Canonical transformations and classical mechanics

The concept of a canonical transformation is fundamental to the Hamiltonian formulation of classical mechanics, the formulation which 

The classical equations of motion for any generalized set of coordinates can be obtained from Hamilton s principle which states that the motion of a system from time to time is such that the line integral 

Schematic representations of actual and varied paths linking initial and final states. In (a) the time end-points are fixed and the variations in the path vanish at Ihese points. In (b) the time end-points are varied, as is the path at these points. 

Using the methods developed in Chapter 5, one rids the expression of variations in 4,- using an integration by parts 

Since each Sqt is assumed to vanish at the time end-points, the condition that the variation in the action integral vanish, i.e. that qj(t) is such as to minimize the action, is given by 

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