Lagrangian Dynamics of Spinning Pole-Dipole-Quadrupole Particles in Metric-Affine Geometries

Abstract: We construct the Lagrangian formulation of a micro-structured spinning, dilating and shearing (deformable) test body, moving in arbitrary non-Riemannian backgrounds possessing all geometrical entities of curvature, torsion and non-metricity. We start with a Lagrangian of a generic form that depends on the particle's velocity, its material frame and its absolute derivative, and the background geometry consisting of a metric and an independent affine connection. Performing variations of the path and the material frame, we derive the equations of motion for the particle that govern the evolution of its momentum and hypermomentum in this generic background. The reported equations of motion generalize those of a spinning particle (Mathisson \cite{Mathisson:1937zz}, Papapetrou \cite{Papapetrou:1951pa}, Dixon \cite{Dixon:1974xoz}) by the inclusion of the dilation and shear (hadronic) currents of matter. Using the derived equations of motion, a generalized conserved quantity is also found. Further conserved quantities that can be obtained by appropriate supplementary conditions are also discussed.