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Mechanics pp 77-165 | Cite as

The Principles of Canonical Mechanics

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Abstract

Canonical mechanics is a central part of general mechanics, where one goes beyond the somewhat narrow framework of Newtonian mechanics with position coordinates in the three-dimensional space, towards a more general formulation of mechanical systems belonging to a much larger class. This is the first step of abstraction, leaving behind ballistics, satellite orbits, inclined planes, and pendulum-clocks; it leads to a new kind of description that turns out to be useful in areas of physics far beyond mechanics. Through d’Alembert’s principle we discover the concept of the Lagrangian function and the framework of Lagrangian mechanics that is built onto it. Lagrangian functions are particularly useful for studying the role symmetries and invariances of a given system play in its description. By means of the Legendre transformation we are then led to the Hamiltonian function, which is central to the formulation of canonical mechanics, as developed by Hamilton and Jacobi.

Keywords

Harmonic Oscillator Poisson Bracket Lagrangian Function Hamiltonian Function Canonical Transformation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. Rümann, H.: Konvergente Reihenentwicklungen in der Störungstheorie der Himmelsmechanik, Selecta Mathematica V ( Springer, Berlin, Heidelberg 1979 )Google Scholar
  2. Rümann, H.: Non-degeneracy in the Perturbation Theory of Integrable Dynamical Systems, in Dodson, M.M., Vickers, J.A.G. (eds.) Number Theory and Dynamical Systems, London Mathematical Society Lecture Note Series 134 (Cambridge University Press 1989 )Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  1. 1.Institut für Physik, Theoretische ElementarteilchenphysikJohannes Gutenberg-UniversitätMainzGermany

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