Re: transformations
From: James Dolan (jdolan_at_math-cl-n03.math.ucr.edu)
Date: 09/01/04
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Date: Wed, 1 Sep 2004 08:57:17 +0000 (UTC)
in article <41357a71@news.uni-ulm.de>,
ivan pongrac <ivan.pongrac@student.uni-ulm.de> wrote:
|James Dolan <jdolan@math-cl-n03.math.ucr.edu> wrote:
|
|> tnorx <tnorx@rediffmail.com> wrote:
|>
|> |1. canonical transformation; very general coordinate
|> |transformations, leaving an equation form invariant regarding its
|> |variables?
|>
|> roughly, these are linear transformations represented by matrixes
|> whose scalar entries are taken from the rig b of "costs" with
|> "addition" being minimization of costs and "multiplication" being
|> addition of costs. (roughly, a "rig" is a domain of quantities
|> within which there's a good concept of "addition" and a compatibly
|> good concept of "multiplication", allowing you to carry out all the
|> basic processes of linear algebra such as matrix multiplication.)
|> these matrixes are called "generating functions of canonical
|> transformations" in goldstein's classical mechanics book, for
|> example. because addition in the rig b is the operation of
|> cost-minimization, b-linearity is a sort of embodiment of the
|> optimization principles of physics such as "the principle of least
|> action".
|
|I must admit I don't know much, alas, about abstract algebra (had to
|look up that "rig" = semi-ring but have problems to imagine what you
|mean with "costs").
it shouldn't be so hard to imagine. the principle of least action
(and related ideas) started with vague philosophical musings about the
universe being run as though by a stingy god who (in some sense)
always uses the most economical means of getting the universe from
configuration x to configuration y over a given time interval.
we can take the rig b to be the set (-infinity,+infinity]; that is,
the ordinary real numbers together with positive infinity. negative
costs mean that god makes a profit on the deal; positive infinity
means that even god can't afford it.
|So all canonical transformations are based on matrices, which
|represent linear transformations, and canonical transformations are a
|subset of linear transformations?
very roughly, yes. there's a huge amount of fine print involved,
though, in part because the approach that i'm describing _isn't_ the
way that the subject is usually formalized. there's a long tradition
of (misleadingly) denying that the principle of least action really
works, and claiming that instead you should use the "principle of
stationary action". if you follow that approach then the subject
isn't strictly a kind of linear algebra anymore, but it's almost
impossible to understand unless you're aware of the secret analogy
with linear algebra.
if you'd like to learn more about the approach that i'm talking about,
then you should probably start by calculating an example of
"multiplying together a pair of matrixes". for example, take the
first matrix to be the function f(x,y)=x^2-xy+2y^2 and the second one
to be the function g(y,z)=3y^2+2yz+z^2, and then calculate their
product matrix h(x,z). in other words, you interpret f(x,y) as "the
cost of getting from x to y over time interval t1" and g(y,z) as "the
cost of getting from y to z over time interval t2", and then calculate
h(x,z) as "the minimum over all possible intermediate stopover points
y of the cost of getting from x to y over time interval t1 and then
from y to z over time interval t2". you should find that your
calculation can be interpreted as a matrix multiplication calculation,
and that it mostly agrees with the concept of "composition of
generating functions of canonical transformations" (or whatever it's
called) in books such as goldstein's classical mechanics book (even
though those books mostly hide the fact that a minimization problem is
being solved, by disguising the solution process as an unmotivated
exercise in differential calculus).
|> |1. Legendre transformation; transition from Lagrange- to
|> |Hamilton-Mechanics; is this the same or a subset of a contact
|> |transformation? (I assume a subset.)
|>
|> roughly, this is a very special canonical transformation, playing a
|> similar role with respect to the rig of costs as laplace transform
|> and/or fourier transform play with respect to the rigs of real
|> and/or complex numbers.
|
|So Laplace and Fourier transformations are just very special
|transformations with respect to some very special rigs?
yes, the laplace and fourier transforms are certain very special and
important linear transformations on certain function spaces.
|> |2. point transformations? difference to contact transformations?
|>
|> you can think of these as the very special linear transformations
|> that map basis vectors to basis vectors. this property is so
|> special that it pretty much doesn't matter what rig you think of
|> the transformations as being linear with respect to.
|>
|> |3. contact transformation; introduced by Sophus Lie to
|> |geometrically analyze differential equations, leading to contact
|> |geometry; what is the relationship between contact geometry and
|> |symplectic geometry?
|>
|> roughly, these are linear transformations represented by matrixes
|> whose scalar entries are taken from the two-element sub-rig of b
|> {zero cost, infinite cost}. the relationship between contact
|> geometry and symplectic geometry is an example of the relationship
|> between linear algebra over two different rigs, one a sub-rig of
|> the other. (there are some extra twists to the relationship in
|> this case, though.)
|
|So therefore contact and symplectic geometry are called "twins". If
|symplectic geometry is seen as being the mathematical basis for
|classical mechanics, so I assume that also contact geometry has some
|very general applications in physics.
well, there's probably a lot to say about this, but one thing to say
is that when you have a rig and a sub-rig of it, then linear
transformations over the big rig can be interpreted as a special case
of linear transformations over the sub-rig. for example, the real
numbers form a sub-rig of the rig of complex numbers, and thus
complex-linear transformations can be interpreted as a special case of
real-linear trandformations. similarly, {zero cost, infinite cost}
forms a sub-rig of b, and thus canonical transformations can be
interpreted as a special case of contact transformations.
|> if these aren't the answers you were expecting, it might be because
|> there are some very simple and powerful approaches to these topics
|> that aren't commonly taught.
|
|Ah well... Can all transformations, in a most general way, be seen
|as only an application of geometry?
i'm not sure i understand this question. also it sounds so general
that even if i did understand it i don't know if i could give a useful
answer.
|And Geometry, as Galilei, Hilbert and others have pointed out, is one
|of the (maybe even "the") most fundamental concepts of nature.
-- [e-mail address jdolan@math.ucr.edu]
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