A nilpotent algebra approach to Lagrangian mechanics and constrained motion

Lagrangian mechanics is extended to the so-called nilpotent Taylor algebra ({mathbb {T}}). It is shown that this extension yields a practical computational technique for the evaluation and analysis of the equations of motion of general constrained dynamical systems. The underlying ({mathbb {T}})-algebra utilized herein permits the analysis of constrained dynamical systems without the need for analytical or symbolic differentiations. Instead, the algebra produces the necessary exact derivatives inherently through binary operations, thus permitting the numerical analysis of constrained dynamical systems using only the defining scalar functions (the Lagrangian ({mathcal {L}}) and the imposed constraints). The extension of the Lagrangian framework to the ({mathbb {T}})-algebra is demonstrated analytically for a problem of constrained motion in a central field and numerically for the calculation of Lyapunov exponents of N-pendulum systems.