Building on this Phys.SE post I am interested in how non-inertial frames can be considered in Lagrangian mechanics. My understanding is that changing the reference frame causes a transformation of the Lagrangian that will transform in such a way as to account for the apparent fictitious forces if the resultant frame is non-inertial. Colloquially we might say that it has all the information in it from the start to describe a system. Therefore a Lagrangian is not unique and transforms like a scalar. Can we describe the Euler-Lagrange equation (for discrete particles) as covariant? (normally this is the topic of field equations, which are covariant).
Bringing this back to more familiar topics in Lagrangian mechanics, should we view this as a point or a gauge transformation?
On the one hand a point transformation is the changing of the coordinate system, say $(q,\dot q)\rightarrow (Q,\dot Q)$, this will in the most general sense change the Lagrangian and leave equations of motion invariant.
A gauge transformation however has the same coordinates and the same general form of the Lagrangian except that there is a total time derivative added on the end.
Therefore how should one, if indeed one can, consider inertial frames in analytical mechanics in terms of point/gauge transformations?
Thank you for your thoughts.
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