Geometric Hamilton-Jacobi theory for systems with external forces

In this paper, we develop a Hamilton¿Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems, and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton¿Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton¿Jacobi problem for a ¿aplygin system to the Hamilton¿Jacobi problem for a forced Lagrangian system.We thank the referee for his/her constructive comments. The authors acknowledge financial support from the Spanish Ministry of Science and Innovation (MICINN), under grants PID2019- 106715GB-C21 and “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019- 000904-S). Manuel Lainz wishes to thank MICINN and the Institute of Mathematical Sciences (ICMAT) for the FPI-Severo Ochoa predoctoral contract PRE2018-083203. Asier López- Gordón would like to thank MICINN and ICMAT for the predoctoral contract PRE2020- 093814. He is also grateful for enlightening discussions on fibre bundles with his friend and colleague Alejandro Pérez-González