Discrete System of Point Particles

The time integral over the Lagrangian between two fixed times fi and f2 defines the action 

Note that only those variations of the path are considered which leave the starting and end points invariant, i.e., there is no variation at the boundaries of the path. 

We will now explicitly execute these infinitesimal variations 

The solution of this set of / second-order differential equations is uniquely determined by 2/ integrational constants, which may be incorporated either via the fixed boundary conditions of the variational procedure q h) q h)) or via the initial conditions q ti),q ti)). The quantity 

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