Euler–Poincaré reduction and the Kelvin–Noether theorem for discrete mechanical systems with advected parameters and additional dynamics

Abstract The Euler–Poincaré equations, firstly introduced by Henri Poincaré in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various frameworks, such as semidirect products, advected parameters, and field theory, with broad applications across physics and engineering. In this paper, we develop a discrete variational framework by establishing the discrete Euler–Poincaré reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics. The reduction is achieved through the use of group difference maps, specifically those defined using the Cayley transform or the matrix exponential. Furthermore, we extend the Kelvin–Noether theorem and its discrete analogue to characterize the Kelvin–Noether quantities for the resulting continuous and discrete Euler–Poincaré equations. As a specific application, we derive both the continuous and discrete Euler–Poincaré equations for the dynamics of underwater vehicles. Numerical simulations validate the proposed discrete scheme, demonstrating excellent long-term preservation of both total energy and Kelvin–Noether quantities. These results highlight the potential of the framework for high-fidelity simulation, control, and navigation of underwater vehicles and other complex mechanical systems.