What is a reference frame in field theory and mechanics

Non-relativistic mechanics as like as classical field theory is formulated in terms of fibre bundles.

In classical field theory on a fibre bundle Y->X, a reference frame is defined as an atlas of this fibre bundle, i.e. a system of its local trivializations.

If it is a gauge theory, Y->X is a fibre bundle with a structure Lie group G, and a reference frame equivalently defined as a system of local sections of an associated principal bundle P->X.

In particular, in gravitation theory on a world manifold X, a reference frame is a system of local frame fields, i.e. local sections of a fibre bundle P->X of linear frames in the tangent bundle TX of X.  This also is the case of relativistic mechanics. Therefore, one can treat the components of a tangent reference frame as relativistic velocities of some observers.

There are some reasons to assume that a world manifold X is parallelizable, i.e. its tangent bundle is trivial, and there exists a global section of a frame bundle P->X, i.e. a global reference frame. In this case, by virtue of the well-known theorem, there exists a flat connection K on a world manifold X which is trivial (K=0) with respect to this reference frame, and vice versa. Consequently, a reference frame on a parallelizable world manifold can be defined as a flat connection. The corresponding covariant differential provides relative velocities with respect to this reference frame.

Lagrangian and Hamiltonian non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles Q->R over the time axis R. Such fibre bundle is always trivial. Therefore, a reference frame in non-relativistic mechanics can be defined both as a trivialization of a fibre bundle Q->R and a connection K on this fibre bundle. Absolute velocities are represented by elements v of the jet bundle JQ of Q, and their covariant differential v-K are relative velocities with respect to a reference frame K.

This description of a reference frame as a connection enables us to formulate non-relativistic mechanics with respect to any reference frame and arbitrary reference frame transformations.

References:

G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced Classical field Theory (WS, 2009)

G.Giachetta, L.Mangiarotti, G.Sardanashvily,  Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)

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