Hamilton’s Principle, Lagrange’s Method, and Ship Motion Theory

Abstract

Lagrange's equations of motion, describing the motion of several bodies on or below a free surface, are here derived from Hamilton's variational principle. The Lagrangian density is obtained by extending Luke's principle to the wave-radiation problem, and the hydrodynamical loads on the bodies are expressed in terms of the Lagrangian density and its derivatives with respect to the generalized coordinates of the bodies. First we consider a forced harmonic oscillation without a forward speed and then we discuss the case of the same oscillatory motion superimposed on arbitrary steady motion. In both cases we employ Lagrange's method to derive the transfer functions between the generalized forces and the amplitudes of the harmonic motions, in terms of added mass, damping, and the hydrostatic restoring coefficients. The case of a steady forward motion, for which the transfer function is already known, is obtained as a particular case of the general solution.