This chapter presents some important applications of the above developed mathematical tools in physics. The concept of fibred manifolds and the corresponding adapted structures is very natural for the use in physics and in technical applications as well. We present examples of two types: The use of fibred manifolds and their prolongations as underlying geometrical structures as well as the vector fields and differential forms adapted to the fibred structure for formulation of some key variational physical theories, including finding symmetries of Lagrangians and corresponding Noether currents. Inverse problem: It appears that important physical theories (as classical mechanics, relativistic mechanics, wave mechanics, classical electrodynamics, quantum mechanics are variational). On the basis of well-known equations of motion of some of them we prove their variationality, including the constructions of corresponding Vainberg-Tonti Lagrangian as well as the minimal order Lagrangian. In addition basic ideas of may be the simplest version of the string theory as the variational one is presented.

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Variational Physics

  • Jana Musilová,
  • Pavla Musilová,
  • Olga Rossi

摘要

This chapter presents some important applications of the above developed mathematical tools in physics. The concept of fibred manifolds and the corresponding adapted structures is very natural for the use in physics and in technical applications as well. We present examples of two types: The use of fibred manifolds and their prolongations as underlying geometrical structures as well as the vector fields and differential forms adapted to the fibred structure for formulation of some key variational physical theories, including finding symmetries of Lagrangians and corresponding Noether currents. Inverse problem: It appears that important physical theories (as classical mechanics, relativistic mechanics, wave mechanics, classical electrodynamics, quantum mechanics are variational). On the basis of well-known equations of motion of some of them we prove their variationality, including the constructions of corresponding Vainberg-Tonti Lagrangian as well as the minimal order Lagrangian. In addition basic ideas of may be the simplest version of the string theory as the variational one is presented.