Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2001-03-23
Nonlinear Sciences
Chaotic Dynamics
6 pages, no figures. To appear in Proceedings of the 39th IEEEE Conference on Decision and Control
Scientific paper
In this paper we consider the Hamiltonian formulation of the equations of incompressible ideal fluid flow from the point of view of optimal control theory. The equations are compared to the finite symmetric rigid body equations analyzed earlier by the authors. We discuss various aspects of the Hamiltonian structure of the Euler equations and show in particular that the optimal control approach leads to a standard formulation of the Euler equations -- the so-called impulse equations in their Lagrangian form. We discuss various other aspects of the Euler equations from a pedagogical point of view. We show that the Hamiltonian in the maximum principle is given by the pairing of the Eulerian impulse density with the velocity. We provide a comparative discussion of the flow equations in their Eulerian and Lagrangian form and describe how these forms occur naturally in the context of optimal control. We demonstrate that the extremal equations corresponding to the optimal control problem for the flow have a natural canonical symplectic structure.
Bloch Anthony M.
Crouch Peter E.
Holm Darryl D.
Marsden Jerrold E.
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