In quantum mechanics, a physical quantity is expressed as an operator G, then the unitary transformation coresponding to the physical quantity is expressed as exp(-iG/ħt), being also an operator, where t is the tranformation parameter. G is actually the conservative quantity corresponding to the transformation generated by G. In classical mechanics, an one-parameter Lie group action is expressed as exp(tX), X being an element of the Lie algebra of the Lie group in question. According to the Noether theorem, the invariance of a physical system under a Lie group action gives a conservative quantity. However, I found it's like X is not the correspponding conservative quantity, like the situation in quantum mechanics. For example, you can see an element of the Lie algebra of SO(3) is not (a quantity proportional to) the angular momentum. So what is the relation between the Lie algebra elements and the corresponding conservative quantity? Is there a generic formula to express the relation?