Can theoretical mechanics be derived from differential geometry? I've heard this somewhere, I don't know how True it is, can somebody clarify the relationship between theoretical mechanics and differential geometry? Can theoretical mechanics really be derived from differential geometry really?  can somebody suggest a reading for this topic? lectures or other things will do as well. Thanks
 A: OP's question is partially a matter of terminology, but one can certainly built a good case arguing that fundamental physics is (or is at least written in the language of) geometry. E.g. many fundamental physical systems have geometric & covariant action formulations, and the corresponding Euler-Lagrange equations have often an interpretation as, say, the geodesic equations, or generalizations thereof. 
A: Physical theories are derivable from mathematics, but only to the extent that you know which Lagrangian (or Hamiltonian) to use, which comes from physical insight. Also, how does one know when the theory, birthed only from differential geometry, represents reality? WELL, its predictions must be compared to experiment - this is physics!
Examples:
1) classical mechanics can be derived from the differential geometry of a symplectic manifold and Poisson algebra, but you need to guess the correct Hamiltonian for the system you want (the math can't tell you that alone, you must consult experimentation and/or physical intuition).
2) general relativity can be derived from a Lagrangian via the Euler-Lagrange equations, but you must use the correct Lagrangian! To find the correct Lagrangian, you must again use physical intuition - that is, can empirically validated formulas be derived from the Lagrangian (i.e. Poisson equation in weak-field, low-speed limit)? And of course trial-and-error helps. 
If it's a model of reality, then reality must be consulted at some point - this is the essence of empiricism, and it's the reason that modern science is where it is today (rather than steeped in scholasticism like in the humanities).
