Invariants for dissipative nonlinear systems by using rescaling
A rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second‐order differential equation with a friction term proportional to the velocity, β(t)v. The transformation is of the form (x,t)→(ξ,θ), where x=ξC(t)+α(t), dθ=d t/A 2(...
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| Veröffentlicht in: | J. Math. Phys. (N.Y.); (United States) 1985-01, Vol.26 (1), p.68-73 |
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| container_title | J. Math. Phys. (N.Y.); (United States) |
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| creator | Feix, Marc R. Lewis, H. Ralph |
| description | A rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second‐order differential equation with a friction term proportional to the velocity, β(t)v. The transformation is of the form (x,t)→(ξ,θ), where x=ξC(t)+α(t), dθ=d
t/A
2(t). This rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant. The invariants are found explicitly for a power‐law potential, γ(t)x
m+1/(m+1), and an arbitrary coefficient of friction β(t). We show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist. For certain parameters, the asymptotic solution is a self‐similar solution that is an attractor for all initial conditions. The technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications. |
| doi_str_mv | 10.1063/1.526750 |
| format | Article |
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t/A
2(t). This rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant. The invariants are found explicitly for a power‐law potential, γ(t)x
m+1/(m+1), and an arbitrary coefficient of friction β(t). We show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist. For certain parameters, the asymptotic solution is a self‐similar solution that is an attractor for all initial conditions. The technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications.</description><identifier>ISSN: 0022-2488</identifier><identifier>EISSN: 1089-7658</identifier><identifier>DOI: 10.1063/1.526750</identifier><identifier>CODEN: JMAPAQ</identifier><language>eng</language><publisher>Melville, NY: American Institute of Physics</publisher><subject>990200 - Mathematics & Computers ; ANALYTICAL SOLUTION ; DIFFERENTIAL EQUATIONS ; DISSIPATION FACTOR ; DISTURBANCES ; EQUATIONS ; Exact sciences and technology ; FLUID MECHANICS ; FRICTION ; GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE ; Geometry, differential geometry, and topology ; HYDRODYNAMICS ; MAGNETOHYDRODYNAMICS ; Mathematical methods in physics ; MECHANICS ; NUMERICAL SOLUTION ; Physics ; POTENTIALS ; SPACE-TIME ; TRANSFORMATIONS ; VELOCITY</subject><ispartof>J. Math. Phys. (N.Y.); (United States), 1985-01, Vol.26 (1), p.68-73</ispartof><rights>American Institute of Physics</rights><rights>1985 INIST-CNRS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c349t-20c63b581a15597b1114dd4cbcee42f80fdfa53a5c936f3525e910ac75aae2fa3</citedby><cites>FETCH-LOGICAL-c349t-20c63b581a15597b1114dd4cbcee42f80fdfa53a5c936f3525e910ac75aae2fa3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://pubs.aip.org/jmp/article-lookup/doi/10.1063/1.526750$$EHTML$$P50$$Gscitation$$H</linktohtml><link.rule.ids>314,780,784,885,1559,4024,27923,27924,27925,76390</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=9274913$$DView record in Pascal Francis$$Hfree_for_read</backlink><backlink>$$Uhttps://www.osti.gov/biblio/5990522$$D View this record in Osti.gov$$Hfree_for_read</backlink></links><search><creatorcontrib>Feix, Marc R.</creatorcontrib><creatorcontrib>Lewis, H. Ralph</creatorcontrib><creatorcontrib>Los Alamos National Laboratory, Center for Nonlinear Studies and CTR Division, Los Alamos, New Mexico 87545</creatorcontrib><title>Invariants for dissipative nonlinear systems by using rescaling</title><title>J. Math. Phys. (N.Y.); (United States)</title><description>A rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second‐order differential equation with a friction term proportional to the velocity, β(t)v. The transformation is of the form (x,t)→(ξ,θ), where x=ξC(t)+α(t), dθ=d
t/A
2(t). This rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant. The invariants are found explicitly for a power‐law potential, γ(t)x
m+1/(m+1), and an arbitrary coefficient of friction β(t). We show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist. For certain parameters, the asymptotic solution is a self‐similar solution that is an attractor for all initial conditions. The technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications.</description><subject>990200 - Mathematics & Computers</subject><subject>ANALYTICAL SOLUTION</subject><subject>DIFFERENTIAL EQUATIONS</subject><subject>DISSIPATION FACTOR</subject><subject>DISTURBANCES</subject><subject>EQUATIONS</subject><subject>Exact sciences and technology</subject><subject>FLUID MECHANICS</subject><subject>FRICTION</subject><subject>GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE</subject><subject>Geometry, differential geometry, and topology</subject><subject>HYDRODYNAMICS</subject><subject>MAGNETOHYDRODYNAMICS</subject><subject>Mathematical methods in physics</subject><subject>MECHANICS</subject><subject>NUMERICAL SOLUTION</subject><subject>Physics</subject><subject>POTENTIALS</subject><subject>SPACE-TIME</subject><subject>TRANSFORMATIONS</subject><subject>VELOCITY</subject><issn>0022-2488</issn><issn>1089-7658</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1985</creationdate><recordtype>article</recordtype><recordid>eNp90E1Lw0AQBuBFFKxV8CcE8aCH1P1MsieR4keh4EXPy2SzW1faTdiJhf77pkR6ETzNYR7eYV5CrhmdMVqIBzZTvCgVPSETRiudl4WqTsmEUs5zLqvqnFwgflPKWCXlhDwu4hZSgNhj5tuUNQExdNCHrctiG9chOkgZ7rB3G8zqXfaDIa6y5NDCsFxdkjMPa3RXv3NKPl-eP-Zv-fL9dTF_WuZWSN3nnNpC1KpiwJTSZc0Yk00jbW2dk9xX1DcelABltSi8UFw5zSjYUgE47kFMyc2Y22IfDNrQO_tl2xid7Y3SmirOB3Q3IptaxOS86VLYQNoZRs2hHcPM2M5Ab0faweETnyDagEeveSk1EwO7H9nh4tBKG49k26ZjnOka_5_9c34PNpN_mg</recordid><startdate>198501</startdate><enddate>198501</enddate><creator>Feix, Marc R.</creator><creator>Lewis, H. Ralph</creator><general>American Institute of Physics</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>OTOTI</scope></search><sort><creationdate>198501</creationdate><title>Invariants for dissipative nonlinear systems by using rescaling</title><author>Feix, Marc R. ; Lewis, H. Ralph</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c349t-20c63b581a15597b1114dd4cbcee42f80fdfa53a5c936f3525e910ac75aae2fa3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1985</creationdate><topic>990200 - Mathematics & Computers</topic><topic>ANALYTICAL SOLUTION</topic><topic>DIFFERENTIAL EQUATIONS</topic><topic>DISSIPATION FACTOR</topic><topic>DISTURBANCES</topic><topic>EQUATIONS</topic><topic>Exact sciences and technology</topic><topic>FLUID MECHANICS</topic><topic>FRICTION</topic><topic>GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE</topic><topic>Geometry, differential geometry, and topology</topic><topic>HYDRODYNAMICS</topic><topic>MAGNETOHYDRODYNAMICS</topic><topic>Mathematical methods in physics</topic><topic>MECHANICS</topic><topic>NUMERICAL SOLUTION</topic><topic>Physics</topic><topic>POTENTIALS</topic><topic>SPACE-TIME</topic><topic>TRANSFORMATIONS</topic><topic>VELOCITY</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Feix, Marc R.</creatorcontrib><creatorcontrib>Lewis, H. Ralph</creatorcontrib><creatorcontrib>Los Alamos National Laboratory, Center for Nonlinear Studies and CTR Division, Los Alamos, New Mexico 87545</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>OSTI.GOV</collection><jtitle>J. Math. Phys. (N.Y.); (United States)</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Feix, Marc R.</au><au>Lewis, H. Ralph</au><aucorp>Los Alamos National Laboratory, Center for Nonlinear Studies and CTR Division, Los Alamos, New Mexico 87545</aucorp><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Invariants for dissipative nonlinear systems by using rescaling</atitle><jtitle>J. Math. Phys. (N.Y.); (United States)</jtitle><date>1985-01</date><risdate>1985</risdate><volume>26</volume><issue>1</issue><spage>68</spage><epage>73</epage><pages>68-73</pages><issn>0022-2488</issn><eissn>1089-7658</eissn><coden>JMAPAQ</coden><abstract>A rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second‐order differential equation with a friction term proportional to the velocity, β(t)v. The transformation is of the form (x,t)→(ξ,θ), where x=ξC(t)+α(t), dθ=d
t/A
2(t). This rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant. The invariants are found explicitly for a power‐law potential, γ(t)x
m+1/(m+1), and an arbitrary coefficient of friction β(t). We show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist. For certain parameters, the asymptotic solution is a self‐similar solution that is an attractor for all initial conditions. The technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications.</abstract><cop>Melville, NY</cop><pub>American Institute of Physics</pub><doi>10.1063/1.526750</doi><tpages>6</tpages></addata></record> |
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| ispartof | J. Math. Phys. (N.Y.); (United States), 1985-01, Vol.26 (1), p.68-73 |
| issn | 0022-2488 1089-7658 |
| language | eng |
| recordid | cdi_pascalfrancis_primary_9274913 |
| source | AIP Digital Archive |
| subjects | 990200 - Mathematics & Computers ANALYTICAL SOLUTION DIFFERENTIAL EQUATIONS DISSIPATION FACTOR DISTURBANCES EQUATIONS Exact sciences and technology FLUID MECHANICS FRICTION GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE Geometry, differential geometry, and topology HYDRODYNAMICS MAGNETOHYDRODYNAMICS Mathematical methods in physics MECHANICS NUMERICAL SOLUTION Physics POTENTIALS SPACE-TIME TRANSFORMATIONS VELOCITY |
| title | Invariants for dissipative nonlinear systems by using rescaling |
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