The role of the concept of symmetry in physics is historically illustrated starting from the connection between symmetry properties and conservation laws in classical mechanics up to the recent developments of the concept of spontaneous symmetry breaking and of hidden symmetry in quantum mechanics. (i) Newton's First Law of Motion states that an iso-lated material body (of su±ciently small size) main-tains a state of rest or of uniform motion in a straight line. The quantum statistical treatment of the symmetry number is briefly reviewed in the Introduction for comparison with the classical statistical treatment. It is perhaps surprising that until now there has been no attempt to classify and analyse all the symmetries relevant to music perception. From the dynamical origin of the symmetry properties of some mechanical systems,like … Since the second law of thermodynamics states that entropy increases as time flows toward the future, in general, the macroscopic universe does not show symmetry under time reversal. In quantum mechanics, phase symmetry is the fact that the state of a system can be multiplied by a length 1 complex number without changing the probabilities of observables. The concept of a localized system is discussed as a convenient method to calculate the classical partition function. A rather strong concept of symmetry is introduced in classical mechanics, in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey. Is there an analogous symmetry concept in classical physics? The general concept of symmetry plays an important role in many areas of science, including classical mechanics, quantum mechanics and crystallography. The symmetry number in classical statistical mechanics is then defined, and it is applied to some examples. T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : ↦ −. Boldface symbols will denote vec- tors, as usual. Symmetry and relativity: From classical mechanics to modern particle physics* Z. J. Ajaltouni Laboratoire de Physique Corpusculaire de Clermont-Ferrand IN2P3/CNRS Université Blaise Pascal F-63177, Aubière Cedex, France; [email protected] Received 17 December 2013; revised 17 January 2014; accepted 24 January 2014 quantum-mechanics classical-mechanics hilbert-space symmetry complex-numbers. Symmetry in Classical Mechanics I n r ec alig f m t, w k s p-by progression from elementary considerations to a com-prehensive formalism.