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kawabata

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Some points are not clear to me in classical mechanics. I would be grateful if any of you could make them clear.

1. "In general, if a system subject to r non-holonomic conditions has f degrees of freedom in

*finite motion*, it has only f-r degrees of freedom in

*infinitesimal motion*." [Arnold Sommerfeld, Mechanics, 1964, p50.]

** I searched and couldn't find any definition for finite and infinitesimal motion in that book. What do we understand if a motion is finite or infinitesimal so that the mentioned difference is proven? What is the theoretical basis of this statement?

2. I encountered a differentiation on degree of freedom (dof) in the Internet. "Controllable dof" and "effective dof". It is said that a car has 2 controllable dof, which are steering and driving, but 3 effective dof (some says "total dof"), which are x, y, and orientation. It seems that these definitions are relatively new in literature. I don't know where I can refer to. Considering the controllability in modern control theory, is there a mathematical explanation of this "controllable" word?