JGP - Vol. 6, n. 2, 1989 Hamiltonian formulation of adiabatic free boundary Euler flows ARTHUR MAZER, TUDOR RATIU Department ofMathematics University ofArizona, Tucson, AZ 85721 Department of Mathematics University of Califomia, Santa Cmi, CA 95064 Abstract. A Hamiltonian formulation of adiabatic free boundary inviscid fluid flow using only physical variables is presented in both the material and spatial formula- tion. Using the symmetry of particle relabeling, we derive the noncanonicalPoisson bracket in Eulerian representation as a reduction from the canonical bracket in La- grangian representation. When the free boundary of the fluid is given as the zero set of afunction draggedalong by the fluid flow, there is anotherbracket due to Abarbanel et al. [Physics of Fluids, (vol. 31), (2802), (1988)]. his shown that this formulation ~covers~ the present one by proving that the natural restriction map is Poisson. It is also shown that the potential vortycity and the conserved quantities found by Abar- band and Holm [Physics of Fluids (vol. 30), (3369), (1987)] are also conserved in the free boundary case. 1. INTRODUCTION This paper extends previous results of Lewis et al. [I] to the case of ideal adiabatic self-gravitating flow with surface tension, obtaining a Hamiltonian formulation of this problem. This is a step in a larger program concerned with the stability and bifurcation analysis of fluid equilibria. For the case of a fluid with surface tension such a program was carried out hi Lewis et al. [2] and Lewis [3]. Other relevant problems would be the study of weather systems, the free boundary being the outer surface of the atmosphere, or oceanographic problems where the water surface is the free boundary. Irrotational free Key-Words: Hamiltonian formulation, Euler flows 1980 MSC.’ 76 B 99