Definition of the Lagrangian density

Morse and Feshbach [499] have discussed the variation approach to a description of equations of motion for diffusion. Their approach is straightforward and is generalised here to consider the cases where there is an energy of interaction, U, between the pair of particles, separated by a distance r at time t. It is relatively easy to extend this to a many-body situation. The usual Euler form of the equation of motion is the Debye— Smoluchowski equation, which has been discussed in much detail before, viz. 

Providing the approximate Euler density, p, satisfies the initial and boundary conditions, this expression can be integrated by parts and simplified considerably. The case where the approximate Euler density docs not completely satisfy the boundary and initial conditions is discussed in Sect. 2.3, It gives 

For the deviation of the invariant 6 i from its minimum value St to be very small, for an arbitrary but non-zero 8p, the term in square brackets must be zero, whence 

Equation (256) serves to define the Lagrangian density, L, corresponding to Euler density p. 

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