The
variational principle of least action becomes here, without difficulties, a postulate.
These are based on either Dyson-Schwinger equations [2-7] or functional renormalization group flow equations [8, 9], or they exploit the
variational principle in either the Hamiltonian [10, 11] or covariant [12, 13] formulation of gauge theory.
Blum and Elze studied the semiquantum chaos in a double-well potential with the time dependence of
variational principle [16].
Based on the elastic
variational principle [20], the governing equation is as follows:
It is worth mentioning that Caristi's fixed point theorem is equivalent to the Ekeland
variational principle [8].
In [14] a
variational principle for Lagrangian densities containing derivatives of real order was formulated and the invariance of this principle is studied in two characteristic cases.
[2] Bernard.f.Schutz, (1970), Perfect Fluid in General Relativity: Velocity Potentials and a
Variational Principle, Phys.
Lemma 4 (Ekeland's
variational principle; see [23]).
Of course the first integrals we have found here from a
variational principle all satisfy (33a) and (33b).
In the present paper, using a smooth version of [8, Theorem 2.1] which is a more precise version of Ricceri's
Variational Principle [25, Theorem 2.5], requiring that the nonlinear term f has a suitable oscillating behavior at infinity, we establish the existence of a precise interval A such that for every [lambda] [member of] [LAMBDA] and for every [L.sup.2]-Caratheodory function g satisfying a certain growth at infinity, choosing [mu] sufficiently small, the problem (1.1) admits a sequence of generalized solutions which is unbounded in the space E which will be introduced later (Theorem 3.1).
We shall do this by combining and extending two recent developments: a recent reformulation of the enhanced sampling problem into a powerful
variational principle that opens a wealth of possibilities and provides a novel and fruitful standpoint for new developments; and a procedure for extracting rates from enhanced runs.
Thus, referring to reference [3] for applying least square method to establish "partial and temporary unified theory of natural science so far" including all the equations of natural science so far (in which, the theory of everything to express all of natural laws, described by Hawking that a single equation could be written on a T-shirt, is partially and temporarily realized in the form of "partial and temporary unified
variational principle of natural science so far"), Eq.(5) (one kind of "certainty-uncertainty principles" with general form) can be restricted (or constrained) by principle (law) of conservation of energy as follows
