Making sense of generalized coordinates
dfan
completely swamped by all the variables and functions (the fact that
they came in latin and greek, regular and italic, and lowercase and
uppercase, with a random glyph thrown in for good measure) and it took
me a long time to get an intuitive sense of what was really being said.
It wasn't until I made the following picture that I really grasped
what was going on, so here it is for the benefit of people reading SICM
for the first time (you'll have to read this in a monospaced font for
it to look right):
CONFIGURATION SPACE
(coordinate-free)
+------+ +-------+
------->|config| -------> | all | -------
/ \gamma | | \T | derivs| \L \
/ +------+ +-------+ \
/ | | \ integrate
t | \chi | \chart(\chi) >-----------> S
(action)
\ | | /
\ V v /
\ +------+ +-------+ /
------->|vector| -------> | all | -------
q | repr.| \Gamma | derivs| L(\chi)
+------+ +-------+
COORDINATE SPACE
(specific representation)
The real points are:
1) The action of a path of a system is completely coordinate-free; God
can move along the upper pathway of the diagram all the way from time
to action without ever choosing how to represent the state of a system
or performing any computation.
2) (Corollary) You're completely free to choose any representation (the
\chi function here) you want when deciding how to construct the lower
pathway of the diagram, and it won't make any difference to the action.
So you're free to choose the one that makes your computation most
convenient.
I'm not an actual physicist, just a self-studier, so if I'm misleading
the new folks, corrections and clarifications are welcome.
Dan
Tobin Fricke
> The first time I read section 1.3 on generalized coordinates I was
> completely swamped by all the variables and functions (the fact that
functions and their domains and ranges. I was confused by the place of
gamma: my understanding is that it's a function from time to configuration
space, where the point in configuration space is given in a completely
abstract way (i.e. an element of the set of configuration points)? By
contrast, q is a mapping from t to coordinate space, say, R^n? If this is
the case, what does it mean to take derivatives of gamma, as in the local
tuple?
Tobin
Jose A. Ortega Ruiz
> Nice diagram. I was working on a sort of "glossary" myself of all the
> functions and their domains and ranges. I was confused by the place of
> gamma: my understanding is that it's a function from time to configuration
> space, where the point in configuration space is given in a completely
> abstract way (i.e. an element of the set of configuration points)? By
> contrast, q is a mapping from t to coordinate space, say, R^n? If this is
> the case, what does it mean to take derivatives of gamma, as in the local
> tuple?
>
http://mitpress.mit.edu/SICM/book-Z-H-10.html#footnote_Temp_34.
The 'derivative' of gamma, \D\gamma is defined as an operator which
maps real-valued functions on the configuration space to their
derivatives along \gamma in a point \gamma(t) (again a real-valued
function from the configuration space to the real numbers). (In the
languange of differential manifols, \D\gamma is the tangent vector to
the curve.) That is, if f is a function from the configuration space
(M) to the reals (|R):
f: M ---> |R
p ---> f(p) \in |R
then \D\gamma(f) is again a function from M to |R whose value at the
point p = \gamma(t0) is defined to be d/dt (f(\gamma(t))) computed at
t0.
Df: M ---> |R
p = \gamma(t0) ---> d/dt (f(\gamma(t))) (t = t0)
In other words, when you have a real-valued function f defined over the
configuration space, you can think of it of a function from t to the
reals using gamma. Then you define the derivative of f along a curve
\gamma by first transforming it into a function from t to the reals
(using \gamma) and then deriving it with respect to t (i.e., along the
curve---note that now you have a function from |R to |R, so
derivatives are well defined). The operator that converts functions of
coordinates to their derivatives along a path \gamma in such a way is
\D\gamma. Each curve (along with its parametrization) defines a
different \D\gamma.
Any modern book on differential geometry will explain much better than
me these issues at the very beginning, when defining vectors.
Hope this helps,
jao
--
I shall be telling this with a sigh / Somewhere ages and ages hence: / Two
roads diverged in a wood, and I -- / I took the one less traveled by, / And
that has made all the difference. -Robert Frost, poet (1874-1963)