  The recent interest in structure preserving stochastic Lagrangian and
Hamiltonian systems raises questions regarding how such models are to be
understood and the principles through which they are to be derived. By
considering a mathematically sound extension of the Hamilton-Pontryagin
principle, we derive a stochastic analogue of the Euler-Lagrange equations,
driven by independent semimartingales. Using this as a starting point, we can
apply symmetry reduction carefully to derive non-canonical stochastic
Lagrangian / Hamiltonian systems, including the stochastic Euler-Poincar\'e /
Lie-Poisson equations, studied extensively in the literature. Furthermore, we
develop a framework to include dissipation that balances the
structure-preserving noise in such a way that the overall stochastic dynamics
preserves the Gibbs measure on the symplectic manifold, where the dynamics
effectively take place. In particular, this leads to a new derivation of
double-bracket dissipation by considering Lie group invariant stochastic
dissipative dynamics, taking place on the cotangent bundle of the group.
