canonical_transf

canonical_transf - CANONICAL TRANSFORMATION THEORY A...

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CANONICAL TRANSFORMATION THEORY A canonical transformation may express new displacements and momenta as functions of both the original displacements and momenta, but is restricted such that it preserves the Hamiltonian form of the differential equations. Original variables ("old coordinates") H = H( q , p ) d q /dt = H( q , p )/ p f ( q , p ,t) d p /dt = – H( q , p )/ q + e ( q , p ,t) The functions f ( q , p ,t) and e ( q , p ,t) contain the non-conservative and forcing terms and are known as "canonical forces" (an unfortunate historical misnomer). d q /dt = d p /dt 0 1 1 0 H/ q H/ p f e + Symplectic notation: f e q p H = H( r ) G = r = 0 1 1 0 J = The 2n x 2n matrix J is the so-called symplectic matrix. Because J J = – 1 it is analogous to the complex number j (the square root of –1). It is also analogous to a
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canonical_transf - CANONICAL TRANSFORMATION THEORY A...

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