Action Concept and Hamilton’s Principle

The approach using ‘variational principles’ for solving problems of physics, particularly problems related to mechanics, has been proposed by the mathematicians of continental Europe from the early stages of the development of the subject. All the scholars depended on their philosophical insights and proposed various forms of the approach based on the idea that ‘the nature follows the path of least resistance’. After some initial erroneous starts finally the scholars like Euler and Lagrange proposed a variational approach called the ‘principle of least action’. An interested reader can refer to a book on the history of the conceptual evolution of mechanics in the Eighteenth-century Europe to get a detailed account. It was Hamilton who first showed that an integration with respect to time can lead to a variational form of d’Alembert’s principled’Alembert’s principle. The motion of a dynamical system can be represented by a C-curve as shown earlier. This is the locus of the configuration point from the start of the motion A to the end B as shown in Fig. 4.1.