Mathematical Researches

fa ویژگی های فضای مدارهای تشدیدهای همیلتونی نامعین Properties of Orbits Space of Indefinite Hamiltonian Resonances هندسه سیستم های دینامیکی Geometry of dynamical systems مقاله استخراج شده از پایان نامه Research Paper در این مقاله به تشدیدهای همیلتونی نامعین و به‌خصوص فرم نرمال توابع همیلتونی مربوطه برحسب مختصات عمل-زاویه می‌پردازیم. آنگاه همچنین برخی مدل‌های مختلف از فیزیک و مکانیک گذرنده از تشدیدهای همیلتونی نامعین، را معرفی می‌کنیم. ویژگی‌های فضای حالت و فضای مدارهای مربوط به تشدیدهای همیلتونی نامعین به‌خصوص ویژگی‌های توپولوژیکی و خاصیت بی‌کرانی آن‌ها را به‌عنوان نتیجه اصلی مقاله بررسی خواهیم‌ کرد.    ABSTRACT                                       The study of Hamiltonian systems around the elliptic equilibrium points, which is a non-generic subject in the study of Hamiltonian systems, has received attention in recent decades. Such systems appear in many applied models, including molecular physics, galactic dynamics, and mechanics. Consider a Hamiltonian of <m:omath><m:r>n</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:7.15pt; height:15pt" >  degrees of freedom, whose quadratic part is as follows, <table align="right" class="MsoTableGrid" dir="rtl" style="border-collapse:collapse; border:none"> <tbody> <tr> <td style="width:48px; padding:0in 7px 0in 7px" valign="top">(1)</td> <td style="width:576px; padding:0in 7px 0in 7px" valign="top"><m:omathpara><m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Η</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:r>q</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>.</m:r><m:r>p</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r><m:f><m:fpr><m:ctrlpr></m:ctrlpr></m:fpr><m:num><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>1</m:r></m:num><m:den><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:den></m:f><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="undOvr"><m:ctrlpr></m:ctrlpr></m:limloc></m:chr></m:narypr><m:sub><m:r>j</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=1</m:r></m:sub><m:sup><m:r>n</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>ω</m:r></m:e><m:sub><m:r>j</m:r></m:sub></m:ssub><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>q</m:r></m:e><m:sub><m:r>j</m:r></m:sub><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sup></m:ssubsup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>+</m:r><m:ssubsup><m:ssubsuppr><m:ctrlpr></m:ctrlpr></m:ssubsuppr><m:e><m:r>p</m:r></m:e><m:sub><m:r>j</m:r></m:sub><m:sup><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sup></m:ssubsup></m:e></m:d></m:e></m:nary><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,      </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>ω</m:r></m:e><m:sub><m:r>j</m:r></m:sub></m:ssub><m:r>ϵ</m:r><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="i"></m:sty></m:scr></m:rpr>Z</m:r><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,  </m:r><m:r>j</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=1.….</m:r><m:r>n</m:r><m:r>.</m:r></m:omath></m:omathpara><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:273.4pt; height:42pt" > </td> </tr> </tbody> </table> The vector <m:omath><m:r>ω</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>≔</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>ω</m:r></m:e><m:sub><m:r>1</m:r></m:sub></m:ssub><m:r>,</m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>ω</m:r></m:e><m:sub><m:r>2</m:r></m:sub></m:ssub><m:r>, …, </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>ω</m:r></m:e><m:sub><m:r>n</m:r></m:sub></m:ssub></m:e></m:d></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:103.5pt; height:15pt" >  is called frequency vector (related to <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Η</m:r></m:e><m:sub><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></m:sub></m:ssub></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:12.75pt; height:15pt" > ) and its components are called frequency. If all frequencies are non-zero, <m:omath><m:r>ω</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:9pt; height:15pt" >  is called non-degenerate, and if at least one of the frequencies is zero, we say <m:omath><m:r>ω</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:9pt; height:15pt" >  is degenerate. Definition. Any integer-valued vector perpendicular to the frequency vector <m:omath><m:r>ω</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:9pt; height:15pt" >  is called a resonance or annihilator vector. In other words, <m:omath><m:r>k</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>∈</m:r><m:f><m:fpr><m:type m:val="skw"><m:ctrlpr></m:ctrlpr></m:type></m:fpr><m:num><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:r><m:rpr><m:scr m:val="double-struck"><m:sty m:val="p"></m:sty></m:scr></m:rpr>Z</m:r></m:e><m:sup><m:r>n</m:r></m:sup></m:ssup></m:num><m:den><m:d><m:dpr><m:begchr m:val="{"><m:endchr m:val="}"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>0</m:r></m:e></m:d></m:den></m:f></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:55.15pt; height:23.65pt" > is a resonant vector for the frequency vector <m:r></m:r><m:omath>ω</m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:9pt; height:15pt" >  if <m:omathpara><m:omathparapr><m:jc m:val="center"></m:jc></m:omathparapr><m:omath><m:d><m:dpr><m:begchr m:val="〈"><m:endchr m:val="〉"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>k</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>,</m:r><m:r>ω</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r><m:nary><m:narypr><m:chr m:val="∑"><m:limloc m:val="undOvr"><m:ctrlpr></m:ctrlpr></m:limloc></m:chr></m:narypr><m:sub><m:r>j</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=1</m:r></m:sub><m:sup><m:r>n</m:r></m:sup><m:e><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>k</m:r></m:e><m:sub><m:r>j</m:r></m:sub></m:ssub><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>ω</m:r></m:e><m:sub><m:r>j</m:r></m:sub></m:ssub></m:e></m:nary><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=0.</m:r></m:omath></m:omathpara><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:109.9pt; height:42pt" > If there is such an annihilator vector for <m:omath><m:r>ω</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:9pt; height:15pt" > , we call the <m:omath><m:r>ω</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:9pt; height:15pt" >  resonance frequency vector. If at least one of the components of the non-degenerate resonance frequency vector <m:omath><m:r>ω</m:r></m:omath><img alt="" id="_x0000_i1025" src="data:image/png;base64,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" style="width:9pt; height:15pt" >  is negative, the Hamiltonian resonance is called indefinite. Material and methods In this paper, we focus on indefinite Hamiltonian resonances, so that by introducing some important physical models, we review their occurrence in different parts of science including the satellite problem, plasma interactions, Pais-Ollenbeck oscillators (PU) and a problem of cosmology. Then, by using the normal form of indefinite Hamiltonian resonances, we will discuss the space structure of the corresponding vector field. Results and discussion As we mentioned there are many models passing indefinite Hamiltonian resonances. We clear the space of orbits of indefinite Hamiltonian resonances topologically. Specifically, we will see it can be unbounded in the comparison to space of orbits of other Hamiltonian resonances.   تشدید همیلتونی, تشدید نامعین, فرم نرمال, مختصات عمل-زاویه, فضای مدارها. Hamiltonian resonance, Indefinite resonance, Normal form, Action-angle coordinates, Space of orbits 19 34 http://mmr.khu.ac.ir/browse.php?a_code=A-10-1715-3&slc_lang=fa&sid=1 Reza Mazrooei-Sebdani رضا مزروعی سبدانی mazrooei@iut.ac.ir 10031947532846006822 10031947532846006822 Yes Isfahan University of Technology دانشگاه صنعتی اصفهان Omid Toghraei امید طغرایی o.toghraei@math.iut.ac.ir 10031947532846006823 10031947532846006823 No Isfahan University of Technology دانشگاه صنعتی اصفهان