Re: Why momentum is a cotangent vector?

In article <ep837j$pbk$1@xxxxxxxxxxxx>, sirix says...

Edward Green wrote:
Sorry :-) Phase space in Hamiltonian mechanics is a cotangent bundle. And
elements of fibers are called momentum vectors.

I have no special knowledge here, but in general, co-vectors are linear
maps on the vectors. So, simply identify what your "tangent" vectors
are in this context, and then read off why momentum would be a linear
map on them. Easy! :-)

This is actually my question: why momementum is a linear map on a space of
velocities?

It depends on how exactly you are defining momentum. One definition,
from Lagrangian mechanics, is this: The "Lagrangian" L(x,v,t) is a
real-valued function of position x velocity v and time t.
(I'm using x as shorthand for (x^1, x^2, x^3) and v as shorthand
for (v^1,v^2,v^3)). In the simplest case of free particles in
Newtonian physics, L is just 1/2 m |v|^2. So when we hold x and t
fixed, L is a function from velocities to reals. But it's *not* a
linear function, however. To get a linear function out of it, we
take a limiting procedure: Let w be an arbitrary velocity vector
(not necessarily equal to v) and compute

F(x,v,t,w)
= limit as s --> 0 of 1/s (L(x,v + sw,t) - L(x,v,t))

In the limit as s --> 0, this becomes a linear function of w
(it depends on x,v, and t, as well, but it need not depend
linearly on these parameters). Every linear function of vectors
can be written as a covector. So we can write

F(x,v,w,t) = p_j(x,v,t) w^j

We call the covector p_j the "canonical momentum" corresponding
to the Lagrangian L. So momentum is a covector, a linear function
on vectors.

--
Daryl McCullough
Ithaca, NY

.

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