exact solvability underscores the potential's role in quantum theory, from basic models to advanced applications. These quantum analogs influence modern research in quantum chaos and semiclassical methods, where the Coulomb and harmonic potentials represent the sole central force systems achieving maximal superintegrability, ensuring all bound states are fully integrable without chaotic behavior. In the Einstein-Brillouin-Keller (EBK) quantization scheme, a semiclassical approximation, the action integrals over tori yield precisely the exact quantum energies for these potentials, validating the connection between classical closed orbits and quantum degeneracy.^[26]