Variational Approach to the Inviscid Compressible Fluid

Abstract

It is well-known that not every partial differential equation admits a variational formula. A rigorous proof of the existence of a variational principle is very difficult. In this paper, the semi-inverse method proposed by Ji-Huan He is used to construct a variational principle for a one-dimensional inviscid compressible fluid.

Variational Principles for Vibrating Carbon Nanotubes Modeled as Cylindrical Shells Based on Strain Gradient Nonlocal Theory

“…(58) are not equal indicates that a variational principle cannot be derived for the differential Eqs. (49) and (53) in their present form. This difficulty can be overcome by transforming the dependent variables u 1 , u 2 , w 1 and w 2 as follows 17…”

“…(49) and (53) correspond to the Euler-Lagrange equations of the variational functional (54) given by…”

“…Comparing Eqs. (49) and (53) with (55)-(56), we observe that the following equations have to be satisfied for Euler-Lagrange Eqs. (55) and (56) to represent the governing Eqs.…”

“…In the case of the double-walled nanotube, the semi-inverse method developed by He 45 47 is employed which was applied previously to several problems of mathematical physics to derive the variational formulation of problems formulated in terms of differential equations. [48][49][50][51][52] 2. SINGLE-WALLED NANOTUBES…”

He’s variational method for the Benjamin–Bona–Mahony equation and the Kawahara equation

“…Tao [14,15] found that He's variational method was very simple and effective. Other applications of He's variational method are available in Refs.…”