Poisson Brackets in Continuous Media

Abstract

Now that we have defined the necessary thermodynamic quantities in chapter 4, we can turn back to the consideration of the dynamics of various physical systems. In order to apply a Poisson bracket to macroscopic transport phenomena, it is first necessary to rewrite the bracket (3.3-3) in a form which is suitable for continuum-mechanical considerations. As the number of particles increases to infinity, the transition is made from the specification of a very large number of discrete particle trajectories, xi(t), i=1,2,...,N, N→∞, to the determination of a single, continuous, vector function, Y(r,t), indicating the position of a fluid particle at time t which at a reference time t=0 was at position r, i.e., Y(r,0)=r. This is called a Lagrangian or material description. Alternatively, an Eulerian or spatial description can be used according to which the flow kinematics are completely specified through the determination of the velocity vector field, v(x,t), indicating the velocity of a fluid particle at a fixed spatial position, x, and time, t. (Truesdell [1966, p. 17] notes that the Lagrangian/Eulerian terminology is erroneous, however.) In this chapter, we shall use both descriptions to tackle the problem of ideal (inviscid) fluid flow and to arrive at a Poisson bracket for each case. The dissipative system will be considered in chapter 7. Once again, the concept of time and length scales is very important in determining when the experimenter views the system in consideration as a continuum entity. In chapter 3, we studied the dynamics of a system of discrete particles bouncing around with our time scale implicitly set on the order of the mean free time of the particles between collisions, ζ, and the length scale on the order of the mean free path, λ. As the number of particles approaches infinity, however, we note that certain averages, such as the velocity and the energy of the system, are practically constant on such a small time scale. Since the number of particles is so large, it is almost impossible to get any detailed information about the system as a whole by looking at individual particles because the number of the degrees of freedom is horrendous.