When transforming a field in classical field theory the transformation of the four-gradient of this field follows automatically. At least this is what i have learned in my lectures.

This circumstance kind of contradicts my understanding of the Lagrange formalism in classical mechanics. In classical mechanics the generalized coordinates and their derivatives are understood as coordinates in configuration space. Therefore i can perform a transformation in configuration space only concerning the generalized coordinates and not their derivatives.

So there is probably a strong difference between the Lagrange formalism of fields and the Lagrange formalism of classical mechanics which I do not understand yet. If anyone could share some insights I would be very thankful.

When transforming a field in classical field theory the transformation of the four-gradient of this field follows automatically. At least this is what i have learned in my lectures.

This circumstance kind of contradicts my understanding of the Lagrange formalism in classical mechanics. In classical mechanics the generalized coordinates and their derivatives are understood as coordinates in configuration space. Therefore i can perform a transformation in configuration space only concerning the generalized coordinates and not their derivatives.

So there is probably a strong difference between the Lagrange formalism of fields and the Lagrange formalism of classical mechanics which I do not understand yet.

Question: Why is it possible to transform $q(t)$ in classical mechanics without transforming $\dot{q}(t)$, but it is not possible to transform $\phi(x)$ without transforming $\partial_\mu \phi(x)$?