Non-Noetherian Conserved Quantity? [Archive]

Daryl McCullough

Apr7-04, 09:23 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>\n\nI posted this to sci.math, but I think it got a little lost in the noise.\n\nI stumbled across a family of conserved quantities in classical Lagrangian\nmechanics, and I\'m wondering whether these are well-known, and whether there is\na way to interpret them as Noether conserved charges that follow from symmetries\nof the Lagrangian.\n\nLet the Lagrangian have the following form:\n\nL = A(x,v_x) + 1/2 B(x) v_y^2 - C(y)/B(x)\n\n(where v_q is the velocity for coordinate q). Then the quantity Q defined by\n\nQ = 1/2 (B(x) v_y)^2 + C(y)\n\nis conserved. To see this, note that the Lagrangian equations of motion for\nv_y is\n\nd/dt(B(x) v_y) = - C\'(y)/B(x)\n\nMultiplying both sides by B(x) v_y gives\n\nB(x) v_y d/dt(B(x) v_y) = - v_y C\'(y)\n\n(where C\'(y) is dC/dy)\n\nThe left-hand side is equal to\n\nd/dt(1/2 (B(x) v_y)^2)\n\nThe right-hand side is equal to\n\n- d/dt(C(y))\n\nSo the combination\n\nQ = 1/2 (B(x) v_y)^2 + C(y)\n\nsatisfies\n\ndQ/dt = 0\n\nQ is conserved, but it is not obviously a Noetherian conserved charge\ndue to a symmetry of the Lagrangian.\n\n--\nDaryl McCullough\nIthaca, NY\n\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>I posted this to sci.math, but I think it got a little lost in the noise.

I stumbled across a family of conserved quantities in classical Lagrangian
mechanics, and I'm wondering whether these are well-known, and whether there is
a way to interpret them as Noether conserved charges that follow from symmetries
of the Lagrangian.

Let the Lagrangian have the following form:

L = A(x,v_x) + 1/2 B(x) v_y^2 - C(y)/B(x)

(where v_q is the velocity for coordinate q). Then the quantity Q defined by

Q = 1/2 (B(x) v_y)^2 + C(y)

is conserved. To see this, note that the Lagrangian equations of motion for
v_y is

d/dt(B(x) v_y) = - C'(y)/B(x)

Multiplying both sides by B(x) v_y gives

B(x) v_y d/dt(B(x) v_y) = - v_y C'(y)

(where C'(y) is dC/dy)

The left-hand side is equal to

d/dt(1/2 (B(x) v_y)^2)

The right-hand side is equal to

- d/dt(C(y))

So the combination

Q = 1/2 (B(x) v_y)^2 + C(y)

satisfies

dQ/dt =

Q is conserved, but it is not obviously a Noetherian conserved charge
due to a symmetry of the Lagrangian.

--
Daryl McCullough
Ithaca, NY

Roland Franzius

Apr7-04, 03:11 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>Daryl McCullough wrote:\n\n> I posted this to sci.math, but I think it got a little lost in the noise.\n>\n> I stumbled across a family of conserved quantities in classical Lagrangian\n> mechanics, and I\'m wondering whether these are well-known, and whether there is\n> a way to interpret them as Noether conserved charges that follow from symmetries\n> of the Lagrangian.\n>\n> Let the Lagrangian have the following form:\n>\n> L = A(x,v_x) + 1/2 B(x) v_y^2 - C(y)/B(x)\n>\n> (where v_q is the velocity for coordinate q). Then the quantity Q defined by\n>\n> Q = 1/2 (B(x) v_y)^2 + C(y)\n>\n> is conserved. To see this, note that the Lagrangian equations of motion for\n> v_y is\n>\n> d/dt(B(x) v_y) = - C\'(y)/B(x)\n>\n> Multiplying both sides by B(x) v_y gives\n>\n> B(x) v_y d/dt(B(x) v_y) = - v_y C\'(y)\n>\n> (where C\'(y) is dC/dy)\n>\n> The left-hand side is equal to\n>\n> d/dt(1/2 (B(x) v_y)^2)\n>\n> The right-hand side is equal to\n>\n> - d/dt(C(y))\n>\n> So the combination\n>\n> Q = 1/2 (B(x) v_y)^2 + C(y)\n>\n> satisfies\n>\n> dQ/dt = 0\n>\n> Q is conserved, but it is not obviously a Noetherian conserved charge\n> due to a symmetry of the Lagrangian.\n\nIn many situations one has a decompositon of the Hamiltonian into a net\nof partial conserved quadratic momenta, sometimes with additonal\npotential of product type. Eg in spherical coordinates the Hamiltonian\nfor free particles or with specialy designed potentials may be\ndecomposed according to\n\n\nH = H_r + f(r)( H_theta) + g(theta) H_phi))\n\nThen obviously H_phi and ( H_theta) + g(theta) H_phi)) are conserved\nbecause they commute with H.\n\nIn your case call B(x) a radius r and y an angle \\phi. You found a split\nof the total energy into a radial part and an independent conserved\nangular part of the form\n\nE = A(r,\\dot r) + 1/2 r^2 v\\phi + V(\\phi)\n\nin analogy to the angular momentum splitting with V=0\n\nE = 1/2 vr^2 + 1/2 r^2 v\\phi^2\n\nWrite down the Hamiltonian with vr = a(r,p_r) the inverse function of\np_r = dA(r,vr)/v_r\n\nH = p_r a(r,p_r) + 1/B ( 1/2 p_phi^2 + 1/B V(phi) )\n\nand you see without calculation that the angular energy is conserved.\n\nThe feature of the existence of a such a decomposition is a consequence\nof the existence of a commuting set of conserved quantities which are\nquadratic in canonical momenta and depend on a product potential\nspecialy fitted with the metric factors not to disturb the separating\nprocess. In quantum mechanics this leads to a decompositon of the\nHilbert space into series of tensor products of Hilbert spaces of one\nvariable.\n\nIn this situation the representation of the symmetry group is governed\nby a series of casimir operators. Additional potentials working in the\none variable Hilbert spaces separately do not disturb the splitting one\nsees best for free particles.\n\n\n--\n\nRoland Franzius\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>Daryl McCullough wrote:

> I posted this to sci.math, but I think it got a little lost in the noise.
>
> I stumbled across a family of conserved quantities in classical Lagrangian
> mechanics, and I'm wondering whether these are well-known, and whether there is
> a way to interpret them as Noether conserved charges that follow from symmetries
> of the Lagrangian.
>
> Let the Lagrangian have the following form:
>
> L = A(x,v_x) + 1/2 B(x) v_y^2 - C(y)/B(x)
>
> (where v_q is the velocity for coordinate q). Then the quantity Q defined by
>
> Q = 1/2 (B(x) v_y)^2 + C(y)
>
> is conserved. To see this, note that the Lagrangian equations of motion for
> v_y is
>
> d/dt(B(x) v_y) = - C'(y)/B(x)
>
> Multiplying both sides by B(x) v_y gives
>
> B(x) v_y d/dt(B(x) v_y) = - v_y C'(y)
>
> (where C'(y) is dC/dy)
>
> The left-hand side is equal to
>
> d/dt(1/2 (B(x) v_y)^2)
>
> The right-hand side is equal to
>
> - d/dt(C(y))
>
> So the combination
>
> Q = 1/2 (B(x) v_y)^2 + C(y)
>
> satisfies
>
> dQ/dt =
>
> Q is conserved, but it is not obviously a Noetherian conserved charge
> due to a symmetry of the Lagrangian.

In many situations one has a decompositon of the Hamiltonian into a net
of partial conserved quadratic momenta, sometimes with additonal
potential of product type. Eg in spherical coordinates the Hamiltonian
for free particles or with specialy designed potentials may be
decomposed according to

H = H_r + f(r)( H_{theta}) + g(\theta) H_{phi}))

Then obviously H_{phi} and ( H_{theta}) + g(\theta) H_{phi})) are conserved
because they commute with H.

In your case call B(x) a radius r and y an angle \phi. You found a split
of the total energy into a radial part and an independent conserved
angular part of the form

E = A(r,\dot r) + 1/2 r^2 v\phi + V(\phi)

in analogy to the angular momentum splitting with V=0

E = 1/2 vr^2 + 1/2 r^2 v\phi^2

Write down the Hamiltonian with vr = a(r,p_r) the inverse function of
p_r = dA(r,vr)/v_r

H = p_r a(r,p_r) + 1/B ( 1/2 p_{phi}^2 + 1/B V(\phi) )

and you see without calculation that the angular energy is conserved.

The feature of the existence of a such a decomposition is a consequence
of the existence of a commuting set of conserved quantities which are
quadratic in canonical momenta and depend on a product potential
specialy fitted with the metric factors not to disturb the separating
process. In quantum mechanics this leads to a decompositon of the
Hilbert space into series of tensor products of Hilbert spaces of one
variable.

In this situation the representation of the symmetry group is governed
by a series of casimir operators. Additional potentials working in the
one variable Hilbert spaces separately do not disturb the splitting one
sees best for free particles.

--

Roland Franzius

Boris Bralo

Apr8-04, 05:50 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>\n\nHi\n\n>\n>\n> I posted this to sci.math, but I think it got a little lost in the noise.\n>\n> I stumbled across a family of conserved quantities in classical Lagrangian\n> mechanics, and I\'m wondering whether these are well-known, and whether\n> there is a way to interpret them as Noether conserved charges that follow\n> from symmetries of the Lagrangian.\n\nThere is related quantity (maps into lagrangian symetry) in general\nrelativity. Kerr-Newman solution of the Einstein\'s equations has symetry\n(Killing field in GR lingua ) commonly denoted "a" - it\'s related to\norbital momentum.\n\n--\nBoris\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>Hi

>
>
> I posted this to sci.math, but I think it got a little lost in the noise.
>
> I stumbled across a family of conserved quantities in classical Lagrangian
> mechanics, and I'm wondering whether these are well-known, and whether
> there is a way to interpret them as Noether conserved charges that follow
> from symmetries of the Lagrangian.

There is related quantity (maps into lagrangian symetry) in general
relativity. Kerr-Newman solution of the Einstein's equations has symetry
(Killing field in GR lingua ) commonly denoted "a" - it's related to
orbital momentum.

--
Boris

Charles Torre

Apr8-04, 06:34 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>daryl@atc-nycorp.com (Daryl McCullough) wrote:\n>\n> I stumbled across a family of conserved quantities in classical Lagrangian\n> mechanics, and I\'m wondering whether these are well-known, and whether there is\n> a way to interpret them as Noether conserved charges that follow from symmetries\n> of the Lagrangian.\n>\n> Let the Lagrangian have the following form:\n>\n> L = A(x,v_x) + 1/2 B(x) v_y^2 - C(y)/B(x)\n>\n> (where v_q is the velocity for coordinate q). Then the quantity Q defined by\n>\n> Q = 1/2 (B(x) v_y)^2 + C(y)\n>\n> is conserved.\n\nEvery differential conservation law for a system of\nEuler-Lagrange equations comes from a symmetry.\n\nIt\'s too early in the morning for reliable computations,\nbut consider the infinitesimal transformation:\n\ndx = 0, dy = Bv_y.\n\nHere "d" denotes the infinitesimal variation, usually\ndenoted "delta". It looks like this infinitesimal\ntransformation changes L by a total derivative (so it\'s a\n"divergence symmetry") with the corresponding conserved\ncharge being your Q.\n\ncharlie\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>daryl@atc-nycorp.com (Daryl McCullough) wrote:
>
> I stumbled across a family of conserved quantities in classical Lagrangian
> mechanics, and I'm wondering whether these are well-known, and whether there is
> a way to interpret them as Noether conserved charges that follow from symmetries
> of the Lagrangian.
>
> Let the Lagrangian have the following form:
>
> L = A(x,v_x) + 1/2 B(x) v_y^2 - C(y)/B(x)
>
> (where v_q is the velocity for coordinate q). Then the quantity Q defined by
>
> Q = 1/2 (B(x) v_y)^2 + C(y)
>
> is conserved.

Every differential conservation law for a system of
Euler-Lagrange equations comes from a symmetry.

It's too early in the morning for reliable computations,
but consider the infinitesimal transformation:

dx = 0, dy = Bv_y.

Here "d" denotes the infinitesimal variation, usually
denoted "\delta". It looks like this infinitesimal
transformation changes L by a total derivative (so it's a
"divergence symmetry") with the corresponding conserved
charge being your Q.

charlie