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| Lesson Objectives |
- To introduce configuration space of a system;
- To introduce paths in configuration space and to define action as a functional of paths in configuration space;
- To formulate Hamilton's action principle and and to prove that it is equivalent to Euler Lagrange equations of motion.
- 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 |
- \(\S1\) Configuration Space Action functional
- \(\S1.1\) Defining The Action functional.
- \(\S1.2\) Paths in configuration space .
- \(\S1.3\) Specifying Coordinates at the end points .
- \(\S1.4\) The Actional Functional
- \(\S2\) Hamilton’s Principle
- \(\S2.1\) Infinitesimally close paths .
- \(\S2.2\) Computing variation of action.
- \(\S2.3\) Condition for an extremum.
- \(\S2.4\) Action Principle — The Statement.
- \(\S3\) Weiss Action Principle
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EndNotes |
- For action principle see Ref. [1,2,3] ; Weiss action principle can be found in see Ref. [3].
- For an account of different variational principles and historical developments see Lanczos [5], Yourgrau and Mandelstam [6].
| References |
- 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)
- Calkin, M.G. Lagrangian and Hamiltonian Mechanics, World Scientific Publishing Co. Pte. Ltd. (1996);
- Sudarshan E.C.G. and Mukunda N., Classical Dynamics : A Modern Perspectives , John Wiley and Sons Inc New York (1974)
- Goldstein, Herbert, et al, Classical Mechanics,
- Lanczos Cornelius The Variational Principles of Mechanics University of Toronto Press, Toronto, Canada (1949)
- Yourgrau W. and Mandelstam S. Variational Principles in Dynamics And Quantum Theory Pitman Publishing- Corporation2 West 45th Street, New York (1960)
Download: cm-lsn-03001.pdf (123.03 KB)
