Re: Tangent bundle geometry and Lagrangian dynamics
- From: markwh04@xxxxxxxxx
- Date: Wed, 4 Jul 2007 11:55:41 +0000 (UTC)
On Jul 3, 11:45 am, senderi...@xxxxxxxxx wrote:
I definitely need to get comfy with jet bundles; hopefully wikipedia
will suffice. I am somewhat familiar with the Frohlicher-Nijenhuis
(and Nijenhuis-Richardson) bracket; I'd be very interested to see how
that fits into the story (what are the vector-valued differential
forms involved?).
A simple example is the gauge field! The potentials A^a_{mu} have an
upper index corresponding to a Lie algebra basis, and a lower space-
time index corresponding to a space-time one-form. So, the entity
written out fully is:
A = A^a_{mu} dx^{mu} (x) Y_a
where (Y_a) is the Lie algebra basis, and (x) denotes tensor product.
This is a vector-valued one-form. The "vector" is the upper index, and
the form is encapsulated by the lower indices.
The FN bracket applied to the form d + A gives you [d+A,d+A] = F,
where F is the (vector-valued) 2-form representing the gauge force.
I'd be curious to see how (and if!) the most mathematically
sophisticated working physicists are using some of these advanced
notions to do real research in theoretical physics (not "mathematical
physics"!)
This is one of those cases where a well-developed mathematics has not
yet fully broached the confines of the Physics community and where, on
account of that, a lot of subtleties (and resolutions to problems) are
being completely missed out on.
The covariant Hamiltonian approach is one of those big gaps, as a case
in point. There aren't even that many people in the Physics community,
I would venture, who are even aware that you can write down the
symplectic structure of field theory without having to break spacetime
into 3+1 or (for that matter) without even having to worry about
whether there even is a time-like direction in the first place! (Not
that you want to go to that extreme, but it underscores the point).
There's a big subtlety in the way the Poisson bracket would generalize
-- an ambiguity, in fact. When you work out the Lagrangian for
electromagnetism (keeping intact Maxwell's notation of B,D,E,H), you
find that the forms (D.dS - H.dr ^ dt), (E.dr^dt + B.dS) (using vector-
like notation with dr=(dx,dy,dz) and dS=(dy^dz,dz^dx,dx^dy)) appear.
The replacement of the Poisson bracket has to be an (n-1)-form over
base space, capturing the essence of the variational of the
Lagrangian:
delta(L) = D.delta(E) - H.delta(B).
The boundary term can be written out as an (n-1) form. This
generalizes the (p dq) form seen in symplectic mechanics. But it can
be written out as an (n-1) form in several ways. One way involves the
(D,H) 2-form above combined with the potential 1-form. Another way
makes uses 3--forms for (D,H) and 0-forms for the potentials. The
different parsing amount to different definitions of the (p's,q's)
and, potentially, different approaches to the field theory.
The one that Physicsts use, in effect, is the (3+0) decomposition.
However, the one that naturally stands out is the (2+1) decomposition.
The distinction is subtle since, in both cases, your q's involve the
potentials. But in one case, the configuration variables are the
actual components, while in the other case, it's the entire 1-form.
.
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- From: senderista
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