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What is the mathematical basis for studying classical mechanics?
If you have studied calculus, you'd better know the concepts of multivariate calculus, ordinary differential equations and matrices. If you know the operation of tensor, you can look at rigid body mechanics.

Others, if more in-depth, need a lot, such as symmetry problems, group knowledge and so on.

Of course, beginners will have more original integrals, and ordinary differential equations will be enough.