The main reasons why connections play a prominent role in many theoretical models field models lie in the fact that they enable us to deal with well (globally, invariantly) defined objects.
Connections in classical field theory have been discussed (Why connections in classical field theory?).
Classical non-relativistic mechanics is formulated as a particular field theory on smooth fibre bundles Q->R over the time axis R (Mechanics as particular classical field theory). Its velocity phase space is the first order jet bundle JQ->R. Its momentum phase space is the vertical cotangent bundle V*Q of Q. The concept of a connection is the central ingredient in this geometric formulation as follows.
(i) An essential difference between classical mechanics and field theory lies in the fact that connections on a fibre bundle Q->R are flat and, therefore, they fail to be dynamic variables. They describe non-relativistic reference frames. This fact enables us to define relative velocities and accelerations, and describe non-relativistic mechanics with respect to different reference frames.
In particular, one can define a free motion equation and the geodesic reference frame for it which is called the inertial reference frame. However, an absolute inertial frame fails to be defined.
(ii) Equations of motion of non-relativistic mechanics almost always are of second order. Second order dynamic equations on a fiber bundle Q->R are conventionally defined as the holonomic connections on the jet bundle JQ->R. These equations also are represented by connections on the jet bundle JQ->Q and, due to the canonical imbedding of JQ to the tangent bundle TQ, they are proved to be equivalent to non-relativistic geodesic equations on TQ.
(iii) In Hamiltonian non-relativistic mechanics on the momentum phase space V*Q,
Hamiltonian connections on V*Q->R define the
L.Mangiarotti, G.Sardanashvily, Gauge Mechanics (WS, 1998)
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Geometric Formulation of Classical and Quantum Mechanics (WS, 2010)
G.Giachetta, L.Mangiarotti, G.Sardanashvily, Advanced mechanics. Mathematical introduction, arXiv: 0911.0411