{"title":"The Hamiltonian equations of motion. Poisson brackets","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0010","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0010","url":null,"abstract":"This chapter addresses invariance of the Hamiltonian function under a given transformation and the conservation law, the Hamiltonian function for the beam of light, the motion of a charged particle in a nonuniform magnetic field, and the motion of electrons in a metal or semiconductor. The chapter also discusses the Poisson brackets and the model of the electron and nuclear paramagnetic resonances, the Poisson brackets for the components of the particle velocity, and the “hidden symmetry” of the hydrogen atom.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115167355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Scattering in a given field. Collision between particles","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0016","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0016","url":null,"abstract":"This chapter addresses the differential and total cross section for the scattering of particles by central field, the scattering of particles by the fixed to ellipsoid, and the small angles scattering of particles by central field as well as a dipol. The authors also discuss the cross section for the process where a particle falls towards the centre of the field, decay of particles and the distribution of the secondary particle, and the change in intensity of a beam of particles travelling through a volume filled with absorbing centres.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114348781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Hamilton–Jacobi equation","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0025","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0025","url":null,"abstract":"This chapter discusses the motion of particles which are scattered by and fall towards the center of the dipol, the motion of a particle in the Coulomb and the constant electric fields, and a particle inside a smooth elastic ellipsoid. The chapter also addresses the trajectory of a particle moving in the field of two Coulomb centres and a beam of electrons inside a short magnetic lens.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115228242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Motion of a particle in three-dimensional fields","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0015","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0015","url":null,"abstract":"In the central field, the energy and angular momentum are conserved. It allows for the reduction of this problem to the problem of the motion of the particle in the effective one-dimensional field. Here the motion of a particle in Coulomb field or in the field of the isotropic harmonic oscillation with small perturbations are the most important ones. The authors discuss how the motion of a particle in the given central field can be described qualitatively for different values of the angular momentum and of the energy. Several problems deal with the motion of a particle in the Coulomb field under influence of weak constant uniform electric or magnetic fields (the classical analog of the Stark or Zeeman effect). In addition, the authors consider the motion of a charged particle in the field of the magnetic monopole and magnetic dipole. The motion of the Earth–Moon system in the field of the Sun is considered in some approximation. The displacement of the Coulomb orbit under the influence of a small force of radiation damping.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117152082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-linear oscillations","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0008","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0008","url":null,"abstract":"This chapter addresses the distortion in the free and forced oscillations of a harmonic oscillator caused by the presence of the anharmonic terms in the potential energy, a simple model related to the coupling of the longitudinal and flexural oscillations inmolecules, and two oscillators with a weak non-linear coupling (the so-called Fermi resonance). The chapter also examines non-linear resonances, the parametric resonances, drift of the orbit centre for a charged particle in the weakly inhomogeneous magnetic field, and a mechanical model of phase transitions of the second kind.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"200 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132128410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Small oscillations of systems with several degrees of freedom","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0006","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0006","url":null,"abstract":"This chapter addresses the free and forced oscillations of simple systems (with two or three degrees of freedom), the free oscillations of systems with the degenerate frequencies, and the eigen-oscillations of the electromechanical systems. This chapter also studies the oscillations of more complex systems using orthogonality of eigenoscillations and the symmetry properties of the system, the free oscillations of an anisotropic charged oscillator moving in a uniform constant magnetic field, and the perturbation theory adapted for the small oscillations. Finally, the chapter addresses oscillations of systems in which gyroscopic forces act and the eigen-oscillations of the simple molecules.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132958230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Canonical transformations","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0011","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0011","url":null,"abstract":"This chapter addresses the canonical transformation defined by the given generating function, the rotation in the phase space as a canonical transformation, and themovement of the system as a canonical transformation. The chapter also discusses using the canonical transformations for solving the problems of the anharmonic oscillations and using the canonical transformation to diagonalize the Hamiltonian function of an anisotropic charged harmonic oscillator in a magnetic field. Finally, the chapter addresses the canonical variables which reduce the Hamiltonian function of the harmonic oscillator to zero and using them for consideration of the system of the harmonic oscillators with the weak nonlinear coupling.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"559 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132690767","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Small oscillations of systems with one degree of freedom","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0018","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0018","url":null,"abstract":"This chapter addresses the small free oscillations for particle moving near the minimum of the potential energy, an oscillator with friction under action of a given force, and the stable oscillations of an oscillator which is acted upon by a periodic force. The authors also discuss the differential cross section for the oscillator which excited to an given energy by a fast particle and a harmonic oscillator in the field of the travelling wave.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122590349","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Integration of one-dimensional equations of motions","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0014","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0014","url":null,"abstract":"If the potential energy is independent of time, the energy of the system remains constant during the motion of a closed system. A system with one degree of freedom allows for the determination of the law of motion in quadrature. In this chapter, the authors consider motion of the particles in the one-dimensional fields. They discuss also how the law and the period of a particle moving in the potential field change due to adding to the given field a small correction.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127305672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rigid-body motion. Non-inertial coordinate systems","authors":"G. Kotkin, V. Serbo","doi":"10.1093/oso/9780198853787.003.0009","DOIUrl":"https://doi.org/10.1093/oso/9780198853787.003.0009","url":null,"abstract":"This chapter addresses the inertia tensor and its relation with the mass quadrupole moment tensor, the principal axes and the principal moments of inertia, evolution of the period of the Earth’s rotation around its axis due to the action of tidal forces, and the motion of the gyrocompass at a given latitude. The chapter also addresses precession of a symmetric top, the stability of rotations of an asymmetric top, “motion” of a plane disk which rolls in the field of gravity over a smooth horizontal plane, and the displacement from the vertical of a particle which is dropped from a given height with zero initial velocity. Finally, the chapter discusses the Lagrange point in the Sun-Jupiter system.","PeriodicalId":201389,"journal":{"name":"Exploring Classical Mechanics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131122044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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