  We describe a general framework for studying duality between different phase
spaces which share the same symmetry group $\mathrm{H}$. Solutions
corresponding to collective dynamics become dual in the sense that they are
generated by the same curve in $\mathrm{H}$. Explicit examples of phase spaces
which are dual with respect to a common non trivial coadjoint orbit
$\mathcal{O}_{c,0}(\mathbf{\alpha},1) \subset\mathfrak{h}^{\ast}$ are
constructed on the cotangent bundles of the factors of a double Lie group
$\mathrm{H}=\mathrm{N}\Join\mathrm{N}^{\ast}$. In the case $\mathrm{H}=LD$, the
loop group of a Drinfeld double Lie group $D$, a hamiltonian description of
Poisson-Lie T-duality for non trivial monodromies and its relation with non
trivial coadjoint orbits is obtained.
