Given a Riemannian manifold M and “mechanical” Hamiltonian function H on T* M we construct invariant Lagrange submanifolds of dissipative systems on T* M whose cooordinate presentation has the form

q' = dH/dp(p,q) , p' = - dH/dq(p,q) - ap

where a is a positive constant.
The construction uses the curvature of the flows on the cotangent bundle and ideas of the partially hyperbolic dynamics.