Estudio de sistemas mecánicos disipativos mediante mecánica geométrica lagrangiana

Abstract

This thesis presents a critical analysis of the standard Lagrangian formalism, identifying its structural limitations in describing mechanical systems with dissipation. Mathematical extensions, such as integrating factors, are explored, which, while allowing for a variational treatment, often compromise the physical interpretation of the resulting quantities. To overcome these shortcomings, a theoretical framework based on contact geometry is introduced, where dissipation is modeled as an intrinsic property of the extended state space (TQ × ℝ) rather than as an external force. Within this formalism, the contact Euler-Lagrange equations are derived, a generalization that naturally accounts for dissipative terms. The applicability of the approach is demonstrated with the Damped Harmonic Oscillator, and a generalization of Noether's theorem for contact geometry is applied to predict and construct invariants of motion (conserved quantities) in systems with symmetries, even in the presence of dissipation.