We present an approach to the canonical quantization of systems with equations of motion that are historically called non-Lagrangian equations. Our viewpoint of this problem is the following: despite the fact that a set of differential equations cannot be directly identified with a set of Euler-Lagrange equations, one can reformulate such a set in an equivalent first-order form which can always...

3 Lagrange’s Equations of Motion 9 3.1 Lagrange’s Equations Via The Extended Hamilton’s Principle . . . . . . . . . . . . . . . . 9 3.2 Rayleigh’s Dissipation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Kinematic Requirements of Lagrange’s Equation . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Lagrange Equation Examples . . . . . . . . . . . . . . . ...

We reformulate the one-dimensional complex Ginzburg-Landau equation as a fourth order ordinary differential equation in order to find stationary spatiallyperiodic solutions. Using this formalism, we prove the existence and stability of stationary modulated-amplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.

Let Φ(u, v) = ∑ ∞ m=0 ∑ ∞ n=0 cmnu v. Bouwkamp and de Bruijn found that there exists a power series Ψ(u, v) satisfying the equation tΨ(tz, z) = log (∑ ∞ k=0 t k k! exp(kΦ(kz, z)) ) . We show that this result can be interpreted combinatorially using hypergraphs. We also explain some facts about Φ(u, 0) and Ψ(u, 0), shown by Bouwkamp and de Bruijn, by using hypertrees, and we use Lagrange inversi...

A certain functional–difference equation that Runyon encountered when analyzing a queuing system was solved in a combined effort of Morrison, Carlitz, and Riordan. We simplify that analysis by exclusively using generating functions, in particular the kernel method, and the Lagrange inversion formula.

A prismatic beam made of a behaviorally nonlinear material is analyzed under aharmonic load moving with a known velocity. The vibration equation of motion is derived usingHamilton principle and Euler-Lagrange Equation. The amplitude of vibration, circular frequency,bending moment, stress and deflection of the beam can be calculated by the presented solution.Considering the response of the beam,...

In (Kosmol, 1991; Kosmol and Pavon, 1993) a new elementary approach to optimal control problems relying on the Lagrange lemma was described which appears to be technically, and conceptually, much simpler than existing methods, and, furthermore, provides a unified variational approach. In (Kosmol and Pavon, 1992) this method was further clarified and developed for linear Lagrangefunctionals. We ...

In this work, we report a hybrid (MPI/OpenMP) parallelization strategy for the minimum action method recently proposed in [17]. For nonlinear dynamical systems, the minimum action method is a useful numerical tool to study the transition behavior induced by small noise and the structure of the phase space. The crucial part of the minimum action method is to minimize the Freidlin–Wentzell action...

The instantonic approach for a φ4 model potential is reexamined in the asymptotic limit. The path integral of the system is derived in semiclassical approximation expanding the action around the classical background. It is shown that the singularity in the path integral, arising from the zero mode in the quantum fluctuation spectrum, can be tackled only assuming a finite (although large) system...

We present the derivation of the discrete Euler–Lagrange equations for an inverse spectral element ocean model based on the shallow water equations. We show that the discrete Euler–Lagrange equations can be obtained from the continuous Euler–Lagrange equations by using a correct combination of the weak and the strong forms of derivatives in the Galerkin integrals, and by changing the order with...