A theorem first formulated by Sturm (1841) and related to the least curvature principle of Heinrich Hertz Eric Weisstein's World of Biography and Gauss. Eric Weisstein's World of Biography It states that, if a given set of impulses is applied to different points of a system in motion (either holonomic or nonholonomic), then the kinetic energy of the resulting motion is greater than the kinetic energy which the system would acquire under the action of the same impulses and constraints and of any additional constraints due to the reactions of perfectly smooth or perfectly rough fixed surfaces, or rigid connections between the particles of the system (Whittaker 1944, p. 260).
Another "Bertrand's theorem" states that the only two central force laws expressible as functions of r that give rise to closed orbits independent of initial conditions are linear and inverse square, i.e., Ar and (Bertrand 1873; Goldstein 1980, pp. 90-94). This follows from that fact that an orbit with force law f is almost closed if is rational and
Here "closed" means closed with respect to perturbations where the second Taylor term can be ignored. Expanding to further terms gives Bertrand's theorem, which says the only closed orbits occur for and 4.
Central Force, Closed Orbit, Constraint, Impulse, Kinetic Energy, Orbital Stability
References
Bertrand, J. "Mécanique analytique." C. R. Acad. Sci. 77, 849-853, 1873.
Goldstein, H. "Conditions for Closed Orbits (Bertrand's Theorem)" and "Proof of Bertrand's Theorem." §3-6 and Appendix A in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 90-94 and 601-605, 1980.
Sturm, J. C. F. Comptes Rendus, Vol. 13. p. 1046, 1841.
Whittaker, E. T. "Bertrand's Theorem." §108 in A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies. New York: Dover, pp. 260-261, 1944.
