Talk:Curve (physics)
I didn't know physicists had a different definition of "curve" than mathematicians. I'm not exactly sure how the ordering jives with the topology. Are you saying the curve is a totally ordered set with the induced topology (induced from the total order)? What is this "betweenness" relation, how is it definied? The definitions of concepts like "simple closed curve" seem totally different than what I take that to mean in math. Is a circle parametrised once around starting where it ends a simple closed curve? It doesn't seem so, because the endpoint (starting and endpoint) isn't "between" two other non-endpoints, and a circle parametrised once around can't have an ordering on it that induces the topology (the starting/endpoint would be less than or equal to some other point, which is less than or equal to itself, so by antisymmetry of the total order, the curve would reduce to a single point). I have a lot of trouble following this article.
As for: I didn't know physicists had a different definition of "curve" than mathematicians.
- The definition by which physicists decide whether a given set constitutes a "(simple) curve" differs necessarily from that of mathematicians since the latter requires a particular topology to be given initially with the set under consideration, i.e. an assignment which of its subsets were to be called "open" (namely, in order to evaluate whether an interval of real numbers can be continuously and injectively mapped to the given set), while the sets that are typically considered in physics are without any initial assignment of a particular topology. Instead, there one establishes for instance the order (vs. simultaneity) of the given elements, and if applicable their durations and/or distances pairwise to each other. However,
As for: The definitions of concepts like "simple closed curve" seem totally different than what I take that to mean in math.
- The topology which is induced by the order and distance relations between the elements of a given set, as described (presently) in the article to be talked about, and which establishes the given set as aparticular topological space, allows for instance an interval of real numbers to be mapped continuously and injectively to this topological space. It thus identifies the set under consideration as a "simple curve" in the sense described (presently) in the article defining "curve" in Mathematics/Topology as well.
As for: Are you saying the curve is a totally ordered set with the induced topology (induced from the total order)?
- IIUUC, no: total order of the given set (as required for a set to constitute a "(simple) curve (in physics)") and the thus induced topology is itself not sufficient, the described distance relations are essential as well. For instance:
- Consider a totally ordered set with an element G, for which there exist elements A and Z such that G is between A and Z (i.e. wrt. the given total order), and for which there exist elements between A and G as well as between G and Z. For all pairs of elements of the set under consideration let the distance values be different from zero.
- If there exists an element K, between G and Z, such that the distance between G and K is less than the distance between G and each element between A and G, then the set under consideration is not a "(simple) curve (in physics)"; instead it is said to have a "gap", "between A and G", "next to G".
As for: What is this "betweenness" relation, how is it definied?
- One important operation in physics by which to attempt ordering a given set (possibly totally) and thus possibly to establish a "betweenness" relation of its elements, is through measuring and comparing the pairwise durations of its elements to each other (which incidentally does not necessaarily imply any particular distance relations of these elements to each other).
As for: Is a circle parametrised once around starting where it ends a simple closed curve? It doesn't seem so, because [...]
- A circle constitutes a Curve (physics) regardless of any incidental parametrization. To order its elements (by their distance relations alone, provided they were all found to be pairwise simultaneous), one may select any one particular element as "start" and order all elements without ending.
- However, for a given "start" element the thus obtained order is not unique: for any two elements P and Q which are distinct from the "start", by one "order sense" P is between "start" and Q, and the other puts Q between "start" and P. A circle is consequently identified as a "closed curve".
Best regards, Frank W ~@) R 19:33, 21 Sep 2003 (UTC).
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