  We prove that some paths of contactomorphisms of $\mathbb{R}^{2n} \times S^1$
endowed with its standard contact structure are geodesics for different norms
defined on the identity component of the group of compactly supported
contactomorphisms and its universal cover. We characterize these geodesics by
giving conditions on the Hamiltonian functions that generate them. For every
norm considered we show that the norm of a contactomorphism that is the
time-one of such a geodesic can be expressed in terms of the maximum of the
absolute value of the corresponding Hamiltonian function. In particular we
recover the fact that these norms are unbounded
