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I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of situations ?....I am talking about classical systems here.

Also, the additivity of energy becomes obvious from the property of the additivity of Lagrangian itself (as given in Landau's Classical Mechanics page-14). Then, is it necessary that Lagrangian should have the additive property or is it just always defined with the additive property ?

Thanks

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