TO LOCAL COORDINATES AS A CANONICAL TRANSFORNATION

It is shown that the conversion from m aterial to local coordinates in continuum mechanics can be considered as a restricted canonical transformation. As a simple example the longitudinal motion of an elastic bar is discussed. § 1. Material and l ocal coordinates It is well known that the motion of a continuous medium can be described either with respect to a fixed system of local coordinates or with respect 1o material coordinates moving with the medium. For a medium without any dissipation it is often not difficult to write the material equations in Lagrangian or Hamiltonian form. As an example we mention a paper on fluid dynamics by Eckart I1J. One would expect then that this must oe possible also for the equations in local coordinates. Moreover, one then would like to interpret the eonversion from material to local coordinates as a canonical transformation. In this note we will, in order to confine the diseussion to essentials first investigate this problem for one dimensional motions. In local coordJnates the motion is known when the function ra(x, t) has been determined. Here m is some material eoordinate. A simple choice for m is the total mass to the left of the polnt x. For the denslty and veloclty v of the medium we obviously have: