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.

我们证明了具有标准接触结构的\({\mathbb {R}}^{2n}\times S^1\) 的一些接触同态路径是在紧支撑群的恒等分量上定义的不同范数的测地线接触同态及其普遍覆盖。我们通过给出生成这些测地线的哈密顿函数的条件来表征这些测地线。对于所考虑的每个范数，我们表明接触同胚的范数（即这种测地线的时间范数）可以用相应哈密顿函数的绝对值的最大值来表示。特别是我们恢复了这些规范是无限的这一事实。