It will certainly not satisfy all your needs, but you cannot avoid spending some time with "Structure and Interpretation of Classical Mechanics" and its compani

It will certainly not satisfy all your needs, but you cannot avoid spending some time with "Structure and Interpretation of Classical Mechanics" and its companion scheme code. There is also a neat appendix, slightly simpler and more modern, aimed at the mathematical details, titled "Functional Differential Geometry", by the same authors.