Integrating factors and adjoint equations are determined for linear and nonlinear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by...

We show that solutions of the Bach equation exist which are not conformal Einstein spaces. In connection with fourth order gravitational field equations, cf. e.g. [1, 2] where the breaking of conformal invariance was discussed, the original Bach equation, Bij = 0 , (1) enjoys current interest. Eq. (1) stems from a Lagrangian L = 1 2 √ −gCijklC , and variation gives, cf. Bach [3], 1 √−g δL/δg ij...

The relative classical motion of membranes is governed by the equation (w c β cr )a = R α δγβr xpa, where w is the hessian. This is a generalization of the geodesic deviation equation and can be derived from the lagrangian p · r. Quantum mechanically the picture is less clear. Some quantizations of the classical equations are attempted so that the question as to whether the Universe started wit...

Noether symmetries provide conservation laws that are admitted by Lagrangians representing physical systems. For partial differential equation possessing Lagrangians these symmetries are obtained by the invariance of the corresponding action integral. In this paper we provide a systematic procedure for determiningNoether symmetries and conserved vectors for a Lagrangian constructed from a Loren...

It is well known that the Schrödinger equation is only suitable for the particle in common potential V (~r, t). In this paper, a general Quantum Mechanics is proposed, where the Lagrangian is the general form. The new quantum wave equation can describe the particle which is in general potential V (~r, ~̇r, t). We think these new quantum wave equations can be applied in many fields. PACS numbers:...

A Semi-Lagrangian Computation of Front Speeds G-equation in ABC and Kolmogorov Flows with Estimation via Ballistic Orbits

A closed set of exact equations describing statistical theory of turbulent self-diffusion by multivariate-normal turbulent velocity field is derived. In doing so, we first suggest exact formulas for correlations 〈fi(p)fj(p ′)R[f ]〉, 〈g(p)R[f ]〉 and 〈g(p)fj(p ′)R[f ]〉 when the functional R[f ] is functional of functions fi’s having multivariate-normal distribution, g and fi’s have joint normal d...

The structure formation scenario based on gravitational instability has been studied for a long time. The Lagrangian perturbative method for the cosmological fluid describes the nonlinear evolution of density fluctuation rather well. Zel’dovich [1] proposed a linear Lagrangian approximation for dust fluid. This approximation is called the Zel’dovich approximation (ZA) [1–6]. ZA describes the ev...