Lagrangian classical physics

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

All aspects of Newtonian mechanics can equally well be formulated within the more general Lagrangian framework based on a single scalar function, the Lagrangian. These formal developments are essential prerequisites for the later discussion of relativistic mechanics and relativistic quantum field theories. As a matter of fact the importance of the Lagrangian formalism for contemporary physics cannot be overestimated as it has strongly contributed to the development of every branch of modem theoretical physics. We will thus briefly discuss its most central formal aspects within the framework of classical Newtonian mechanics. 

L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics, 3rd ed. Vol.l, Pergamons, New York, 1988. [(An excellent treatment of Lagrangian mechanics is given in this text.) Other excellent theoretical mechanics texts include H. Goldstein, Classical Mechanics, 2nd ed., Addison-Wesley, MA, 1980. E.A. Desloge, Classical Mechanics, Vols. 1 and 11, Krieger Publishing, FL, 1989.]... 

This conservation law is known as the analogous classical fluid theory [20]. An interesting feature of it is that the term in brackets in 7, is the Lagrangian of a particle of mass m moving in the potential V + Vq. This suggests that this Lagrangian may acquire a physical significance as a component of the current. 

Recently, a new type of phase separation called viscoelastic phase separation was observed in polymer solutions or dynamically asymmetric fluid mixtures [1-3]. It is an interesting feature of this phenomenon that network-like domains of more viscous phase emerge in a transient regime. It has little been understood what ingredient of physics is crucial to this phenomenon. Various numerical approaches have been made for the phase separation phenomena in binary fluid systems in the last decade [4-6]. Most of these studies have been concerned with classical fluids and have not involved viscoelasticity. A new numerical model was recently proposed by the author [7] based upon the two-fluid model [8,9] using the method of smoothed-particle hydrodynamics (SPH) [10,11]. In this model the Lagrangian picture for fluid is adopted and the viscoelastic effect can easily be incorporated. In this paper we carry out a computer simulation for the viscoelastic phase separation in polymer solutions with this model. 

A course in classical mechanics is an essential requirement of any first degree course in physics. In this volume Dr Brian Cowan provides a clear, concise and self-contained introduction to the subject and covers all the material needed by a student taking such a course. The author treats the material from a modern viewpoint, culminating in a final chapter showing how the Lagrangian and Hamiltonian formulations lend themselves particularly well to the more modem areas of physics such as quantum mechanics. Worked examples are included in. the text and there are exercises, with answers, for the student. 

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