Abstract Geometrical methods play an important role in classical dynamics, most notably in the dynamical preservation of symplecticity in Hamiltonian systems. In this way, dynamics derived from a Hamiltonian function admit energy-conserving integration schemes with an error that remains bounded in the infinite time limit. Furthermore, the existence of this scalar first integral greatly simplifies analysis of a system's dynamics. On the other hand, active systems imply power-injection at the microscale and thus dynamics that can not be encapsulated within the Hamiltonian formalism. It is therefore surprising that Hamiltonian functions should occured in a variety of active matter problems. Even more surprising is that these appearances often manifest in overdamped systems, where viscous damping forces dominate inertia driven effects.

Abstract Starting at the equations of motion for active systems, where constituent active particles provide energy input at the microscopic level, the emergence of symplectic structure is investigated. In the presence of overdamped forces and torques, inertial forces are negligible and time-evolution become purely kinematical in nature. This dissipative structure precludes the existence of a dynamical Hamiltonian and there is therefore no a priori reason for the existence of a conserved symplectic form. Yet in many active systems, pairs of kinematic variables may be identified as conjugate to one another, and thus a kinematically symplectic structure may be conserved throughout time-evolution. Particular examples of kinematic symplecticity include both two- and infinite-body sedimenting active systems. Here, the constituent active particles break microscopic rotational symmetry through a preferred axis of translation due to their internal activity manifesting as self-propulsion in a dissipative environment while sedimenting through a viscous medium under the influence of an external gravitational force. This produces hydrodynamically mediated forces and torques that induce collective interactions. Kinematic Hamiltonians are identified within the equations of motion and are exploited in order to understand the periodic limit cycle behavior observed in the presence of dissipation. In the two-body system, the presence of gravitational torque and hydrodynamic interaction with a nearby boundary result in transient decay to a stable limit cycle which provides a model for “dancing Volvox”. In the case of the sedimenting infinite lattice, dubbed the “active Cosserat crystal”, the presence of activity allows for travelling waves of position and orientation and the two-dimensional crystal acts as an elastic medium with Lamé constants that depend on activity. In the presence of a plane boundary, we show that the active Cosserat crystal can undergo a dynamical phase transition upon tuning of the activity, resulting in the formation of new dissipative steady states. The first of these is the “active Peiels transition”, whereby the lattice spontaneously dissociated into multimeric layers. The second is a spontaneously generated active oscillation, whereby travelling waves of position and orientation are sustained through active energy injection.

Abstract We then investigate the occurance of dynamical symplecticity for arbitrary inertial active systems. The additional microscopic structure of these active particles endows the underlying state space of these systems with non-trivial geometry. Following the erlangen program of Felix Klein, the Lie group that acts transitively on this state space is denoted as the principle group of the geometry and it is within this group that the equations of motion are constructed. This provides tremendous advantages since the algebraic structure of the Lie group itself allows the equations of motion to be formulated in a compact and expressive manner while time-evolution is guaranteed to remain on manifold. The virtual power principle is employed in order to construct dynamical equations of motion, which immediately admits a notion of generalized force as conjugate to the generalized velocity. Integrability criteria for these generalized forces are derived and explicit conditions are written down for the experimentally relevant system of rigid bodies, where the generalized forces include forces and torques. For integrable generalized forces we construct a Hamiltonian function that encapsulates the dynamical evolution.