On a new type of solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point)
In this paper, we proceed to develop a new approach which was formulated first in Ershkov (Acta Mech 228(7):2719–2723, 2017 ) for solving Poisson equations: a new type of the solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point) is suggested in the current research...
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| Veröffentlicht in: | Acta mechanica 2019-03, Vol.230 (3), p.871-883 |
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| creator | Ershkov, Sergey V. Leshchenko, Dmytro |
| description | In this paper, we proceed to develop a new approach which was formulated first in Ershkov (Acta Mech 228(7):2719–2723,
2017
) for solving Poisson equations: a new type of the solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point) is suggested in the current research. Meanwhile, the Euler–Poisson system of equations has been successfully explored for the existence of analytical solutions. As the main result, a new ansatz is suggested for solving Euler–Poisson equations: the Euler–Poisson equations are reduced to a system of three nonlinear ordinary differential equations of first order in regard to three functions
Ω
i
(
i
=
1
,
2
,
3
); the proper elegant approximate solution has been obtained as a set of quasi-periodic cycles via re-inversing the proper elliptical integral. So the system of Euler–Poisson equations is proved to have analytical solutions (in quadratures) only in classical simplifying cases: (1) Lagrange’s case, or (2) Kovalevskaya’s case or (3) Euler’s case or other well-known but particular cases. |
| doi_str_mv | 10.1007/s00707-018-2328-7 |
| format | Article |
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2017
) for solving Poisson equations: a new type of the solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point) is suggested in the current research. Meanwhile, the Euler–Poisson system of equations has been successfully explored for the existence of analytical solutions. As the main result, a new ansatz is suggested for solving Euler–Poisson equations: the Euler–Poisson equations are reduced to a system of three nonlinear ordinary differential equations of first order in regard to three functions
Ω
i
(
i
=
1
,
2
,
3
); the proper elegant approximate solution has been obtained as a set of quasi-periodic cycles via re-inversing the proper elliptical integral. So the system of Euler–Poisson equations is proved to have analytical solutions (in quadratures) only in classical simplifying cases: (1) Lagrange’s case, or (2) Kovalevskaya’s case or (3) Euler’s case or other well-known but particular cases.</description><identifier>ISSN: 0001-5970</identifier><identifier>EISSN: 1619-6937</identifier><identifier>DOI: 10.1007/s00707-018-2328-7</identifier><language>eng</language><publisher>Vienna: Springer Vienna</publisher><subject>Classical and Continuum Physics ; Control ; Differential equations ; Dynamical Systems ; Engineering ; Engineering Fluid Dynamics ; Engineering Thermodynamics ; Heat and Mass Transfer ; Mathematical analysis ; Nonlinear differential equations ; Nonlinear equations ; Ordinary differential equations ; Original Paper ; Quadratures ; Rigid structures ; Rotating bodies ; Rotation ; Solid Mechanics ; Theoretical and Applied Mechanics ; Vibration</subject><ispartof>Acta mechanica, 2019-03, Vol.230 (3), p.871-883</ispartof><rights>Springer-Verlag GmbH Austria, part of Springer Nature 2018</rights><rights>COPYRIGHT 2019 Springer</rights><rights>Acta Mechanica is a copyright of Springer, (2018). All Rights Reserved.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c394t-ca00e8e987ee142fb3c90de7aa1c145bce60af6921d528ff12534bda7dd061343</citedby><cites>FETCH-LOGICAL-c394t-ca00e8e987ee142fb3c90de7aa1c145bce60af6921d528ff12534bda7dd061343</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00707-018-2328-7$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00707-018-2328-7$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Ershkov, Sergey V.</creatorcontrib><creatorcontrib>Leshchenko, Dmytro</creatorcontrib><title>On a new type of solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point)</title><title>Acta mechanica</title><addtitle>Acta Mech</addtitle><description>In this paper, we proceed to develop a new approach which was formulated first in Ershkov (Acta Mech 228(7):2719–2723,
2017
) for solving Poisson equations: a new type of the solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point) is suggested in the current research. Meanwhile, the Euler–Poisson system of equations has been successfully explored for the existence of analytical solutions. As the main result, a new ansatz is suggested for solving Euler–Poisson equations: the Euler–Poisson equations are reduced to a system of three nonlinear ordinary differential equations of first order in regard to three functions
Ω
i
(
i
=
1
,
2
,
3
); the proper elegant approximate solution has been obtained as a set of quasi-periodic cycles via re-inversing the proper elliptical integral. So the system of Euler–Poisson equations is proved to have analytical solutions (in quadratures) only in classical simplifying cases: (1) Lagrange’s case, or (2) Kovalevskaya’s case or (3) Euler’s case or other well-known but particular cases.</description><subject>Classical and Continuum Physics</subject><subject>Control</subject><subject>Differential equations</subject><subject>Dynamical Systems</subject><subject>Engineering</subject><subject>Engineering Fluid Dynamics</subject><subject>Engineering Thermodynamics</subject><subject>Heat and Mass Transfer</subject><subject>Mathematical analysis</subject><subject>Nonlinear differential equations</subject><subject>Nonlinear equations</subject><subject>Ordinary differential equations</subject><subject>Original Paper</subject><subject>Quadratures</subject><subject>Rigid structures</subject><subject>Rotating bodies</subject><subject>Rotation</subject><subject>Solid Mechanics</subject><subject>Theoretical and Applied Mechanics</subject><subject>Vibration</subject><issn>0001-5970</issn><issn>1619-6937</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNqNkc9OHDEMxqOqSN0CD9BbpF7gMNTO_MnMESFokZDgUM5RNnG2oUuyJDO0e-Md-oY8CdlOJU5IlSVbsb5f4vhj7BPCCQLIL7kkkBVgX4la9JV8xxbY4VB1Qy3fswUAYNUOEj6wjznflZOQDS7Yz-vANQ_0i4_bDfHoeI7rRx9WfJOiITsl4i4mfj6tKT0__bmJPucYOD1MevQxZH6U_Mpbvox2y1Mc_3Z5fKTExx-F9b_J8k30YTw-YHtOrzMd_qv77Pbi_PvZt-rq-uvl2elVZeqhGSujAainoZdE2Ai3rM0AlqTWaLBpl4Y60K4bBNpW9M6haOtmabW0Fjqsm3qffZ7vLV94mCiP6i5OKZQnlcC2adoOEYrqZFat9JqUDy6OSZsSlu69iYGcL_3TDqREQGz_FyhLFi32sJsDZ8CkmHMipzbJ3-u0VQhq55qaXVPFNbVzTcnCiJnJRRtWlF5nfxt6ARP-mvs</recordid><startdate>20190301</startdate><enddate>20190301</enddate><creator>Ershkov, Sergey V.</creator><creator>Leshchenko, Dmytro</creator><general>Springer Vienna</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>KR7</scope></search><sort><creationdate>20190301</creationdate><title>On a new type of solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point)</title><author>Ershkov, Sergey V. ; Leshchenko, Dmytro</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c394t-ca00e8e987ee142fb3c90de7aa1c145bce60af6921d528ff12534bda7dd061343</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Classical and Continuum Physics</topic><topic>Control</topic><topic>Differential equations</topic><topic>Dynamical Systems</topic><topic>Engineering</topic><topic>Engineering Fluid Dynamics</topic><topic>Engineering Thermodynamics</topic><topic>Heat and Mass Transfer</topic><topic>Mathematical analysis</topic><topic>Nonlinear differential equations</topic><topic>Nonlinear equations</topic><topic>Ordinary differential equations</topic><topic>Original Paper</topic><topic>Quadratures</topic><topic>Rigid structures</topic><topic>Rotating bodies</topic><topic>Rotation</topic><topic>Solid Mechanics</topic><topic>Theoretical and Applied Mechanics</topic><topic>Vibration</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ershkov, Sergey V.</creatorcontrib><creatorcontrib>Leshchenko, Dmytro</creatorcontrib><collection>CrossRef</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>Civil Engineering Abstracts</collection><jtitle>Acta mechanica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ershkov, Sergey V.</au><au>Leshchenko, Dmytro</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On a new type of solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point)</atitle><jtitle>Acta mechanica</jtitle><stitle>Acta Mech</stitle><date>2019-03-01</date><risdate>2019</risdate><volume>230</volume><issue>3</issue><spage>871</spage><epage>883</epage><pages>871-883</pages><issn>0001-5970</issn><eissn>1619-6937</eissn><abstract>In this paper, we proceed to develop a new approach which was formulated first in Ershkov (Acta Mech 228(7):2719–2723,
2017
) for solving Poisson equations: a new type of the solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point) is suggested in the current research. Meanwhile, the Euler–Poisson system of equations has been successfully explored for the existence of analytical solutions. As the main result, a new ansatz is suggested for solving Euler–Poisson equations: the Euler–Poisson equations are reduced to a system of three nonlinear ordinary differential equations of first order in regard to three functions
Ω
i
(
i
=
1
,
2
,
3
); the proper elegant approximate solution has been obtained as a set of quasi-periodic cycles via re-inversing the proper elliptical integral. So the system of Euler–Poisson equations is proved to have analytical solutions (in quadratures) only in classical simplifying cases: (1) Lagrange’s case, or (2) Kovalevskaya’s case or (3) Euler’s case or other well-known but particular cases.</abstract><cop>Vienna</cop><pub>Springer Vienna</pub><doi>10.1007/s00707-018-2328-7</doi><tpages>13</tpages></addata></record> |
| fulltext | fulltext |
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| ispartof | Acta mechanica, 2019-03, Vol.230 (3), p.871-883 |
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| subjects | Classical and Continuum Physics Control Differential equations Dynamical Systems Engineering Engineering Fluid Dynamics Engineering Thermodynamics Heat and Mass Transfer Mathematical analysis Nonlinear differential equations Nonlinear equations Ordinary differential equations Original Paper Quadratures Rigid structures Rotating bodies Rotation Solid Mechanics Theoretical and Applied Mechanics Vibration |
| title | On a new type of solving procedure for Euler–Poisson equations (rigid body rotation over the fixed point) |
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