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If we have a $\phi: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$, $\phi = \phi(t, \mathbf{q},\alpha)$ one-parameter group of infinitesimal transformation which is $\mathcal{C}^2$ with respect to $t$ and $\mathbf{q}$, is it sufficient to $\phi$ to be differentiable with respect to $\alpha$ to write $$\phi = \phi(t, \mathbf{q}, 0) + \alpha \frac{\partial \phi}{\partial \alpha}(t, \mathbf{q}, 0) + o(\alpha)$$ if we want $\eta (t, \mathbf{q}) = \frac{\partial \phi}{\partial \alpha}(t, \mathbf{q}, 0)$ to be $\mathcal{C}^1$ with respect to $t$ and $\mathbf{q}$ ? I suspect this term is $\mathcal{C}^2$ with respect to $t$ and $\mathbf{q}$...

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