{"title":"能量守恒运动方程","authors":"N. A. Vinokurov","doi":"10.1134/S1028335824600238","DOIUrl":null,"url":null,"abstract":"<p>The conventional derivation of the equations of motion in mechanics and the field equations in field theory is based on the principle of least action with a proper Lagrangian. For a time-independent Lagrangian, the function of coordinates and velocities, called energy, is constant. This paper presents a different approach – derivation of the general form of the equations of motion that ensure the constancy of the energy given as a function of generalized coordinates and corresponding velocities. It is shown that these are the Lagrange equations with additional gyroscopic forces. The derivation explicitly uses the important fact that the energy is given as a function on the tangent bundle of the configuration manifold. The Lagrangian is derived from a known energy function. It is proposed to develop generalized Hamilton and Lagrange equations without using variational principles. The new technique is used to derive some equations.</p>","PeriodicalId":533,"journal":{"name":"Doklady Physics","volume":"69 10-12","pages":"108 - 113"},"PeriodicalIF":0.5000,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Energy Conservation Equations of Motion\",\"authors\":\"N. A. Vinokurov\",\"doi\":\"10.1134/S1028335824600238\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The conventional derivation of the equations of motion in mechanics and the field equations in field theory is based on the principle of least action with a proper Lagrangian. For a time-independent Lagrangian, the function of coordinates and velocities, called energy, is constant. This paper presents a different approach – derivation of the general form of the equations of motion that ensure the constancy of the energy given as a function of generalized coordinates and corresponding velocities. It is shown that these are the Lagrange equations with additional gyroscopic forces. The derivation explicitly uses the important fact that the energy is given as a function on the tangent bundle of the configuration manifold. The Lagrangian is derived from a known energy function. It is proposed to develop generalized Hamilton and Lagrange equations without using variational principles. The new technique is used to derive some equations.</p>\",\"PeriodicalId\":533,\"journal\":{\"name\":\"Doklady Physics\",\"volume\":\"69 10-12\",\"pages\":\"108 - 113\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2025-07-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Doklady Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S1028335824600238\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MECHANICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Doklady Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S1028335824600238","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MECHANICS","Score":null,"Total":0}
The conventional derivation of the equations of motion in mechanics and the field equations in field theory is based on the principle of least action with a proper Lagrangian. For a time-independent Lagrangian, the function of coordinates and velocities, called energy, is constant. This paper presents a different approach – derivation of the general form of the equations of motion that ensure the constancy of the energy given as a function of generalized coordinates and corresponding velocities. It is shown that these are the Lagrange equations with additional gyroscopic forces. The derivation explicitly uses the important fact that the energy is given as a function on the tangent bundle of the configuration manifold. The Lagrangian is derived from a known energy function. It is proposed to develop generalized Hamilton and Lagrange equations without using variational principles. The new technique is used to derive some equations.
期刊介绍:
Doklady Physics is a journal that publishes new research in physics of great significance. Initially the journal was a forum of the Russian Academy of Science and published only best contributions from Russia in the form of short articles. Now the journal welcomes submissions from any country in the English or Russian language. Every manuscript must be recommended by Russian or foreign members of the Russian Academy of Sciences.