Bertrand's theorem

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In classical mechanics, Bertrand's theorem states that 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.[1][2]

Joseph Bertrand

The first such potential is an inverse-square central force such as the gravitational or electrostatic potential:

with force .

The second is the radial harmonic oscillator potential:

with force .

The theorem is named after its discoverer, Joseph Bertrand.

Derivation

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Small changes in the power of the force scaling with distance will lead to significantly different kind of orbits.

All attractive central forces can produce circular orbits, which are naturally closed orbits. The only requirement is that the central force exactly equals the centripetal force, which determines the required angular velocity for a given circular radius. Non-central forces (i.e., those that depend on the angular variables as well as the radius) are ignored here, since they do not produce circular orbits in general.

The equation of motion for the radius   of a particle of mass   moving in a central potential   is given by motion equations

 

where  , and the angular momentum   is conserved. For illustration, the first term on the left is zero for circular orbits, and the applied inwards force   equals the centripetal force requirement  , as expected.

The definition of angular momentum allows a change of independent variable from   to  :

 

giving the new equation of motion that is independent of time:

 

This equation becomes quasilinear on making the change of variables   and multiplying both sides by   (see also Binet equation):

 

As noted above, all central forces can produce circular orbits given an appropriate initial velocity. However, if some radial velocity is introduced, these orbits need not be stable (i.e., remain in orbit indefinitely) nor closed (repeatedly returning to exactly the same path). Here we show that a necessary condition for stable, exactly closed non-circular orbits is an inverse-square force or radial harmonic oscillator potential. In the following sections, we show that those two force laws produce stable, exactly closed orbits (a sufficient condition) [it is unclear to the reader exactly what is the sufficient condition].

Define   as

 

where   represents the radial force. The criterion for perfectly circular motion at a radius   is that the first term on the left be zero:

  (1)

where  .

The next step is to consider the equation for   under small perturbations   from perfectly circular orbits. On the right, the   function can be expanded in a standard Taylor series:

 

Substituting this expansion into the equation for   and subtracting the constant terms yields

 

which can be written as

  (2)

where   is a constant.   must be non-negative; otherwise, the radius of the orbit would vary exponentially away from its initial radius. (The solution   corresponds to a perfectly circular orbit.) If the right side may be neglected (i.e., for small perturbations), the solutions are

 

where the amplitude   is a constant of integration. For the orbits to be closed,   must be a rational number. What's more, it must be the same rational number for all radii, since   cannot change continuously; the rational numbers are totally disconnected from one another. Using the definition of   along with equation (1),

 

Since this must hold for any value of  ,

 

which implies that the force must follow a power law

 

Hence,   must have the general form

  (3)

For more general deviations from circularity (i.e., when we cannot neglect the higher-order terms in the Taylor expansion of   ),   may be expanded in a Fourier series, e.g.,

 

We substitute this into equation (2) and equate the coefficients belonging to the same frequency, keeping only the lowest-order terms. As we show below,   and   are smaller than  , being of order  .  , and all further coefficients, are at least of order  . This makes sense, since   must all vanish faster than   as a circular orbit is approached.

 
 
 

From the   term, we get

 

where in the last step we substituted in the values of   and  .

Using equations (3) and (1), we can calculate the second and third derivatives of   evaluated at  :

 
 

Substituting these values into the last equation yields the main result of Bertrand's theorem:

 

Hence, the only potentials that can produce stable closed non-circular orbits are the inverse-square force law ( ) and the radial harmonic-oscillator potential ( ). The solution   corresponds to perfectly circular orbits, as noted above.

Classical field potentials

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For an inverse-square force law such as the gravitational or electrostatic potential, the potential can be written

 

The orbit u(θ) can be derived from the general equation

 

whose solution is the constant   plus a simple sinusoid:

 

where e (the eccentricity), and θ0 (the phase offset) are constants of integration.

This is the general formula for a conic section that has one focus at the origin; e = 0 corresponds to a circle, 0 < e < 1 corresponds to an ellipse, e = 1 corresponds to a parabola, and e > 1 corresponds to a hyperbola. The eccentricity e is related to the total energy E (see Laplace–Runge–Lenz vector):

 

Comparing these formulae shows that E < 0 corresponds to an ellipse, E = 0 corresponds to a parabola, and E > 0 corresponds to a hyperbola. In particular,   for perfectly circular orbits.

Harmonic oscillator

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To solve for the orbit under a radial harmonic-oscillator potential, it's easier to work in components r = (x, y, z). The potential can be written as

 

The equation of motion for a particle of mass m is given by three independent Euler equations:

 
 
 

where the constant   must be positive (i.e., k > 0) to ensure bounded, closed orbits; otherwise, the particle will fly off to infinity. The solutions of these simple harmonic oscillator equations are all similar:

 
 
 

where the positive constants Ax, Ay and Az represent the amplitudes of the oscillations, and the angles φx, φy and φz represent their phases. The resulting orbit r(t) = [x(t), y(y), z(t)] is closed because it repeats exactly after one period

 

The system is also stable because small perturbations in the amplitudes and phases cause correspondingly small changes in the overall orbit.

References

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  1. ^ Bertrand J (1873). "Théorème relatif au mouvement d'un point attiré vers un centre fixe". C. R. Acad. Sci. 77: 849–853.
  2. ^ Johnson, Porter Wear (2010-02-24). Classical Mechanics With Applications. World Scientific. pp. 149–. ISBN 9789814304153. Retrieved 2 December 2012.

Further reading

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