{"title":"On Tautochronic Motions","authors":"A. G. Petrov","doi":"10.1134/S106456242470220X","DOIUrl":null,"url":null,"abstract":"<p>Linear motion of a point particle influenced by two forces varying according to power laws with arbitrary exponents is considered. Exponents are found for which the governing equation is nonlinear and the oscillation period is independent of the initial data (tautochronic motion). The equations are brought to Hamiltonian form, and the Hamiltonian normal form method is used to prove that there exist only two variants of tautochronic motion, namely, when the exponents are equal to 1 and –3 (variant 1) and when the exponents are equal to 0 and –1/2 (variant 2). For the other power laws, the motion of the point particle is not tautochronic. The Hamiltonian normal form of tautochronic motion is the Hamiltonian of a linear oscillator. The canonical transformation reducing the original Hamiltonian to normal form is expressed in terms of elementary functions. Hamiltonians of tautochronic motions can be used to test computer codes for calculating Hamiltonian normal forms.</p>","PeriodicalId":531,"journal":{"name":"Doklady Mathematics","volume":"110 1","pages":"312 - 317"},"PeriodicalIF":0.5000,"publicationDate":"2024-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Doklady Mathematics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S106456242470220X","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Linear motion of a point particle influenced by two forces varying according to power laws with arbitrary exponents is considered. Exponents are found for which the governing equation is nonlinear and the oscillation period is independent of the initial data (tautochronic motion). The equations are brought to Hamiltonian form, and the Hamiltonian normal form method is used to prove that there exist only two variants of tautochronic motion, namely, when the exponents are equal to 1 and –3 (variant 1) and when the exponents are equal to 0 and –1/2 (variant 2). For the other power laws, the motion of the point particle is not tautochronic. The Hamiltonian normal form of tautochronic motion is the Hamiltonian of a linear oscillator. The canonical transformation reducing the original Hamiltonian to normal form is expressed in terms of elementary functions. Hamiltonians of tautochronic motions can be used to test computer codes for calculating Hamiltonian normal forms.
期刊介绍：
Doklady Mathematics is a journal of the Presidium of the Russian Academy of Sciences. It contains English translations of papers published in Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences), which was founded in 1933 and is published 36 times a year. Doklady Mathematics includes the materials from the following areas: mathematics, mathematical physics, computer science, control theory, and computers. It publishes brief scientific reports on previously unpublished significant new research in mathematics and its applications. The main contributors to the journal are Members of the RAS, Corresponding Members of the RAS, and scientists from the former Soviet Union and other foreign countries. Among the contributors are the outstanding Russian mathematicians.
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