Symplectic structures for dummies [Archive]

Gerard Westendorp

May29-04, 11:58 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Does anyone feel like explaining why symplectic structures\nare important in dynamics?\n\nthanks,\nGerard\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Does anyone feel like explaining why symplectic structures
are important in dynamics?

thanks,
Gerard

Calvin Ritchie

May31-04, 06:26 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nGerard Westendorp wrote:\n\n>Does anyone feel like explaining why symplectic structures\n>are important in dynamics?\n\nFor an unusually clear presentation, see Corben and Stehle, "Classical\nMechanics", 2nd Ed., Chapter 13, Dover Press. Here\'s a very brief and\nsketchy summary of some important points:\n\nIn the Lagrange/Hamilton formalism, n coordinates,q^i, and n momenta, p_i,\nand their time dependences, are related through the 2n Hamilton\'s equations,\n\ndq^i/dt=partial(H) w.r.t. p_i; dp_i/dt=-partial(H) w.r.t. q^i.\n\nIf we invent a set of 2n "generalized coordinates",\n\nz^(2i-1)=q^i; z^(2i)=p_i,\n\nthen the 2n Hamilton\'s equations become\n\ndz^j/dt=A_(jk)*partial(H) w.r.t. z^k,\n\nwhere A is the anti-symmetric matrix having 1s in elements immediately above\nand to the right of the diagonal, and -1s immediately below and to the left\nof the diagonal, and all other elements zero.\n\nA_(jk)=+1 and A_(kj)=-1 if k=j+1; A_(jk)=0 otherwise.\n\nIf you\'re familiar with the Pauli 2x2 matrices, then A is the 2nX2n matrix\nhaving i*Sigma_2 strung, in n 2x2 blocks, along the diagonal.\n\nThe remarkable thing about Hamilton\'s equations written in this "symplectic"\nform is that they are preserved by any transformation of the "coordinates",\nz^j, which preserves the matrix A. Such "symplectic" transformations may\nwell mix what were originally coordinates, q^i, with what were originally\nmomenta, p_j.\n\nIn equations, any 2nX2n matrix T, with transpose T^t, such that\n\nT^t A T = A,\ny^j=T_(jk)z^k,\n\nmakes y^j obey the same Hamiltonian equations as did the z^j.\n\nIn group theory, the matrices T form the "symplectic group", which is\none of the Classical Lie Groups. Corben and Stehle point out, in their eq.\n69.7, that this whole thing is tied directly to the classical Poisson\nbracket. Since that is directly associated with the commutators of QM\noperators, the symplectic transformations have obvious applications in\nquantum, as well as classical, dynamics.\n\nDon Ritchie,\nsubmitted afternoon 29 May 2004.\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Gerard Westendorp wrote:

>Does anyone feel like explaining why symplectic structures
>are important in dynamics?

For an unusually clear presentation, see Corben and Stehle, "Classical
Mechanics", 2nd Ed., Chapter 13, Dover Press. Here's a very brief and
sketchy summary of some important points:

In the Lagrange/Hamilton formalism, n coordinates,q^i, and n momenta, p_i,
and their time dependences, are related through the 2n Hamilton's equations,

dq^i/dt=partial(H) w[/itex].r.t. p_i; dp_i/dt=-partial(H) w.r.t. q^i.

If we invent a set of 2n "generalized coordinates",

z^(2i-1)=q^i; z^(2i)=p_i,

then the 2n Hamilton's equations become

dz^j/dt=A_(jk)*partial(H) w.r.t. z^k,

where A is the anti-symmetric matrix having 1s in elements immediately above
and to the right of the diagonal, and -1s immediately below and to the left
of the diagonal, and all other elements zero.

A_(jk)=+1 and A_(kj)=-1 if k=j+1; A_(jk)=0 otherwise.

If you're familiar with the Pauli 2x2 matrices, then A is the 2nX2n matrix
having i*\Sigma_2 strung, in n 2x2 blocks, along the diagonal.

The remarkable thing about Hamilton's equations written in this "symplectic"
form is that they are preserved by any transformation of the "coordinates",
z^j, which preserves the matrix A. Such "symplectic" transformations may
well mix what were originally coordinates, q^i, with what were originally
momenta, p_j.

In equations, any 2nX2n matrix T, with transpose T^t, such that

[itex]T^t A T = A,y^j=T_(jk)z^k,

makes y^j obey the same Hamiltonian equations as did the z^j.

In group theory, the matrices T form the "symplectic group", which is
one of the Classical Lie Groups. Corben and Stehle point out, in their eq.
69.7, that this whole thing is tied directly to the classical Poisson
bracket. Since that is directly associated with the commutators of QM
operators, the symplectic transformations have obvious applications in
quantum, as well as classical, dynamics.

Don Ritchie,
submitted afternoon 29 May 2004.

Arnold Neumaier

May31-04, 04:20 PM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Gerard Westendorp wrote:\n> Does anyone feel like explaining why symplectic structures\n> are important in dynamics?\n\nEach symplectic structure defines a Poisson bracket, and hence\nprovides the same niceties for Hamiltonian dynamics as for the\nstandard approach in classical mechanics. And there are natural\nquantum analogues, too. A thorough, comprehensive treatment is in\n\nJ.E. Marsden and T.S. Ratiu,\nIntroduction to Mechanics and Symmetry,\nSpringer, New York 1994.\n\nA good, reasonably self-contained article (easier to read than the\nabove book, especially if you know already basics on Lie algebras) is\n\nP. J. Morrison\nHamiltonian description of the ideal fluid\nRev. Mod. Phys. 70, 467=96521 (1998)\n\nUnfortunately, I currently don\'t have the time to explain things\nin detail, but mechanics on symplectic manifolds and - slightly\nmore general - Poisson manifolds is a very nice subject,\ngiving unity to essentially all conservative dynamics found\nanywhere in physics.\n\nIt may take a while to get used to the differential geometry\nterminology; looking at everything from the point of view of the\nalgebra of C^infinity functions on the manifold is a good exercise\nthat helps to check one\'s understanding.\n\n\nArnold Neumaier\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>Gerard Westendorp wrote:
> Does anyone feel like explaining why symplectic structures
> are important in dynamics?

Each symplectic structure defines a Poisson bracket, and hence
provides the same niceties for Hamiltonian dynamics as for the
standard approach in classical mechanics. And there are natural
quantum analogues, too. A thorough, comprehensive treatment is in

J.E. Marsden and T.S. Ratiu,
Introduction to Mechanics and Symmetry,
Springer, New York 1994.

A good, reasonably self-contained article (easier to read than the
above book, especially if you know already basics on Lie algebras) is

P. J. Morrison
Hamiltonian description of the ideal fluid
Rev. Mod. Phys. 70, 467=96521 (1998)

Unfortunately, I currently don't have the time to explain things
in detail, but mechanics on symplectic manifolds and - slightly
more general - Poisson manifolds is a very nice subject,
giving unity to essentially all conservative dynamics found
anywhere in physics.

It may take a while to get used to the differential geometry
terminology; looking at everything from the point of view of the
algebra of C^{infinity} functions on the manifold is a good exercise
that helps to check one's understanding.

Arnold Neumaier