| Topic: |
Science > Physics |
| User: |
"Pavel Pokorny" |
| Date: |
17 Jul 2004 06:06:06 AM |
| Object: |
Jacobi identity in mechanics? |
Dear physics friends
Rossman in his book Lie groups writes:
The Jacobi identity derives its name
from Jacobi's investigation in mechanics (1836).
Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
where [a,b] = a b - b a
a,b,x,y,z are square matrices.
Can you, please tell me, where this can be used in 19th century mechanics?
Thank you
--
Pavel Pokorny
Math Dept, Prague Institute of Chemical Technology
http://www.vscht.cz/mat/Pavel.Pokorny
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| User: "John T Lowry" |
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| Title: Re: Jacobi identity in mechanics? |
17 Jul 2004 08:06:04 AM |
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"Pavel Pokorny" <Pavel.Pokorny@vscht.REMOVEME.cz> wrote in message
news:cdb16u$cp1$2@ns.felk.cvut.cz...
Dear physics friends
Rossman in his book Lie groups writes:
The Jacobi identity derives its name
from Jacobi's investigation in mechanics (1836).
Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
where [a,b] = a b - b a
a,b,x,y,z are square matrices.
Can you, please tell me, where this can be used in 19th century
mechanics?
Thank you
--
Pavel Pokorny
Math Dept, Prague Institute of Chemical Technology
http://www.vscht.cz/mat/Pavel.Pokorny
The Jacobi identity was (and is) used in the Hamilton-Jacobi formulation
of mechanics to come up with a new possibly useful constant of the
motion when two other such constants are already known. You'll find a
discussion in such mechanics texts as those by Goldstein or by Corben
and Stehle.
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| User: "" |
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| Title: Re: Jacobi identity in mechanics? |
17 Jul 2004 03:26:19 PM |
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In article <cdb16u$cp1$2@ns.felk.cvut.cz>, Pavel Pokorny <Pavel.Pokorny@vscht.REMOVEME.cz> writes:
Dear physics friends
Rossman in his book Lie groups writes:
The Jacobi identity derives its name
from Jacobi's investigation in mechanics (1836).
Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
where [a,b] = a b - b a
a,b,x,y,z are square matrices.
Tehy may be square matrices but they don't have to be. Any
mathematical entities for which addition and multiplication is defined
will do.
Can you, please tell me, where this can be used in 19th century mechanics?
Check out Poisson brackets and Canonical transformation. Landau's
"Mechanics" is a possible source.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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| User: "Edward Green" |
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| Title: Re: Jacobi identity in mechanics? |
18 Jul 2004 02:06:57 PM |
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wrote in message news:<LXfKc.45$45.10708@news.uchicago.edu>...
In article <cdb16u$cp1$2@ns.felk.cvut.cz>, Pavel Pokorny <Pavel.Pokorny@vscht.REMOVEME.cz> writes:
Dear physics friends
Rossman in his book Lie groups writes:
The Jacobi identity derives its name
from Jacobi's investigation in mechanics (1836).
Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
where [a,b] = a b - b a
a,b,x,y,z are square matrices.
Tehy may be square matrices but they don't have to be. Any
mathematical entities for which addition and multiplication is defined
will do.
Interesting. So the identity above is actually an observation that
every permutation of the symbols x,y and z occurs twice, once prefixed
by a minus sign: xyz for example occurs as +xyz in the first term,
-xyz in the last.
It's really a statement about commutators and collecting symbols,
rather than a property of the things inside the commutators.
Can you, please tell me, where this can be used in 19th century mechanics?
Check out Poisson brackets and Canonical transformation. Landau's
"Mechanics" is a possible source.
.
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| User: "" |
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| Title: Re: Jacobi identity in mechanics? |
18 Jul 2004 05:53:17 PM |
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In article <eca320d0.0407181106.31a6c7a8@posting.google.com>, (Edward Green) writes:
mmeron@cars3.uchicago.edu wrote in message news:<LXfKc.45$45.10708@news.uchicago.edu>...
In article <cdb16u$cp1$2@ns.felk.cvut.cz>, Pavel Pokorny <Pavel.Pokorny@vscht.REMOVEME.cz> writes:
Dear physics friends
Rossman in his book Lie groups writes:
The Jacobi identity derives its name
from Jacobi's investigation in mechanics (1836).
Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
where [a,b] = a b - b a
a,b,x,y,z are square matrices.
Tehy may be square matrices but they don't have to be. Any
mathematical entities for which addition and multiplication is defined
will do.
Interesting. So the identity above is actually an observation that
every permutation of the symbols x,y and z occurs twice, once prefixed
by a minus sign: xyz for example occurs as +xyz in the first term,
-xyz in the last.
Right on.
It's really a statement about commutators and collecting symbols,
rather than a property of the things inside the commutators.
Yes, exactly. It is perfectly general, quite mindless of what is
inside the commutators.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
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| User: "Edward Green" |
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| Title: Re: Jacobi identity in mechanics? |
19 Jul 2004 05:23:43 PM |
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wrote in message news:<xbDKc.6$25.1349@news.uchicago.edu>...
In article <eca320d0.0407181106.31a6c7a8@posting.google.com>, (Edward Green) writes:
wrote in message news:<LXfKc.45$45.10708@news.uchicago.edu>...
In article <cdb16u$cp1$2@ns.felk.cvut.cz>, Pavel Pokorny <Pavel.Pokorny@vscht.REMOVEME.cz> writes:
Dear physics friends
Rossman in his book Lie groups writes:
The Jacobi identity derives its name
from Jacobi's investigation in mechanics (1836).
Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
where [a,b] = a b - b a
a,b,x,y,z are square matrices.
Tehy may be square matrices but they don't have to be. Any
mathematical entities for which addition and multiplication is defined
will do.
Interesting. So the identity above is actually an observation that
every permutation of the symbols x,y and z occurs twice, once prefixed
by a minus sign: xyz for example occurs as +xyz in the first term,
-xyz in the last.
Right on.
It's really a statement about commutators and collecting symbols,
rather than a property of the things inside the commutators.
Yes, exactly. It is perfectly general, quite mindless of what is
inside the commutators.
Doing a bit of research I see that Lie algebras are typically cited as
primary examples of things-satisfying-the-Jacobi-identity, so the very
simple symbol pushing aspect of the thing is apparently not widely
appreciated! I mean, anything taking a Lie alegebra as its primary
example must be really abstract...
.
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| User: "" |
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| Title: Re: Jacobi identity in mechanics? |
19 Jul 2004 05:55:28 PM |
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In article <eca320d0.0407191423.68192b5d@posting.google.com>, (Edward Green) writes:
mmeron@cars3.uchicago.edu wrote in message news:<xbDKc.6$25.1349@news.uchicago.edu>...
In article <eca320d0.0407181106.31a6c7a8@posting.google.com>, (Edward Green) writes:
mmeron@cars3.uchicago.edu wrote in message news:<LXfKc.45$45.10708@news.uchicago.edu>...
In article <cdb16u$cp1$2@ns.felk.cvut.cz>, Pavel Pokorny <Pavel.Pokorny@vscht.REMOVEME.cz> writes:
Dear physics friends
Rossman in his book Lie groups writes:
The Jacobi identity derives its name
from Jacobi's investigation in mechanics (1836).
Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
where [a,b] = a b - b a
a,b,x,y,z are square matrices.
Tehy may be square matrices but they don't have to be. Any
mathematical entities for which addition and multiplication is defined
will do.
Interesting. So the identity above is actually an observation that
every permutation of the symbols x,y and z occurs twice, once prefixed
by a minus sign: xyz for example occurs as +xyz in the first term,
-xyz in the last.
Right on.
It's really a statement about commutators and collecting symbols,
rather than a property of the things inside the commutators.
Yes, exactly. It is perfectly general, quite mindless of what is
inside the commutators.
Doing a bit of research I see that Lie algebras are typically cited as
primary examples of things-satisfying-the-Jacobi-identity, so the very
simple symbol pushing aspect of the thing is apparently not widely
appreciated! I mean, anything taking a Lie alegebra as its primary
example must be really abstract...
:-))) Yes, I agree. But then, funny as it sounds, said "simple symbol
pushing aspect of the thing" is seldom, if ever, mentioned when the
Jacobi identity is introduced, so I wouldn't put it beyond the realm
of possibility that many "sophisticated users" never stopped to think
of it. Thus, they bring as primary example the case where they
encountered the identity first.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
.
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| User: "Bruce Scott TOK" |
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| Title: Re: Jacobi identity in mechanics? |
18 Jul 2004 10:22:41 AM |
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Mati Meron wrote:
|> In article <cdb16u$cp1$2@ns.felk.cvut.cz>, Pavel Pokorny <Pavel.Pokorny@vscht.REMOVEME.cz> writes:
|> > Dear physics friends
|> >
|> >Rossman in his book Lie groups writes:
|> >
|> > The Jacobi identity derives its name
|> > from Jacobi's investigation in mechanics (1836).
|> >
|> >Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
|> >where [a,b] = a b - b a
|> >a,b,x,y,z are square matrices.
|> >
|> They may be square matrices but they don't have to be. Any
|> mathematical entities for which addition and multiplication is defined
|> will do.
|>
|> >Can you, please tell me, where this can be used in 19th century mechanics?
|> >
|> Check out Poisson brackets and Canonical transformation. Landau's
|> "Mechanics" is a possible source.
Also Goldstein, Classical Mechanics, whose best topic is this one (and
all the Hamiltonian-Jacobi theory for rigid body motion and rotation).
--
cu,
Bruce
drift wave turbulence: http://www.rzg.mpg.de/~bds/
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| User: "Robert J. Kolker" |
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| Title: Re: Jacobi identity in mechanics? |
17 Jul 2004 08:12:18 AM |
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Pavel Pokorny wrote:
Dear physics friends
Rossman in his book Lie groups writes:
The Jacobi identity derives its name
from Jacobi's investigation in mechanics (1836).
Jacobi identity is [[x,y],z] + [[z,x],y] + [[y,z],x] = 0
where [a,b] = a b - b a
a,b,x,y,z are square matrices.
Can you, please tell me, where this can be used in 19th century mechanics?
See
http://farside.ph.utexas.edu/teaching/qm/fundamental/node22.html
Bob Kolker
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