Gauge theory of flows of an ideal fluid

摘要

Fluid mechanics is a field theory in Newtonian mechanics, i.e. a field theory of mass flows of Galilean symmetry. In the gauge theory of theoretical physics, a guiding principle is that laws of physics should be expressed in a form that is independent of any particular coordinate system. Variational formulations of fluid flows are reviewed first from the point of view of gauge theory, and then a new variational formulation is proposed, which leads to a new representation of compressible rotational flows of an ideal fluid. This improves the classical solution of Clebsch (1859). There is a fundamental question how a rotational component of fluid flow can be formulated in the variational framework of an ideal compressible fluid. Present Lagrangian for the action principle consists of main terms of total kinetic energy and internal energy (with negative sign), together with three additional terms yielding the equations of continuity, entropy and the third term which provides rotational component of velocity field. The last term leads to an explicit expression of non-vanishing helicity. Thus, a new expression of velocity field v(x,t) is given in terms of vector potentials of frozen field, i.e. the potentials are convected by the fluid flow under effect of stretching, while the potentials of Clebsch solution are just convected without stretching effect. It is verified that the system of new expressions in fact satisfies the Euler's equation of motion. Associated with two symmetries (translation and space-rotation), there are two gauge fields E and H, which do not exist in the system of discrete masses. One can show that those are analogous to the electric field and magnetic field in the electromagnetism, and fluid Maxwell equations can be formulated for E and H. Sound wave within the fluid is analogous to the electromagnetic wave, in the sense that phase speeds of both waves are independent of wave lengths, i.e. non-dispersive.