The Euler-Lagrange equations of motion for incompressible viscous fluid are derived from Hamilton's principle of least action. The Lagrangian function is composed of the kinetic energy minus the potential energy plus half the time integral of the dissipation function. The method of Lagrange multipliers will be used on the definition of the fluid parcel's velocity, which depends on position coordinate. Since the virtual displacements and virtual velocities will be independent of each other, the time integral of half the dissipation function provides the necessary viscous forces. The Euler-Lagrange equations obtained by requiring the action to be stationary are shown to be equivalent to the Navier-Stokes equation of motion for incompressible viscous fluids with homogeneous constant density, ρ_0. As an application of this theory, the f-plane approximation of the Navier-Stokes equations of motion for incompressible viscous fluids is obtained from the Euler-Lagrange equations. From thermodynamic considerations, the material derivative of the Hamiltonian is shown to be equal to the dissipation function, D, times -3/(2 ρ_0), since the Hamiltonian becomes a different thermodynamic quantity related to the mechanical energy of the fluid.
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