Re: Tangent bundle geometry and Lagrangian dynamics

On Jun 28, 11:15 pm, markw...@xxxxxxxxx wrote:
On Jun 14, 10:27 am, senderi...@xxxxxxxxx wrote:

Well, I've been away from math and physics for the last 8 years or so
(I'm just a humble cubicle slave churning out C++), but I recently
picked up _The Road To Reality_ and the chapter on Lagrangian and
Hamiltonian mechanics contained a statement that got me thinking, to
the effect that Lagrangians are defined on the tangent bundle of the
classical configuration space, and Hamiltonians are defined on its
cotangent bundle. Now, I was familiar with the basics of Hamiltonian
geometry (the god-given one-form on any cotangent bundle, the god-
given symplectic form which is its exterior derivative, how to define
the Hamiltonian vector field and Poisson brackets using the symplectic
form, etc.), but I couldn't think of any analogous "god-given"
geometric structures on the tangent bundle.

When you're talking about the Lagrangian formulation, the most natural
way to approach this was the development carried out in Lecture Notes
in Physics 107.

A field resides on a base space M, locally coordinatized by
(x^1,...,x^n). For statics n = 3, for mechanics n = 1, for dynamics n
= 4. This is actually a completely different point of view from that
taken in Physics and resides across the divide in the Mathematics
community. Whereas in Physics, fields are cajoled into being
mechanics, in the Mathematicians' treatment, mechanics is cajoled into
being field theory. The advantage of the approach is that it lies
beneath the causal structure of the spacetime. It doesn't care how (or
even whether) space divides into space + time. This feature is shared
by the covariant Hamiltonian formulation which arises out of this
general way of doing things (sometimes called the polysymplectic
approach). The other major advantage is that configuration space is
FINITE dimensional.

The Poisson brackets don't survive intact. This needs to be repaired
and generalized since it is only specific to n = 1. The generalized
structures (still an active area of research) come out of symplectic
structures like the Cartan form or the Froelicher-Nijenhuis bracket.
The latter simultaneously generalizes all the constructions seen in
gauge theory (Bianchi identities, field strength, torsion, generalized
Ricci tensor, etc.)

The field, itself, has a finite number of components (q^1,...,q^g),
and so resides in a bundle Y locally coordinatized by (x,q). A field,
itself, is a local map q: (x) -> (x,q(x)), or section.

For Lagrangians, you want fields and velocities, so the velocity
components (v^a_m: a=1,...,g; m=1,...,n) reside in the jet bundle J1Q.
This is a generalization of the notion of tangent spaces suitable to
accommodating the idea of field derivatives. Had you been in mechanics
with n = 1, the jet bundle would just be the vertical tangent space
comprising the derivatives of the q's.

So, then the field q: (x) -> (x,q(x)) is associated with its first
derivative (x,q(x),dq(x)/dx), where the velocity components are
identified as the partial derivatives v^a_m = dq^a/dx^m. This is how
you formalize the kinetic law (v = dq/dx). The corresponding extension
of q is called the jet extension j1q: (x) -> (x,q(x),dq/dx).

This is what the Lagrangian is a function of: L = L(j1q). The
Lagrangian, itself, is a volume form over the base space M, so
integrates to an action.

This whole edifice allows you to come up with a more intuitive parsing
of the Euler-Lagrange equations. The momentum components p^m_a = dL/
dv^a_m and source (or "force") components f_a = dL/dq^a are the
coefficients of the total differential:
dL = p^m_a dv^a_m + f_a dq^a.
The relations may be thought of as the *constitutive relations* for
the field theory. For instance, in the most generalized Lagrangian
Maxwell theory where there's no relations between (D,H) <-> (E,B)
posed at the outset, you start with the q's being the potentials (A,
phi); the v's being the velocities (E, B); and the total Lagrangian
variation given by dL = D.dE - H.dB (+ J.dA - rho dphi). The
constitutive laws are just D = dL/dE, H = -dL/dB. For Lorentz
invariant theories, they'd take on the form
D = epsilon E + lambda B; H = (1/mu) B - lambda E; (epsilon mu = (1/
c)^2)
with a total variation given by
dL = epsilon dI + lambda dJ
where
I = (E^2 - (Bc)^2)/2, J = (E.B)
and L = L(I,J).

The Euler-Lagrange equation is just the law of dynamics, dp^m_a/dx^m =
f_a.

There is no Poisson bracket structure, per se, in here. Instead, you
would work backwards applying the field law and kinetic law to the
total Lagrangian variation:
dL = p^m_a dv^a_m + f_a dq^a = p^m_a d(dq^a/dx^m) + (dp^m_a/dx^m)
dq^a = d/dx^m (p^m_a dq^a).
This highlights the n-1 form
p^m_a dq^a ^ *(dx_m)
is the natural structure to consider.

Thanks, I can't find "Lecture Notes in Physics 107" (aka "A symplectic
framework for field theories") at any bookseller, but I might be able
to check out a copy from a university library (or photocopy what I
need).

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?).

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"!).

.

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