{"title":"Algorithms for Computing Hamiltonian Normal Form","authors":"A. G. Petrov, A. B. Batkhin","doi":"10.1134/S0038094624601853","DOIUrl":null,"url":null,"abstract":"<p>The invariant normalization method proposed by V.F. Zhuravlev, used for calculating normal or symmetrized forms of autonomous Hamiltonian systems, is discussed. The normalizing canonical transformation is represented by a Lie series using a generating Hamiltonian. This method has a generalization proposed by A.G. Petrov, which normalizes not only autonomous but also nonautonomous Hamiltonian systems. The normalizing canonical transformation is represented by a series using a parametric function. For autonomous Hamiltonian systems, the first two approximation steps in both methods are the same, and the remaining steps are different. The normal forms of both methods are identical. A method for testing a normalization program has also been proposed. For this purpose, the Hamiltonian of a strongly nonlinear Hamiltonian system is found, for which the normal form is a quadratic Hamiltonian. The normalizing transformation is expressed in terms of elementary functions.</p>","PeriodicalId":778,"journal":{"name":"Solar System Research","volume":"59 4","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Solar System Research","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S0038094624601853","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"ASTRONOMY & ASTROPHYSICS","Score":null,"Total":0}
引用次数: 0
Abstract
The invariant normalization method proposed by V.F. Zhuravlev, used for calculating normal or symmetrized forms of autonomous Hamiltonian systems, is discussed. The normalizing canonical transformation is represented by a Lie series using a generating Hamiltonian. This method has a generalization proposed by A.G. Petrov, which normalizes not only autonomous but also nonautonomous Hamiltonian systems. The normalizing canonical transformation is represented by a series using a parametric function. For autonomous Hamiltonian systems, the first two approximation steps in both methods are the same, and the remaining steps are different. The normal forms of both methods are identical. A method for testing a normalization program has also been proposed. For this purpose, the Hamiltonian of a strongly nonlinear Hamiltonian system is found, for which the normal form is a quadratic Hamiltonian. The normalizing transformation is expressed in terms of elementary functions.
期刊介绍:
Solar System Research publishes articles concerning the bodies of the Solar System, i.e., planets and their satellites, asteroids, comets, meteoric substances, and cosmic dust. The articles consider physics, dynamics and composition of these bodies, and techniques of their exploration. The journal addresses the problems of comparative planetology, physics of the planetary atmospheres and interiors, cosmochemistry, as well as planetary plasma environment and heliosphere, specifically those related to solar-planetary interactions. Attention is paid to studies of exoplanets and complex problems of the origin and evolution of planetary systems including the solar system, based on the results of astronomical observations, laboratory studies of meteorites, relevant theoretical approaches and mathematical modeling. Alongside with the original results of experimental and theoretical studies, the journal publishes scientific reviews in the field of planetary exploration, and notes on observational results.