Hamiltons Principle and Equations of Motion

Hamilton s principle of least action provides a mechanism for deriving equations of motion from a Lagrangian. Recall from Chap. 1 that the Lagrangian for the N-body system is defined by 

The variational calculus approach to classical mechanics is based on minimizing the action Al over the class G of parameterized curves. This is normally referred to as the principle of least action . It is difficult to provide a physical motivation for this concept, but it is normally taken as a foundation stone for classical mechanics. 

Given F in G with parameterization q(i), t [a, 6], we consider the curve r with parameterization defined by 

0 to 0. In defining variations in this way, we are using, implicitly, the fact that we can add together functions in C°° and multiply them by scalars (e.g. ) and remain in C°°. We also make use of the intuitive concept that (2.3) defines in such a way that it is close to q when s is small. This can be made precise by a little more elaboration, but we forego this here. Effectively, we are using our understanding of C°° as a named function space to restrict attention to variations of the base curve r in a particular direction.  

Using a Taylor series expansion of the Lagrangian, we have 

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