Summary
It is shown that there exist symmetry transformations in phase space that preserve Hamilton’s canonical equations of motion for one Hamiltonian, but not for all. Examples of these « canonoid » transformations are given and their relation to canonical transformations is developed. It is demonstrated that a sufficient condition for a canonoid transformation to be canonical is that it preserve Hamilton’s equations for all Hamiltonians quadratic in theq’s andp’s.
Riassunto
Si dimostra che esistono trasformazioni di simmetria nello spazio delle fasi oho preservano le equazioni canoniche di moto di Hamilton per un hamiltoniano, ma non per tutti. Si danno esempi di questo trasformazioni « canonoidi » e si sviluppa la loro relazione con le trasformazioni canoniche. Si dimostra che una condizione sufficiente perché una trasformazione canonoide sia canonica è che essa preservi le equazioni di Hamilton per tutti gli hamiltoniani quadratici neiq e neip.
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References
E. P. Wigner:Ann. Math.,40, 149 (1939).
D. G. Currie:Journ. Math. Phys.,4, 1470 (1963).
D. G. Currie:Ann. of Phys.,36, 104 (1966).
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This is a simple application of Schur’s Lemma. See,e.g.,E. D. Nering:Linear Algebra and Matrix Theory (New York, 1963).
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Currie, D.G., Saletan, E.J. Canonical transformations and quadratic hamiltonians. Nuov Cim B 9, 143–153 (1972). https://doi.org/10.1007/BF02735514
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DOI: https://doi.org/10.1007/BF02735514