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Lesson/CM03-01 Action Principle

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Lesson Objectives
  1. To introduce configuration space of a system;
  2. To introduce paths in configuration space and to define action as a functional of paths in configuration space;
  3. To formulate Hamilton's action principle and and to prove that it is equivalent to Euler Lagrange equations of motion.
  4. To explain Weiss action principle.

Recall and Discuss                                                                                                                                                                                                     

The state of a system consisting of \(n\) particles and having \(r\) holonomic constraints can specified by a set of \(N=n-r\) independent generalized coordinates \(\mathbf q= (q_1, q_2, q_N)\). The dynamics of the system is governed by Lagrangian \(L\) which is a function of \(2N\) variables \(\mathbf q, \dot{\mathbf q}\). The Euler Lagrange equations of motion \begin{equation}\frac{d}{dt }\Big(\frac{\partial{L}}{\partial{\dot q_k}}\Big)- \frac{\partial{L}}{\partial{q_k}}=0. \end{equation} describe the dynamics of a mechanical system

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Contents

 

EndNotes

  1. For action principle see Ref. [1,2,3] ; Weiss action principle can be found in see Ref. [3].
  2. For an account of different variational principles and historical developments see Lanczos [5], Yourgrau and Mandelstam [6].
 References 
  1. Landau, L. D. and Lifshitz, E. M., Mechanics, Volume 1 of Course of Theoretical Physics, Butterworth-Heinenann Linacre House, Jordan Hill, Oxford 3rd Ed.(1976)

  2.  Calkin, M.G. Lagrangian and Hamiltonian Mechanics, World Scientific Publishing Co. Pte. Ltd. (1996);

  3.  Sudarshan E.C.G. and Mukunda N., Classical Dynamics : A Modern Perspectives , John Wiley and Sons Inc New York (1974)

  4.  Goldstein, Herbert, et al,  Classical Mechanics,

  5.  Lanczos Cornelius  The Variational Principles of Mechanics University of Toronto Press, Toronto, Canada (1949)

  6.  Yourgrau W. and Mandelstam S.  Variational Principles in Dynamics And Quantum Theory Pitman Publishing- Corporation2 West 45th Street, New York (1960)

 

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