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The non-squeezing theorem, also called ''Gromov's non-squeezing theorem'', is one of the most important theorems in symplectic geometry.〔.〕 It was first proven in 1985 by Mikhail Gromov. The theorem states that one cannot embed a sphere into a cylinder via a symplectic map unless the radius of the sphere is less than or equal to the radius of the cylinder. The importance of this theorem is as follows: very little was known about the geometry behind symplectic transformations. One easy consequence of a transformation being symplectic is that it preserves volume.〔D. McDuff and D. Salamon''Introduction to Symplectic Topology'', Cambridge University Press (1996), ISBN 978-0-19-850451-1.〕 One can easily embed a ball of any radius into a cylinder of any other radius by a volume-preserving transformation: just picture ''squeezing'' the ball into the cylinder (hence, the name non-squeezing theorem). Thus, the non-squeezing theorem tells us that, although symplectic transformations are volume-preserving, it is much more restrictive for a transformation to be symplectic than it is to be volume-preserving. == Background and statement == We start by considering the symplectic spaces : the ball of radius ''R'': and the cylinder of radius ''r'': each endowed with the symplectic form : The non-squeezing theorem tells us that if we can find a symplectic embedding ''φ'' : ''B''(''R'') → ''Z''(''r'') then ''R'' ≤ ''r''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Non-squeezing theorem」の詳細全文を読む スポンサード リンク
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