Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity vol 2, 2004, pp. 3-9 Strong forces in celestial mechanics MIRA-CRISTIANA ANISIU VALERIU ANISIU (CLUJ-NAPOCA) (CLUJ-NAPOCA) Abstract Strong forces in celestial mechanics have the property that the particle moving under their action can describe periodic orbits, whose existence follows in a natural way from variational principles. The Newtonian potential does not give rise to strong forces; we prove that potentials of the form 1=r produce strong forces if and only if 2. Perturbations of the Newtonian potential with this property are also examined. KEY WORDS: celestial mechanics; force function MSC 2000: 70F05, 70F15 1 Introduction Strong forces were considered in 1975 by Gordon [2], when he tried to obtain existence results of periodic solutions in the two-body problem by means of variational methods. As it is well-known, the planar motion of a body (e. g. the Earth) around a much bigger one (e. g. the Sun) is classically modelled by the system x1 x•1 = r3 (1) x2 x•2 = r3 1 2 2 with r = x1 + x2, or, alternatively p x• = V; (2) r with the Newtonian potential 1 V = (3) r 2 and x = (x1; x2) R (0; 0) . The potential V has a singularity at the origin of the plane.2 Evenn f if oneg works in a class of `noncollisional' loops (x(t) = (0; 0); t R), the extremal o ered by a variational principle will be the limit6 of a sequence8 2 of such loops, hence we have no guarantee that it will avoid the origin. Gordon remarked that for other type of conservative forces, called by him strong forces, the extremals are not collisional trajectories. 2 Main results We shall consider systems of the type x• = W; (4) r N 2 N where x = (x1; :::; xN ) R ; and W C (R 0 ) is the force function 2 2 nf g (W = V ). We shall denote by the Euclidean norm in RN . The cases physically meaningful are those withj j N 1; 2; 3 . 2 f g De nition 1 (Gordon [2]) The system (4) satis es the strong force (SF) condition if and only if there exists a neighbourhood of the origin 0 of RN N and a function U C2(RN 0 ) such that (i) U(x) 2 as x nf0; g (ii) W (x)! 1U(x) 2!for all x in 0 . jr j N nf g Remark 2 As a matter of fact, one can choose another di erentiable norm instead of the Euclidean one, hence in De nition 1 may be supposed to be N the unit ball x RN : x < 1 : f 2 j j g Remark 3 If the force function W gives rise to a strong force, the function aW (with a > 0) has the same property; this happens also for each function 2 N W1 C (R 0 ) with W1(x) W (x), x 0 : 2 nf g 2 N nf g 2 The example given by Gordon to illustrate the de nition is W (x) = 1= x 2; he remarks also that W (x) = 1= x , corresponding to the Newto- nianj j potential, is not strong, fact which determinesj j him to say that `it is disappointing that the gravitational case is excluded by the SF condition'. Nevertheless he obtained existence results for periodic orbits in strong force elds, and this was the starting point for applying systematically the vari- ational methods in celestial mechanics. It is interesting to mention that, in 1896, Poincare[5] had the same idea of using the least action principle to nd periodic orbits in the planar three-body problem, for a force of the type 1=rn with n 2 (excluding again the Newtonian potential). By that time the variational methods were not formulated in a rigourous way, and there was a strong belief that Newtonian potential governs the motion of celestial bodies, so Poincare'sresult remained for years purely theoretical. 2 N Our rst concern is to nd out which functions W C (R 0 ), 2 nf g 1 W (x) = ; > 0; (5) x j j satisfy the SF condition. Theorem 4 For > 0, W from (5) satis es the SF condition if and only if 2. Proof. Let 2, = x RN : x < 1 and U = ln x . It follows N 2 j j j j that U(x) 2 = 1= x 2 W (x) whatever x in 0 , hence W satis es the SFjr condition.j j j N nf g Let us consider now 0 < < 2. We suppose that there exists a function N U as in De nition 1. We x x0 R 0 ; for ; > 0; < < 1= x0 we evaluate 2 nf g j j U(x0) U(x0) = dU(tx0; x0)dt x0 U(tx0) dt: j j j j jr j Z Z Using (ii) we obtain 1 x0 U(tx0) dt x0 =2 dt = j j jr j j j tx0 Z Z j j 1 =2 1 1 =2 1 1 =2 1 =2 x0 dt = x0 ; j j t =2 j j 1 =2 Z 3 1 =2 1 1 =2 1 =2 hence U(x0) U(x0) x0 (1 =2) : In this j j j j 1 =2 last relation we make 0+ and we obtain the contradiction x0 1 1 =2 ! j j (1 =2) . It follows that, for any 0 < < 2;W does not satisfy the SF condition. 1 In view of Remark 3, we have Corollary 5 Each function W C2(RN 0 ) with W (x) a= x (a > 0; 2) satis es the SF condition.2 nf g j j Example 6 Corollary 5 includes among the functions which satisfy the SF condition those related to various perturbations of the Newtonian force. One of them, with great physical signi cance, corresponds to the Manev potential [4] and is given by 1 3m 1 W = m + ; (6) M r 2c2 r2 m being the gravitational parameter of the two-body system and c the speed of light. This potential is a good substitute for relativity theory at the solar system's level. It was mentioned as a strong force by Anisiu [1]. The ad- vances in the qualitative understanding of the motion in a Manev-type eld are exposed in [3]. A potential of Manev-type was studied by Newton him- self, and he showed that the force generated by such a potential produces a precessionally elliptic orbit. Schwarzschild [6] solved the relativistic analog of the classical Kepler prob- lem and derived the force function 1 b W = GM + ; (7) S r r3 where G is the gravitational constant, M is the mass of the eld-generating body and b a positive constant. The motion in a Schwarzschild eld, with implications in astrophysics, is studied by Stoica and Mioc [7]. We can establish precisely what perturbations of the Newtonian potential are strong or not. Theorem 7 For > 0, the perturbation of the Newtonian force W 2 C2(RN 0 ) given by nf g 1 b f W (x) = + ; b > 0; x x j j j j satis es the SF conditionf if and only if 2. 4 Proof. The fact that W satis es the SF condition for 2 follows directly from Corollary 5. For 0 < 1 and x f 0 (as mentioned in Remark 2, we take 2 N nf g N the unit ball), we have W (x) (1 + b) = x , so W cannot satisfy the SF j j condition, because from Theorem 4 it follows that W1(x) = 1= x does not j j satisfy it. For 1 < < f2, we suppose that theref exists a function U as in N De nition 1. We x x0 R 0 ; for ; > 0; < < 1= x0 we evaluate as in the proof of Theorem2 4 nf g j j 1 b U(x0) U(x0) x0 U(tx0) dt x0 + dt j j j j jr j j j tx0 tx0 Z Z sj j j j 1 =2 1 1 =2 p1 + b 1 =2 1 =2 x0 p1 + b dt = x0 : j j t =2 j j 1 =2 Z It follows that 1 =2 p1 + b 1 =2 1 =2 U(x0) U(x0) x0 j j j j 1 =2 and, making 0+; we obtain a contradiction. It follows that, for any ! 0 < < 2; W does not satisfy the SF condition. By Corollary 5 we have that each force function with W (x) a= x 2 ; j j a > 0, satis esf the SF condition; due to the simplicity of this description, it is sometimes considered as SF de nition. The next example shows that there are SF potentials which do not satisfy the mentioned inequality. Example 8 De ne ' : (0; 1=2] R, '(t) = ln ( ln t). The function ' can be extended to a C3 function! de ned on (0; ) by taking '(t) a 1 2 polynomial of third degree for t > 1=2. Then W (x) := '0 ( x ) satis es the SF condition. Indeed, we can choose U(x) = ' ( x ) jandj we have j j limx 0 U(x) = limt 0+ '(t) = and ! ! 1 2 2 x 2 N U(x) = '0 ( x ) = '0 ( x ) = W (x); x R 0 : jr j j j x j j 2 nf g j j On the other side, let us suppose that for x 1=2;W (x) a= x 2, where 2 2 j j 2 j j a > 0. This would imply '0 (t) a=t ; that is a 1= ln t for each t (0; 1=2], hence a 0; contradiction.