The article considers the homogeneous vortex motion of an inhomogeneous fluid in a stationary ellipsoidal cavity arbitrarily oriented with respect to the direction of a uniform gravity field using the Lagrange variables. Uniform vortex motion of a fluid is a motion in which the rotor speed for all particles has the same value, and depends only on time. The article shows that the equations of homogeneous vortex motion of a heavy inhomogeneous fluid are possible in an ellipsoidal cavity with a linear density distribution and proposes a geometric interpretation of fluid motion.
Using the Lagrange variables, the equations of motion of an inhomogeneous fluid are obtained, coinciding with the equations of a heavy rigid body motion around a fixed point, written in a fixed coordinate system arbitrarily located relative to the direction of a homogeneous gravity field.