v, we d 1 :i 2 f m 1 m 1 r 1 x 1 X2 ) ( 2k ) d 0 1 x 2 d- [ m2 x2 ( ) + llo xl 1 - v~ /c2 and v;/c 2. NOTE MOTION OF THE PERIHELION

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1 NOTE MOTION OF THE PERIHELION Ieronim Mihaila From the formulae of chap. III, neglecting the interaction mass obtain the equations of motion of the sun and planet, namely (1) (2) (3) dpl dp2 - = L, - =- L, dt dt Pt = mtvl + /J.V2, P2 = m2 v2 + /J.Vt, L = 2 c 2 ( 1 - v v I c 2 ).!. all. 1 2 r ar v, we In order to obtain the equations of relative motion of the planet with respect to the sun, we seek a preferential inertial reference frame. To this let us consider the point whose position vector is p, (4) (mt + JJ.)xt + (m2 + JJ.)X2 p = ;:: =-: ;;--;;---= m 1 + m 2 + 2JJ. Since H = c2 (m 1 + m2 + 2JJ.) =canst., we o~tain p = Pt + P2 + m2 + IJ. ( 5) r m 1 + m2 + 2JJ. m 1 + m2 + 2Jl We shall now show that p = canst., if we confme ourselves to the terms in v~ /c2 and v;/c 2. (6) Indeed, equations (1) become 0 0 d 1 :i 2 f m 1 m 1 r 1 x 1 X2 ) ( 2k ) dt [m~ xl(1 +2 ~) + IJ.oX2] = -..!...r-=-2~-;- (1-2-2_c_2 1 + c2r, 2 d 0 1 x 2 d- [ m2 x2 ( ) + llo xl 1 - t 2 c and A.(r) is taken of the form k/r, k being a constant (cf. I. Mihaila, C.R. Acad. Sc. Paris, A, 280, , 1975).

2 119 We obtain = i.e., since the relative motion is nearly circular, For most planets we have m~ I m~ < I i I I c 2, and therefore we may consider. p = canst. (7) In other words the reference frame with the origin at the point of vector p is, within the approximation considered, an inertial frame. In this reference frame we have the relations ( m1 + 1-lo ) r 1 = - ( m2 + 1-lo ) r 2 ' (m1 + f.lo)~1 =- (m2 + f.lo)~2' (8) (m1 + llo)r1 = (m2 + llo)r2 ' r 1, r 2 are the position vectors, and the equations of motion take the form d [' 1 ri 1 fmy fm~ r 1 my ri 2k - r1 ( c2 +--) J =-- ( )(1 + -), gt 2 c 2 r r2 r 2 m~ c2 c2 r (9) d. 1 r 2 1 fm 0 - [ r ( J ) ] dt 2 2 c 2 2 c 2 r Keeping only the second order terms obtain f m 0 r 1 m 0 r2 2k =--1 -(1+--2 _1 )(1+-). r2 r 2 m 0 c2 c2 r in lrllc and subtracting equations (9), we 1 f(m~+m~) r 2k --=------'----:2:------"-'- - ( ) r r c r (10)

3 120 Using the relations. r = 2 we may write r "2 = r c2 Equation ( 10) becomes d 1 r 7 dt [r (1 + 2 c2 )] f(m~ + m~)!_ 2k =- ( ). r2 r c r We obtain (11) or. ( 12).. J.L* r 2CXIJ.* 2{3r 2 r =- ~ r ( 1 - c2 r + 7 ) ' (13) ll* = f(m~ + m~), 2CXIJ.* =- 2k, 2{3 = -3/2. One sees that the motion is plane. Taking the plane of the motion as a reference plane, the equations of motion become (14).. X-- r3 + X, ll*y Y + y, r3 ( 15)

4 121 (16) Because the force (X, Y) is small, we obtain as a first approximation the equations of elliptical motion. By integration we obtain the osculating orbit. The action of the additional (corrective) force may be considered as a perturbation and the motion of the perihelion may be studied by the method of the variation of constants (e.g. see ]. Chazy, La theorie de la Relativite et la Mecanique celeste, t.l, chap. [I), Gauthier-Villars, Paris, 1928). The differential equation of the longitude of the perihelion is of the form dw du = ~ ( 1 - e cos u) ax ay y 1- e- { X + y _ } :_-----'--n--: 2:---a --:: 2:-e-- a; ae ' (17) a is the semi-major axis of the osculating orbit, e the eccentricity, n the mean motion (mean angular velocity) and u the eccentric anomaly. The derivatives X: and y are computed by the formulas of elliptic motion, the time and the eccentric anomaly are considered as variables. The derivatives of the coordinates with respect to e are computed by the same formulas considering the eccentric anomaly and e as variables. In the plane of the motion the formulas of elliptical motion are x=a(cosu-e)cosw-a~2 sinu sinw, y = a( cos u - e) sin w + a~ sin u cos w, r = a( 1 - e cos u), (18) u - e sin u = nt + Q 0 - w,!1. 0 is the mean longitude at t = 0. The equation of the longitude of the perihelion becomes dw du = p.*~ 2{3-Y"A {--- c2 a e 2 1-e cos u + (-2a+ 4~(1-e 2 )}. ( 1-e cos u) 3 2 a- 4{3-2{3( 1 - e 2 ) + + (1-ecosu)2 (19)

5 122 The increment corresponding to the time interval in the eccentric anomaly varies from u to u + 27T is obtained by integrating ( 18) between these limits. We obtain (20) 211'J.L* 8w = (- a+ 2(3). c 2 a(l-e 2 ) In the solar system, after the elimination of planetary perturbations, the perihelion shows a direct motion which is well represented by the relation (21) Comparing (20) and {21) we obtain the condition (22) -a+ 2(3 = 3 Introducing the values ( 13), we obtain (23) and consequently (24) (25) 9 m~ + m~ i\(r) = 2 f -"--rand the corrective term of Newton's law is determined.

6 CONTENTS Page Preface. Introduction. Chapter I : Motion of a Material Point. 1. Inertial Motion The Universe of Minkowski-Einstein. 3. Waves Associated to the Motion. 4. Motion in a Field Anti-Minkowskian Universes. Chapter II : The Mechanics of Stable Particles. 1. Inertial Motion Motion of the Stable Particles in a Field. Chapter III : Invadantive Mechanics of Systems of Material Points. 1. Inertia, Gravity and Expansion System of n Material Points Inertial Mechanics of a Two Body System The Expansion of the System and Hubbles'Law. Chapter IV : Inertial Movement of a System of Two Particles. 1. Definition of the Stable Particles The Inertial System of Two (Stable) Particles. Chapter V : Mechanics of Continuous Systems.. 1. Principles Pure Inertial Motion of the Components. 3. The Relations Imposed by the Internal Constraints.. 4. The Exp;ession.n ( P) for the Potential Field

7 The Equations of Motion The External Constraints The Integral Form of the Equation of Motion Influence of Heat Chapter VI : The Invariantive Cosmology Introduction Invariantive Mechanics of the System Sn Finiteness or Infiniteness of the Universe Newtonian Interpretation. The Two Kinds of Inertial Forces The Relativistic Cosmology Commentary 1 : Spaces and Axioms of Motion Introduction The Spaces The Invariants of the Spaces Rand U The Differential Operator{) and the corresponding Differential Forms Vectorial Spaces and Generated Vector Fields The Differential Forms Defmed by the Generic Operator Inertial Motion of the Material Particle. Definition Principles and Postulates Commentary 2 : Fibred Spaces and the Representation of Motion Generalities The Universe of Mechanics as a Fibred Spaces The Base of the Fibred Space The Fibre Construction of the Generators of the Characteristic Ring of the Base (after Ion Bucur) Final Considerations Commentary 3: Forces and Constraints Generalities Force and Gravity NOTE : Motion of the Perihelion - Ieronim Mihaila. 118