The 8 reference contexts in paper O. Zagryadskij A., О. Загрядский А. (2016) “Бертрановские системы и их фазовое пространство // Bertrand systems and their phase space” / spz:neicon:technomag:y:2014:i:2:p:365-386

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    ŒÔ‰ÂÎÂÌËÂ2.÷ÂÌÚ‡Î ̧Ì ̊È(Ì Á‡‚ËÒˇ ̆ËÈÓÚφ) ÔÓÚÂ̈ˇÎV(u)( ̃V(u)) ̇ÁÓ‚ÂÏ Á‡Ï ͇̊ ̨ ̆ËÏ, ÂÒÎË: 1) ÒÛ ̆ÂÒÚ‚ÛÂÚÌÂÓÒÓ·‡ˇÓ„‡ÌË ̃ÂÌ̇ˇÌÂÍÛ„Ó‚‡ˇÓ·ËÚ‡‚S(S); 2) ‚ÒˇÍ‡ˇÌÂÓÒÓ·‡ˇÓ„‡ÌË ̃ÂÌ̇ˇÓ·ËÚ‡‚S(S) Á‡ÏÍÌÛÚ‡. —ÎÂ‰Û ̨ ̆‡ˇÚÂÓÂχ
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    [5, 7]
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    ÓÔËÒ ̊‚‡ÂÚ‚Ò ÔÓ‚ÂıÌÓÒÚË(·ÂÁ ̋Í‚‡ÚÓÓ‚)Ò ÔÓÚÂ̈ˇ·ÏË, ÔË‚Ó‰ˇ ̆ËÏËÍ Á‡ÏÍÌÛÚ ̊ÏÓ·ËÚ‡Ï,Ú.Â. ‚Ò ԇ ̊(S, V), „ ‰ ÂS| ÔÓ‚ÂıÌÓÒÚ ̧‚‡ ̆ÂÌˡ,‡ V| Á‡Ï ͇̊ ̨ ̆ËÈÔÓÚÂ̈ˇÎ̇ ÌÂÈ.Õ‡ÁÓ‚ÂÏÚ‡ÍËÂÔ‡ ̊·ÂÚ‡ÌÓ‚ÒÍËÏË.
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    — χÚÂχÚË ̃ÂÒÍÓÈÚÓ ̃ÍËÁÂÌˡM4Ô‰ÒÚ‡‚ΡÂÚ ËÁ Ò·ˇÒËÏÔÎÂÍÚË ̃ÂÒÍÓÂÏÌÓ„ÓÓ·‡ÁËÂÒ ÌÂÍÓÈÒËÏÔÎÂÍÚË ̃ÂÒÍÓÈÒÚÛÍÚÛÓÈ ω=dpu∧du+dpφ∧dφ.(8) —ËÏÔÎÂÍÚË ̃ÂÒ͇ˇÒÚÛÍÚÛ‡ÚÂÒÌÓÒ‚ˇÁ‡Ì‡Ò ‚ÂÍÚÓÓÏÍÓÒÓ„Ó„‡‰ËÂÌÚ‡ÓÚ Ò͇ΡÌÓÈÙÛÌ͈ËË F ̇M4Ë ÒÍÓ·ÍÓȜۇÒÒÓ̇
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    . ŒÔ‰ÂÎÂÌËÂ3. ÓÓ‰ËÌ‡Ú ̊‚ÂÍÚÓ‡sgradF(x1, x2, x3, x4)ÓÔ‰ÂΡ ̨ÚÒˇËÁ ÒÓÓÚÌÓ ̄ÂÌˡω(v,sgradF) =v(F), ÍÓÚÓÓ‰ÓÎÊÌÓ· ̊Ú ̧‚ ̊ÔÓÎÌÂÌӉΡ ÔÓËÁ‚ÓÎ ̧ÌÓ„Ó‚ÂÍÚÓÌÓ„Ó ÔÓΡv. ¬ ÎÓ͇Π̧Ì ̊ıÍÓÓ‰Ë̇ڇı (sgradF)i=ωij ∂F ∂xj ,(9) „ ‰ Â(x1, x2, x3, x4)| ÔÂÂÓ·ÓÁ̇ ̃ÂÌˉΡ(θ, φ, pθ, pφ), ‡ωij| χÚˈ‡,Ó·‡Ú̇ˇÍ χÚˈ ωijÙÓÏ ̊(8). ƒÎˇ ÙÛÌ͈ËÈF(x1, x2, x3, x4)ËG(x1, x2, x3, x4)ÒÍӷ͇œÛ‡ÒÒÓ̇ÓÔ‰ÂΡÂÚÒˇ ÒΉÛ
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    ŒÚÒ ̨‰‡Ò‡ÁÛÒΉÛÚ, ̃ÚÓ{F, G}= (sgradE)Tωij(sgradK) = 0. œÓÎÌÓÚ‡sgradKÒΉÛÂÚËÁ ÚÓ„Ó, ̃ÚÓ Â„Ó ËÌÚ„‡Î ̧Ì ̊ÂÎËÌËË| ̋ÚÓ ÓÍÛÊÌÓÒÚ˂ˉ‡{∗}×S1×{0}×{∗}. ”Ú‚ÂʉÂÌË ‰Ó͇Á‡ÌÓ. —ËÒÚÂÏ ̊¡ÂÚ‡Ì‡ÔÓıÓÊË̇ ËÌÚ„ËÛÂÏ ̊ÂÔÓ ÀËÛ‚ËÎÎ ̨„‡ÏËÎ ̧ÚÓÌÓ‚ ̊ÒËÒÚÂÏ ̊
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    . —ӄ·ÒÌÓÍ·ÒÒË ̃ÂÒÍËÏÂÁÛÎ ̧Ú‡Ú‡Ï,Û Ú‡ÍËıÒËÒÚÂÏ(‚ ̇ ̄ÂÏÒÎÛ ̃‡ÂÒËÒÚÂÏÒ ̃ÂÚ ̊ÂıÏÂÌ ̊ÏÙ‡ÁÓ‚ ̊ÏÔÓÒÚ‡ÌÒÚ‚ÓÏ)ÒÛ ̆ÂÒÚ‚ÛÂÚ‰ÓÔÓÎÌËÚÂÎ ̧Ì ̊ÈÍ „‡ÏËÎ ̧ÚÓÌˇÌÛHÔÂ‚ ̊È ËÌÚ„‡ÎG, Ú‡ÍÓÈ, ̃ÚÓHËGÙÛÌ͈ËÓ̇Π̧ÌÓÌÂÁ‡‚ËÒËÏ ̊,̇ıÓ‰ˇÚÒˇ‚ ËÌ‚ÓÎ ̨ˆËË,‡ Ëı ÔÓÚÓÍËsgradH, sgradKÔÓÎÌ ̊.
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    ‚ÂÏˇÔÓ‰ÓÎʇÂÚÒˇ‰Ó ·ÂÒÍÓÌ ̃ÌÓÒÚË. œÂËÓ‰‰‚ËÊÂÌˡÔÓ Ó„‡ÌË ̃ÂÌÌ ̊ÏÓ·ËÚ‡Ï.œÂËÓ‰‰‚ËÊÂÌˡÔÓ Ó·ËÚ‚ ÒÎÛ ̃‡Â ÔÎÓÒÍÓÒÚË,‡ Ú‡ÍÊ‚ ÒÎÛ ̃‡ÂÒÙÂ ̊‰Îˇ Á‡Ï ͇̊ ̨ ̆ËıÔÓÚÂ̈ˇÎÓ‚ËÁ‚ÂÒÚÂÌ.ƒÎˇ ‚ÒÂı Ú‡ÍËıÔÓ‚ÂıÌÓÒÚÂÈË ÔÓËÁ‚ÓÎ ̧Ì ̊ıˆÂÌÚ‡Î ̧Ì ̊ıÔÓÚÂ̈ˇÎӂ̇ ÌËı Òӄ·ÒÌÓÚÂÓÂÏ √Ó‰Ó̇
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    ÔÂËÓ‰‰‚ËÊÂÌˡÔÓ Á‡ÏÍÌÛÚÓÈÚ‡ÂÍÚÓËËÁ‡‚ËÒËÚÚÓÎ ̧ÍÓÓÚ ËÌÚ„‡Î‡EË Ì Á‡‚ËÒËÚÓÚK. ¬ ÒÎÛ ̃‡ÂÒ ÔÎÓÒÍÓÒÚ ̧ ̨ÀÓ·‡ ̃‚ÒÍÓ„Óˇ‚Ì ̊Ȃˉ Ú‡ÍÓÈÁ‡‚ËÒËÏÓÒÚËÔÓÎÛ ̃ÂÌ ¬.¬. ÓÁÎÓ‚ ̊Ï[3]. —ÙÓÏÛÎËÛÂÏÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ÂÂÛÚ‚ÂʉÂÌËÂ.
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    ËÊÂÌˡÔÓ Ó„‡ÌË ̃ÂÌÌ ̊ÏÓ·ËÚ‡Ï.œÂËÓ‰‰‚ËÊÂÌˡÔÓ Ó·ËÚ‚ ÒÎÛ ̃‡Â ÔÎÓÒÍÓÒÚË,‡ Ú‡ÍÊ‚ ÒÎÛ ̃‡ÂÒÙÂ ̊‰Îˇ Á‡Ï ͇̊ ̨ ̆ËıÔÓÚÂ̈ˇÎÓ‚ËÁ‚ÂÒÚÂÌ.ƒÎˇ ‚ÒÂı Ú‡ÍËıÔÓ‚ÂıÌÓÒÚÂÈË ÔÓËÁ‚ÓÎ ̧Ì ̊ıˆÂÌÚ‡Î ̧Ì ̊ıÔÓÚÂ̈ˇÎӂ̇ ÌËı Òӄ·ÒÌÓÚÂÓÂÏ √Ó‰Ó̇[1] ÔÂËÓ‰‰‚ËÊÂÌˡÔÓ Á‡ÏÍÌÛÚÓÈÚ‡ÂÍÚÓËËÁ‡‚ËÒËÚÚÓÎ ̧ÍÓÓÚ ËÌÚ„‡Î‡EË Ì Á‡‚ËÒËÚÓÚK. ¬ ÒÎÛ ̃‡ÂÒ ÔÎÓÒÍÓÒÚ ̧ ̨ÀÓ·‡ ̃‚ÒÍÓ„Óˇ‚Ì ̊Ȃˉ Ú‡ÍÓÈÁ‡‚ËÒËÏÓÒÚËÔÓÎÛ ̃ÂÌ ¬.¬. ÓÁÎÓ‚ ̊Ï
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    . —ÙÓÏÛÎËÛÂÏÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ÂÂÛÚ‚ÂʉÂÌËÂ. ”Ú‚ÂʉÂÌËÂ2 ( ÓÁÎÓ‚).œÛÒÚ ̧̇ ÔÓ‚ÂıÌÓÒÚËCLÒ ÍÓÓ‰Ë̇ڇÏË(θ, φ)Ë ÏÂÚËÍÓÈ (3), „‰Âc=−1,μ= 1, ‰ÂÈÒÚ‚ÛÂڇ̇ÎÓ„Ì ̧ ̨ÚÓÌÓ‚ÒÍÓ„ÓÔÓÚÂ̈ˇ·V1ËÎË ‡Ì‡ÎÓ„√ÛÍÓ‚ÒÍÓ„ÓV2.
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    ‚ÒÍÓ„ÓÔÓÚÂ̈ˇ·.“Ó„‰‡ÔÂËÓ‰T‰‚ËÊÂÌˡÔÓ Á‡ÏÍÌÛÚÓÈÚ‡ÂÍÚÓËË~r(t)Ì Á‡‚ËÒËÚÓÚ ËÌÚ„‡Î‡ÍËÌÂÚË ̃ÂÒÍÓ„ÓÏÓÏÂÌÚ‡KË ‚ ̊ ̃ËÒΡÂÚÒˇÔÓ ÙÓÏÛΠT= πq 2 √ ∆   √ ∆ +c √ E( √ ∆ +c) + 2A + √ ∆−c √ E(c− √ ∆) + 2A  , „‰Â∆ =c2+ 4t. « ‡ Ï Â ̃ ‡ Ì Ë Â 4. ƒ‡ÌÌÓÂÛÚ‚ÂʉÂÌËÂ,Í ‡ Í Ë Â„Ó ‰Ó͇Á‡ÚÂÎ ̧ÒÚ‚Ó‰ÓÒÎÓ‚ÌÓÔÂÂÌÓÒˇÚÒˇ Ë Ì‡ ÒÎÛ ̃‡ÈÔÓ‚ÂıÌÓÒÚËÒ ÔÒ‚‰ÓËχÌÓ‚ÓÈÏÂÚËÍÓÈ. ƒ Ó Í ‡ Á ‡ Ú Â Î ̧ Ò Ú ‚ Ó. —ӄ·ÒÌÓ
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    Ó·ËÚ‡Ò ̋ÌÂ„ËÂÈEË ÍËÌÂÚË ̃ÂÒÍËÏÏÓÏÂÌÚÓÏK Ô‰ÒÚ‡‚ΡÂÚËÁ Ò·ˇÍË‚Û ̨,ÍÓÚÓ‡ˇÔÓ‰Ó·ÌÓÒËÌÛÒÓˉÂÍÓηÎÂÚÒˇÏÂʉۉ‚ÛÏˇÔ‡‡ÎÎÂΡÏËθ= ̃θ1(ÔÂˈÂÌÚ)Ëθ= ̃θ2(‡ÔÓˆÂÌÚ): θ2= E−B μ2K2 − c 2 + √√√√ √E−B μ2K2 − c 2 )2 +t− 2A μ2K2 sin 2(φ+φ0) μ .(11) œË ̋ÚÓÏÁ‡pÓ·ÓÓÚÓ‚‚ÓÍÛ„ÔÓ‚ÂıÌÓÒÚË,Ó·ËÚ‡ÒÓ‚Â ̄ËÚqÍÓη‡ÌËÈ.œÂËÓ‰‰‚ËÊÂÌˡ ÔÓ Ó·ËÚÂT‚q‡Á ·ÓÎ ̧ ̄ÂÔÂËÓ‰‡Ó‰ÌÓ„ÓÍÓη‡Ìˡ ̃T.
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    ‡ Í ÔÓ͇Á ̊‚‡ ̨ÚÛÚ‚ÂʉÂÌˡ4, 5, ‚ÂÏˇ‚‰ÓÎ ̧ËÌÚ„‡Î ̧Ì ̊ıÚ‡ÂÍÚÓËÈÌ ‚Ò„‰‡ÔÓ‰ÓÎʇÂÚÒˇ‰Ó ·ÂÒÍÓÌ ̃ÌÓÒÚË, ̃ÚÓ ‚Π̃ÂÚÌÂÔÓÎÌÓÚÛÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËıÙ‡ÁÓ‚ ̊ıÔÓÚÓÍÓ‚,‡ Í ‡ Í ÒΉÒÚ‚ËÂ| Ì‚ÓÁÏÓÊÌÓÒÚË̇ÔˇÏÛ ̨ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ÌÂÍÓÚÓ ̊ÂÍ·ÒÒË ̃ÂÒÍËÂÂÁÛÎ ̧Ú‡Ú ̊, Ú‡ÍËÂ,Í ‡ Í ÚÂÓÂχÀËÛ‚ËÎΡ
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    .¬ÏÂÒÚÓ ̋ÚÓ„ÓÌÛÊÌÓÔÓÎ ̧ÁÓ‚‡Ú ̧ÒˇËı ‡Ì‡ÎÓ„‡ÏË,̇ÔËÏÂ ÚÂÓÂÏÓȿΠ̄ÍË̇[9]. ¡ÓÎÂÂÚÓ„Ó,Á̇ÌËÂÚÓ ̃ÌÓÈÁ‡‚ËÒËÏÓÒÚËÔÂËÓ‰‡T‰‚ËÊÂÌˡ ÔÓ Á‡ÏÍÌÛÚÓÈÓ·ËÚÂÓÚ ÔÓÎÌÓÈ ̋ÌÂ„ËËEÔÓÏÓ„‡ÂÚˇ‚ÌÓÔÓÒÚÓËÚ ̧ÍÓÓ‰ËÌ‡Ú ̊<‰ÂÈÒÚ‚ËÂÛ„ÓÎ>,‚ ÍÓÚÓ ̊ıÙ‡ÁÓ‚ ̊ÈÔÓÚÓÍ‚ ̊ÔˇÏΡÂÚÒˇË ÏÌÓ„ËÂÓ· ̇ÂÍÚ ̊(ÒËÏÔÎÂÍÚË ̃ÂÒ͇ˇÙÓχ) ÔËÌËχ ̨Ú̇˷ÓÎÂÂÔÓÒÚÓȂˉ. ƒÎˇÓ‰ÌÓÓ‰Ì ̊ıÏÂı‡ÌË ̃ÂÒÍËıÒËÒÚÂÏÒÚÓÂÌËÂÙ‡ÁÓ‚Ó„ÓÔÓÒÚ‡ÌÒÚ‚‡ÌÂÒÂÚÏÌÓ„Ó ËÌÙÓχ
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    ‡ Í ÔÓ͇Á ̊‚‡ ̨ÚÛÚ‚ÂʉÂÌˡ4, 5, ‚ÂÏˇ‚‰ÓÎ ̧ËÌÚ„‡Î ̧Ì ̊ıÚ‡ÂÍÚÓËÈÌ ‚Ò„‰‡ÔÓ‰ÓÎʇÂÚÒˇ‰Ó ·ÂÒÍÓÌ ̃ÌÓÒÚË, ̃ÚÓ ‚Π̃ÂÚÌÂÔÓÎÌÓÚÛÒÓÓÚ‚ÂÚÒÚ‚Û ̨ ̆ËıÙ‡ÁÓ‚ ̊ıÔÓÚÓÍÓ‚,‡ Í ‡ Í ÒΉÒÚ‚ËÂ| Ì‚ÓÁÏÓÊÌÓÒÚË̇ÔˇÏÛ ̨ËÒÔÓÎ ̧ÁÓ‚‡Ú ̧ÌÂÍÓÚÓ ̊ÂÍ·ÒÒË ̃ÂÒÍËÂÂÁÛÎ ̧Ú‡Ú ̊, Ú‡ÍËÂ,Í ‡ Í ÚÂÓÂχÀËÛ‚ËÎΡ[10].¬ÏÂÒÚÓ ̋ÚÓ„ÓÌÛÊÌÓÔÓÎ ̧ÁÓ‚‡Ú ̧ÒˇËı ‡Ì‡ÎÓ„‡ÏË,̇ÔËÏÂ ÚÂÓÂÏÓȿΠ̄ÍË̇
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    . ¡ÓÎÂÂÚÓ„Ó,Á̇ÌËÂÚÓ ̃ÌÓÈÁ‡‚ËÒËÏÓÒÚËÔÂËÓ‰‡T‰‚ËÊÂÌˡ ÔÓ Á‡ÏÍÌÛÚÓÈÓ·ËÚÂÓÚ ÔÓÎÌÓÈ ̋ÌÂ„ËËEÔÓÏÓ„‡ÂÚˇ‚ÌÓÔÓÒÚÓËÚ ̧ÍÓÓ‰ËÌ‡Ú ̊<‰ÂÈÒÚ‚ËÂÛ„ÓÎ>,‚ ÍÓÚÓ ̊ıÙ‡ÁÓ‚ ̊ÈÔÓÚÓÍ‚ ̊ÔˇÏΡÂÚÒˇË ÏÌÓ„ËÂÓ· ̇ÂÍÚ ̊(ÒËÏÔÎÂÍÚË ̃ÂÒ͇ˇÙÓχ) ÔËÌËχ ̨Ú̇˷ÓÎÂÂÔÓÒÚÓȂˉ. ƒÎˇÓ‰ÌÓÓ‰Ì ̊ıÏÂı‡ÌË ̃ÂÒÍËıÒËÒÚÂÏÒÚÓÂÌËÂÙ‡ÁÓ‚Ó„ÓÔÓÒÚ‡ÌÒÚ‚‡ÌÂÒÂÚÏÌÓ„Ó ËÌÙÓχˆËËÓ ÚËÔ‚ÓÁÌË͇ ̨ ̆„Ӊ‚ËÊÂÌˡ.
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