Conformal fractals, dimensions and ergodic theory by Przytycki F., Urbanski M.

By Przytycki F., Urbanski M.

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This is reminiscent of the action-at-a-distance invoked by earlier formulations, in whose context the variational principle originally arose. The approach proposed by Carathéodory is more in accordance with our modern view of local interactions. He replaced the problem of finding an extremum for the action integral by one of finding a local extremal value for a function. Thus, the field concept is at the forefront, playing a major role. Let us recall some important assumptions [1, 13–15] concerning the central problem of variational calculus: a) To find an extremal curve xi = xi (t) that satisfies δ Ldt = 0 requires that we restrict ourselves to a simply-connected domain.

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