$\newcommand{\matrixelement}[3]{\langle#1#2#3\rangle}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}${}$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}${}$\newcommand{\ket}[1]{#1\rangle}$ {} $\newcommand{\bra}[1]{\langle #1}$
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 \(N=nr\) 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
.
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
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, ButterworthHeinenann 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)
Exclude node summary :
Exclude node links:
TABLE OF CONTENTS
Lessons
Problem Solving
Evaluation and Assessment

About Stockpiles:
