The main paradox of KAM-theory for restricted three-body problem (R3BP, celestial mechanics)

Transcription

1 The main paadox of KAM-theoy fo esticted thee-body poblem (R3BP celestial mechanics) Segey V. Eshkov Institute fo Time Natue Exploations M.V. Lomonosov's Moscow State Univesity Leninskie goy 1-1 Moscow Russia Abstact: Hee ae pesented a key points of citicism of KAM (Kolmogoov-Anold- Mose) theoy in the application of main esults to the field of celestial mechanics especially in the case of esticted thee-body poblem. The main paadox of KAM-theoy is that an appopiate Hamilton fomalism should be valid fo the KAM dynamical systems but Hamilton fomalism could not be applied fo esticted thee-body poblem (which is poved to have only the Jacobiantype integal of motion but the integals of enegy momentum ae not invaiants). Besides we should especially note that thee is no analogue of Jacobian-type integal of motion in the case of photogavitational esticted thee-bodt poblem if we take into consideation even a negligible Yakovsky effect. Key Wods: KAM (Kolmogoov-Anold-Mose) theoy Hamilton fomalism Yakovsky effect photogavitational esticted thee-body poblem Jacobian-type integal of motion 1

2 1. Intoduction. Hee ae pesented a key points of citicism about some initial assumptions in KAM- (Kolmogoov-Anold-Mose)-theoy [1-] when the cental KAM-theoem is known to be applied fo eseaches of stability of Sola system in tems of esticted thee-body poblem [3] especially if we conside photogavitational esticted thee-body poblem [4] with additional influence of Yakovsky effect of non-gavitational natue [5]. KAM is the theoy of stability of dynamical systems which should solve a vey specific question in egad to the stability of obits of so-called small bodies in Sola system [1-] in tems of esticted thee-body poblem: indeed dynamics of all the planets is assumed to satisfy to estictions of esticted thee-body poblem (such as infinitesimal masses negligible deviations of the main obital elements etc.). Nevetheless KAM also is known to assume the appopiate Hamilton fomalism in poof of the cental KAM-theoem [1-]: the dynamical system is assumed to be Hamilton system as well as all the mathematical opeations ove such a dynamical system ae assumed to be associated with a pope Hamilton system. Accoding to the Buns theoem [6-7] thee is no othe invaiants except well-known 10 integals fo thee-body poblem (including integal of enegy momentum etc.) this is a classical example of Hamilton system. But in case of esticted thee-body poblem thee is no othe invaiants except only one Jacobian-type integal of motion [3]. Such a contadiction is the main paadox of KAM-theoy: it adopts all the estictions of esticted thee-body poblem but nevetheless it poves to use the Hamilton fomalism which assumes the consevation of all othe invaiants (the integal of enegy momentum etc.).

3 . Equations of motion. Let us conside the system of ODE fo photogavitational esticted thee-body poblem unde the influence of Yakovsky effect at given initial conditions [5]. We conside thee bodies of masses m₁ m₂ and m such that m₁ > m₂ and m is an infinitesimal mass. The two pimaies m₁ and m₂ ae souces of adiation; q₁ and q₂ ae factos chaacteizing the adiation effects of the two pimaies espectively. We assume that m₂ is an oblate spheoid. The effect of oblateness is denoted by the facto A₂. Let i (i =1 ) be the distances between the cente of mass of the bodies m₁ and m₂ and the cente of mass of body m [5]. The unit of mass is chosen so that the sum of the masses of finite bodies is equal to 1. We suppose that m₁ = 1 - μ and m₂ = μ whee μ is the atio of the mass of the smalle pimay to the total mass of the pimaies and 0 μ ½. The unit of distance is taken as the distance between the pimaies. The unit of time is chosen so that the gavitational constant is equal to 1. The thee dimensional esticted thee-body poblem (we take also into consideation the influence of Yakovsky effect) with an oblate pimay m₂ and both pimaies adiating could be pesented in baycentic otating co-odinate system by the equations of motion below [5]: x n y x Y x ( t) y n x y Y y ( t).1 z z Y ( t) z n q (1 ) z x y 1. 1 q A 3

4 - whee Y x (t) Y y (t) Y z (t) ae the pojecting of Yakovsky effect acceleation Y (t) onto the appopiate axis Ox Oy Oz - besides whee n 3 1 A - is the angula velocity of the otating coodinate system and A₂ - is the oblateness coefficient. Hee A AE AP 5R - whee AE is the equatoial adius AP is the pola adius and R is the distance between pimaies. Besides we should note that 1 ( x ) y z ( x 1 ) y z - ae the distances of infinitesimal mass fom the pimaies [5]. We neglect the elativistic Poynting-Robetson effect which may be teated as a petubation fo cosmic dust (o fo small paticles less than 1 cm in diamete) see Chenikov [8] as well as we neglect the effect of vaiable masses of thee-bodies [9]. The possible ways of simplifying of equations (.1): - if we assume effect of oblateness is zeo A₂ = 0 ( n = 1) it means m₂ is non-oblate spheoid (we will conside only such a case below); - if we assume q₁ = q₂ = 1 it means the case of esticted thee-body poblem. 4

5 3. Anold-diffusion. The equations of esticted thee-body poblem ae poved to descibe the system with non-hamilton fomalism. The additional obvious poof could be found in the stuctue of system (.1) if we attentively analyze the ight pat of equations (.1):... n y n x... - but any components of velocity must be excepted fo Hamilton system in the final expessions fo balance of momentum [3]. This is axiom fo the Hamilton systems: the Hamilton systems ae assumed to be the systems without diffusion. That s why Anold [1] was the 1-st in celestial mechanics who suggested to conside the Hamilton systems with weak diffusion which fom so-called Anold web (Fig.1): such a suggestion was vey moden oiginal coection fo KAM methodology in egad to esticted thee-body poblem. Fig.1. The schematic imagination of an Anold web. It means that such a dynamical systems should have a weak Anold-diffusion [1]: the classical invaiants of such a system ae poved not emaining the same (the integal of 5

6 enegy momentum etc.) but all of them ae subjected to a negligible changing (diffusion) duing a lage time-peiod. Besides the esticted thee-body poblem is poved to have a new the only stable invaiant = Jacobian-type integal of motion [3]. Accoding to [3] we could obtain fom the equations of system (.1) a Jacobian-type integal of motion: ( x ) ( y ) ( z ) ( x y z) C (3.1) - whee C is so-called Jacobian constant. As it was poved in [10] such a Jacobiantype integal of motion should not be depending on time fo lage time-peiod. Additionally we should especially note obvious fact: in the case of photogavitational esticted thee-body poblem with Yakovsky effect [5] thee is no analogue of Jacobian-type integal fo ODE system of motion (.1). 4. Conclusion. We discussed a key points of citicism of KAM (Kolmogoov-Anold-Mose) theoy in the application to the field of celestial mechanics especially in the case of esticted thee-body poblem. The main paadox of KAM-theoy is that appopiate Hamilton fomalism should be valid fo the KAM dynamical systems but Hamilton fomalism could not be applied fo esticted thee-body poblem. Nevetheless KAM-theoy tied to pedict the stability fo Sola system duing a lage time-peiod despite of the fact that cental KAM-theoem adopts all the estictions of esticted thee-body poblem (which was chosen as a basis fo the modelling of Sola system). Such a paadox could be successfully solved if we conside Sola system as dynamical system with Anold diffusion. 6

7 Besides we should especially note that thee is no analogue of Jacobian-type integal of motion in the case of photogavitational esticted thee-body poblem if we take into consideation even a negligible Yakovsky effect. Acknowledgements I am thankful to CNews Russia poject (Science & Technology Foum banches Gavitation KAM-theoy ) - fo valuable discussions in pepaing this manuscipt. Especially I am thankful to D. P.V.Fedotov Col. L.Vladimiov-Paaligon - fo valuable suggestions in peliminay discussions of this manuscipt. Refeences: [1] Anold V. (1978). Mathematical Methods of Classical Mechanics. Spinge New Yok. [] Anold V. (1963). Small Diviso Poblems in Classical and Celestial Mechanics. Russ. Math. Suveys [3] Szebehely V. (1967). Theoy of Obits. The Resticted Poblem of Thee Bodies. Yale Univesity New Haven Connecticut. Academic Pess New-Yok and London. [4] Shankaan Shama J.P. and Ishwa B. (011). Equilibium points in the genealized photogavitational non-plana esticted thee body poblem. Intenational Jounal of Engineeing Science and Technology Vol. 3 () pp [5] Eshkov S.V. (01). The Yakovsky effect in genealized photogavitational 3- body poblem. Planetay and Space Science Vol. 73 (1) pp [6] Buns H. (1887). Uеbe die Integale de Vielkoepe-Poblems. Acta math. Bd. 11 p [7] Duboshin G.N. (1968). Nebesnaja mehanika. Osnovnye zadachi i metody. Moscow: Nauka (handbook fo Celestial Mechanics in ussian). [8] Chenikov Y.A. (1970). The Photogavitational Resticted Thee-Body Poblem. Soviet Astonomy Vol. 14 p

8 [9] Jagadish Singh Oni Leke (010). Stability of the photogavitational esticted theebody poblem with vaiable masses. Astophys Space Sci (010) 36: [10] Nekhooshev N.N. (1977). Exponential estimate on the stability time of nea integable Hamiltonian systems Russ. Math. Suveys 3(6)