Newtonian/Gallileann inertia describes very well i.e answers without creating new puzzling questions the phenomena in the domain i.e unpowered motion in free space.
Machs principle consider inertia a cummulativeveffect of the universe'matter on an object.
There is also MOND's (modifyibg Newtonian gravity) inertia modification on the matter term of the Lagrangian action.
Any others?
There are several alternative theories and concepts related to inertia, apart from Mach's and MOND extensions. Some of these alternatives include:
Insofar as the equivalence principle is concerned, i.e. the equality between inertial and gravitational mass, in the Newtonian approximation, it is known that the way to describe deviations is by including fields, beyond the metric, that define the spacetime geometry, but can't be identified as matter because their energy-momentum tensor doesn't satisfy the appropriate energy condition.
MOND is an alternative to dark matter. The existence of dark matter is a consequence of the shape of the rotation curve of galaxies. The known forms of matter of galaxies aren't enough to explain their rotation curves. So there must, either, exist a new form of matter, with only gravitational interactiins with ordinary matter, or the way the rotation curves are derived must be modified. MOND is one such modification.
MOND in fact doesn't have anything to do with inertia as such-it has to do with the principle of equivalence, which, in the Newtonian approximation, implies that the inertial mass-the mass that appears in Newton's 2nd law-is equal to the gravitational mass-the mass that appears in Newton's law of gravitation. Dimensional analysis implies these are proportional, it's an experimental observation, at this approximation, that the relative-dimnionless-coefficient is equal to 1. General relativity provides the theoretical explanation for this property, by showing that, if only the metric defines the spacetime geometry, then the only way to couple matter to the metric, consistently with invariance under diffeomorphisms, is through the energy-momentum tensor of matter, not the fields that define matter themselves. It is possible to construct generalizations of general relativity, where the spacetime geometry is defined by fields beyond the metric, namely scalar fields, vector fields and spinor fields (that can't be interpreted as matter fields, because their energy-momentum tensordoesn't satisfy the appropriate positivity conditions). These additional fields do couple to the fields that define matter and not to the energy-momentum tensor; therefore they lead to violations of the equivalence principle. MOND can be understood as belonging to this class of theories. There are tests of the equivalence principle , however, and these do place constraints on such extensions-and, therefore, on MOND.
Inertia is an interaction of an accelerating mass with the local vacuum that said mass is accelerating through. This is most proven by quarks and gluons accelerating through the gluon gas, inside an atomic nucleus, creating most of the baryonic mass in the universe. This is supported by hundreds of papers that interpret JWST data, as costing cosmology, with the Hubble radius equaling the vacuum speed of light times the local proper time in the cosmic fluid.
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