  Certain dissipative physical systems closely resemble Hamiltonian systems in
$\mathbb{R}^{2n}$, but with the canonical equation for one of the variables in
each conjugate pair rescaled by a real parameter. To generalise these dynamical
systems to symplectic manifolds in this paper we introduce and study the
properties of deformed Hamiltonian vector fields on Lagrangian fibrations. We
describe why these objects have some interesting applications to symplectic
geometry and discuss how their physical interpretation motivates new problems
in mathematics.
