主講人：Professor Tsutomu Kambe(Chern Institute of Mathematics)

Variational formulation of an ideal fluid:Lagrangian description & Eulerian description

Abstract：

Fluid mechanics is a field theory in Newtonian mechanics, i.e. a field theory of mass flows subject to Galilean transformations. In modern physics, a guiding principle is that laws of physics should be expressed in a form that is independent of any particular coordinate system. The equation of motion of an invscid fluid, proposed by Leonhard Euler 250 years ago, is of a form of partial differential equations, which is analogous to that of a point mass. Euler’s equation of motion is valid at each point of space and invariant with respect to Galilean transformations. This was perhaps the first field quation satisfying the gauge principle. However, in most traditional variational formulations of the action principle of fluid flows, there are some parts which appear to be artificial and hence not satisfactory from the view point of modern theoretical physics. Here, a new variational formulation of an ideal fluid is proposed on the basis of the gauge principle. Flow fields are characterized by symmetries of translation and rotation. The Lagrangian functional L is defined in terms of kinetic energy and internal energy satisfying the symmetries. Euler’s equation of motion results from the Euler-Lagrange equation by variations with repsect to particle coordinate ai (i.e. Lagrange’s coordinate). Invariance requirement of L with respect to rotational gauge transformations in the ai-space results in the invariance of vorticity in the space. This implies that the vorticity is a gauge field. Variations in the physical space xi (i.e. Euler’s coordinate) lead to the equations of continuity, entropy and vorticity. Present formulation provides us a basis on which transformation between the Lagrangian and Eulerian spaces is determined uniquely. In most of traditional formualtions, the continuity equation and the entropy equaiton are taken into account as constraints by using Lagrange multipliers whose physical meaning is not clear. Present formulation is consistent in describing flows of an ideal on the whole.

時間：2008 年 11 月 17 日13:20~14:50

地點：新生大樓202室

茶會：15:00於舊數館201室