Relation between fixed constraints and time derivative of the Lagrangian

Question

I have had some trouble interpreting and proving the following statement from Fasano, Marmi's "Analytical Mechanics" (page 139):

"... ${\partial L}/{\partial t} \neq 0$ (1) only if the constraints are in motion (2)."

I do not understand wether this is an equivalence or an implication and, in that case, in which direction it goes.

I think that (1) $\implies$ (2) is not true for the contrapositive would be "if the constraints are fixed then ${\partial L}/{\partial t} = 0$" which is nonsense since the system could still be in motion. Viceversa, I am not sure on how to go about proving (2) $\implies$ (1).

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As always any comment or answer is much appreciated and let me know if I can explain myself clearer!

Answer 1

1. A holonomic constraint $f({\bf X},t)=0$ is by definition in motion/moving iff $\partial f/\partial t \not \equiv 0$, cf. Definition 1.35 on p. 53.

2. Ref. 1 means that $(1)\Rightarrow (2)$. For the implication to be true, Ref. 1 is apparently only considering a Lagrangian of the form $L=T-U$ with a standard kinetic term $T=\sum_{i=1}^N \frac{1}{2}m_i\dot{\bf X}_i^2$ a conservative potential $U({\bf X})$, and only holonomic constraints.

References:

1. A. Fasano & S. Marmi, Analytic Mechanics, 2006; p. 139.