I am doing some exercises in Lagrangian systems in the book Quantum Mechanics for Mathematicians. One exercise says:
Let $f$ be a $C^\infty$ function on a manifold $M$. Show that the Lagrangian systems $(M,L)$ and $(M,L+df)$ (where $df$ is fibre-wise linear function on $TM$) have the same equations of motion.
I do know what $df$ means as a differential form, and I solved the exercise. I just had never read such terminology ("fibre-wise") and I wonder what does that mean.
$TM\xrightarrow\pi M$ is a vector bundle. In particular, this means that for every $p\in M$, the fibre $\pi^{-1}\{p\}=T_pM\subseteq TM$ is a vector space.
Saying that $df$ is fiberwise linear means that for every $p\in M$ the function $(df)_p:T_pM\to\mathbb R$ is linear.