  When space-time is assumed to be non-Riemannian the minimal coupling
procedure (MCP) is not compatible, in general, with minimal action principle
(MAP). This means that the equations gotten by applying MCP to the
Euler-Lagrange equations of a Lagrangian $\cal L$ do not coincide with the
Euler-Lagrange equations of the Lagrangian obtained by applying MCP to $\cal
L$. Such compatibility can be restored if the space-time admits a
connection-compatible volume element. We show how these concepts can alter
qualitatively the predictions of the Einstein-Cartan theory of gravity.
