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Given a finite dimensional manifold $Q$, the cotangent bundle $(T^* Q, \omega)$ is symplectic. Let $M = \{\xi : T ^*Q \to  Q \}$ and $G =\{ \phi : T^*Q\to T^*Q | \phi^*\omega =\omega\} $ be manifolds of displacement and canonical mappings, respectively. $G$ is a Lie group and bundles over $G$ admit global trivializations. The embedding $G\hookrightarrow T^*M$ is Lagrangian, though, not with canonical two-form. It follows that the Tulczyjew symplectic spaces $TT^* M$ and $T T^*G$ are fiberwise isomorphic. This   result is key to establishing relations among different formulations of plasma dynamics in the Tulczyjew's geometric framework for Legendre transformation.