This book presents a unified study of dynamically coupled systems involving a rigid body and an ideal fluid flow from the perspective of Lagrangian and Hamiltonian mechanics. It compiles theoretical investigations on the topic of dynamically coupled systems using a framework grounded in Kirchhoff's equations. The text achieves a balance between geometric mechanics, or the modern theories of reduction of Lagrangian and Hamiltonian systems, and classical fluid mechanics, with a special focus on the applications of these principles. Following an introduction to Kirchhoff's equations of motion, the book discusses several extensions of Kirchhoff's work, particularly related to vortices. It addresses the equations of motions of these systems and their Lagrangian and Hamiltonian formulations. The book is suitable to mathematicians, physicists and engineers with a background in Lagrangian and Hamiltonian mechanics and theoretical fluid mechanics. It includes a brief introductory overview of geometric mechanics in the appendix.

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Kirchhoff's insufficiently-celebrated equations of motion.- The addition of vortices.- Dynamically-coupled rigid body+point vortices in R2.- Dynamically coupled rigid body+vortex rings in R3.- Viscous effects and their modeling.- Miscellaneous extensions.- References.- A brief introduction to geometric mechanics.- Leading order behavior of the velocity field and vector potential field of a curved vortex filament.- Hamiltonian function and vector field in the half-space model for Np = 2.