The extended variational principle fails for non-holonomic case. We use a vertical rolling disk as an example to illustrate this failure. We have studied the fundamental dynamical principle from which the motion of a mechanical system, whether holonomic or non-holonomic, can be derived.

A variational principle governing the dynamics of a vortex filament in unbounded incompressible inviscid fluid flow was suggested by Rasetti and Regge [Physica A 80, 217 (1975)]. This variational principle holds in the approximation taking into account the logarithmically large terms, on the order of ln(La) , L and a being the length and the cross-section radius of the filament, respectively. I...

Assume that A is a bounded selfadjoint operator in a Hilbert space H. Then, the variational principle max v |(Au, v)| (Av, v) = (Au, u) (*) holds if and only if A ≥ 0, that is, if (Av, v) ≥ 0 for all v ∈ H. We define the left-hand side in (*) to be zero if (Av, v) = 0. As an application of this principle it is proved that C = max v∈L2(S) | ∫ S vdt| ∫ S ∫ S v(t)v(s)dsdt 4π|s−t| , (**) where L(S)...

where K, h ∈ L(R), a(x) is a positive bounded function, and f(s) is asymptotically linear with respect to s at infinity, that is f(s)/s goes to a constant as s → +∞. Under suitable assumptions on K, a and f , we prove that the problem (P) has at least two positive solutions for |K|2 and |h|2 small by using Ekeland’s variational principle and Mountain pass theorem.

A variational principle is constructed for gravity coupled to an asymptotically linear dilaton and a p-form field strength. This requires the introduction of appropriate surface terms – also known as ‘boundary counterterms’ – in the action. The variation of the action with respect to the boundary metric yields a boundary stress tensor, which is used to construct conserved charges that generate ...

In this paper we investigate optimal control problems governed by variational inequalities. We present a method for deriving optimality conditions in the form of Pontryagin's principle. The main tools used are the Ekeland's variational principle combined with penalization and spike variation techniques. 1. Introduction. The purpose of this paper is to present a method for deriving a Pontryagin ...