Lagrange's equations for two-dimensional motion of a cylinder in perfect fluid

Lagrange's formulation of system of particles mechanics is based on La-grange's description of motion. When a rigid body moves in a velocity field of fluid it is convenient to use Lagrange's description of the body motion and the Euler one for the velocity field of fluid. In such a description a Lagrangian equal to the kinetic energy minus the potential one does not lead to correct differential equations of motion. In our case the Lagrangian is defined as a function that leads to the correct differential equations for the problem. The problem is discussed for the case of a circular cylinder moving in a space and half-space of fluid. First of all, the resultant hydrodynamic forces acting on the cylinder are calculated on the basis of pressures and the differential equations for the problem are established. The Lagrangian is given for a general case of motion in space. In the case of half-space with a perfect bottom, it is assumed that the radius of the cylinder is small and an approximate description is introduced. The variational formulations based on Hamillton's principle are useful in numerical solutions.