It is also possible to obtain Lagrange's equations from a principle which considers the entire motion of the system between times t₁ and t₂, and small virtual variations of the entire motion from the actual motion.  An approach of this nature is known as an "integral principle."

Before presenting the integral principle, the meaning attached to the statement "motion of the system between times t₁ and t₂" should be stated more precisely.  The instantaneous configuration of a system is described by the values of the n generalized coordinates , and corresponds to a particular point in a n-dimensional Cartesian hyperspace.  This n-dimensional space is therefore known as configuration space.  As time goes on the state of the system changes, and the system point moves in configuration space tracing out a curve, described as "the path of motion of the system." Each point along the path represents the entire system configuration at some given instant of time.

Hamilton's Principle: The motion of the system from time t₁ to time t₂ is such that the line integral

                                                                      (3.18)

where L = T - V, is an extremum for the path of motion.

For an example of the derivation of Hamilton's Principle see Goldstein [1959, 36-38].  With the Hamiltonian the foundation of Relation Dynamics is complete and we are ready to move on to Relation Thermodynamics. 

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