Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but rather upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed, and Hamilton equations, equivalent with the Eul...

This paper is in part a summary of our earlier work [17, 18, 19], and in part an announcement introducing a refinement of the equations for the pair excitation function used in our previous work with D. Margetis. The new equations are Euler-Lagrange equations, and the solutions conserve energy and the number of particles. 2. Introduction The problem, which has received a lot of attention in rec...

We propose an approach to identifying the solutions of the steady incompressible Navier-Stokes equations for high Reynolds numbers. These cannot be obtained as initial-value problems for the unsteady system because of the loss of stability of the latter. Our approach consists in replacing the original steady-state problem for the Navier-Stokes equations by a boundary value problem for the Euler...

This paper gives a few new developments in mechanics, as well as some remarks of a historical nature. To keep the discussion focussed, most of the paper is confined to equations of “rigid body”, or “hydrodynamic” type on Lie algebras or their duals. In particular, we will develop the variational structure of these equations and will relate it to the standard variational principle of Hamilton. E...

This paper presents the dynamic modeling and design of micro motion compliant parallel mechanism with flexible intermediate links and rigid moving platform. Modeling of mechanism is described with closed kinematic loops and the dynamic equations are derived using Lagrange multipliers and Kane’s methods. Euler-Bernoulli beam theory is considered for modeling the intermediate flexible link. Based...

Soliton equations are infinite-dimensional integrable systems described by nonlinear evolution equations. As one of the soliton equations, long wave equation takes on profound significance of theory and reality. By using the method of nonlinearization, the relation between long wave equation and second-order eigenvalue problem is generated. Based on the nonlinearized Lax pairs, Euler-Lagrange f...

Selfdual variational theory – developed in [7] and [8] – allows for the superposition of appropriate boundary Lagrangians with “interior” Lagrangians, leading to a variational formulation and resolution of problems with various linear and nonlinear boundary constraints that are not amenable to standard Euler-Lagrange theory. The superposition of several selfdual Lagrangians is also possible in ...

Fractional mechanics describes both conservative and non-conservative systems. The fractional variational principles gained importance in studying the fractional mechanics and several versions are proposed. In classical mechanics the equivalent Lagrangians play an important role because they admit the same Euler-Lagrange equations. By adding a total time derivative of a suitable function to a g...

We consider a special class of Lagrangians that play a fundamental role in the theory of second order Lagrangian systems: Twist systems. This subclass of Lagrangian systems is deened via a convenient monotonicity property that such systems share. This monotonicity property (Twist property) allows a nite dimensional reduction of the variational principle for nding closed characteristics in xed e...

Properties of Hamiltonian symmetry flows on hyperbolic Euler-type equations are analyzed. Their Lagrangian densities are demonstrated to supply the Hamiltonian operators for subalgebras of their Noether symmetries, while substitutions between Euler-type equations define Miura transformations between the symmetry flows; some Miura maps for Liouvillean Euler-type systems are supplied by their int...