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Abbreviations
- [0,L]:
-
represents the reference configuration of the string
- (z(s), u(s)) :
-
is the location in space of the point parametrized bys. (For notational convenience thex 1−x 2-plane is represented byℂ)
- m(s):
-
is the mass distribution on the string.m(s) is the mass of the string contained in the interval [0,s]. In the most general case it will be a function a) of bounded total variation b) strictly increasing
- N :
-
is the tension in the string. It is a given function with the following properties: a)N:(0, ∞)→— b) strictly increasing and onto. (Further restrictions will be imposed when they are needed)
- ℜ:
-
is the inverse ofN. It has the following properties: a) ℜ:ℝ→(0,∞) b) strictly increasing and onto
- \(\mathfrak{M}\) :
-
is given by\(\mathfrak{M}(t): = \frac{{\mathfrak{N}(t)}}{t}.\mathfrak{M}\) has a singularity att=0
- ω:
-
is the angular velocity of the rotating frame
- β:
-
is given by β(s, c):=β=m(s)−c
- \(A,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{A} \) :
-
is given by\(\begin{gathered} A(k,s): = A = \sqrt {|k|^2 + \beta ^2 } \hfill \\ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{A} (k,\mu ,s): = \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{A} = \sqrt {\frac{{|k|^2 }}{{\mu ^2 }} + \beta ^2 } \hfill \\ \end{gathered} \)
- I :
-
will be a measureable subset of [0,L]
- X :
-
is given by\(\chi {\text{(}}s{\text{): = }}\left\{ {\begin{array}{*{20}c} {{\text{ + 1, }}s \in I,} \\ { - {\text{1, }}s \in I^c } \\ \end{array} } \right.\)
- ϱ1 :
-
(ϱ1,O) is the position ofP 0
- ϱ2 :
-
ϱ2 e iφ is the position ofP 1
- ϱ1 :
-
is given by ϱ0:=−ϱ2 e iφ
- g :
-
the constant of gravity. By incorporating it intom we can treatg as being one
- ζ:
-
\(\xi (s): = \sqrt {|z'(s)|^2 + u'(s)^2 } \) measures the stretching of the string ats
- X :
-
X:=C 1([0,L])×ℝ2
- Y :
-
Y:=ℝ6
- w :
-
\(w: = \gamma - \int\limits_0^s {\omega ^2 zdm(\sigma )} \), new dependent variable replacingz
- ∑:
-
∑∶={(x, y)εX×Y|A(w, s) has a zero on [0,L]} ≡{(x, y)εX×Y|cε[0,L] andw(c)=0} the singular set, where differentiability of the functional maps breaks down
References
Alexander, J.C., Antman, S.S.: Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems. Arch. Rational Mech. Anal.76, 339–354 (1981)
Alexander, J.C., Antman, S.S., Deng, S.T.: Nonlinear eigenvalue problems for the whirling of heavy elastic strings. II. New methods in global bifurcation theory. Proc. Roy. Soc. Edinburgh
Alexander, J.C., Reeken, M.: On the topological structure of the set of generalized solutions of the catenary problem. Proc. Roy. Soc. Edinburgh
Reeken, M.: Exotic equilibrium states of the elastic string. Proc. Roy. Soc. Edinburgh
Reeken, M.: Rotating chain fixed at two points vertically above each other. Rocky Mountain, J. Math.10, (1980)
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Reeken, M. The rotating string. Math. Ann. 268, 59–84 (1984). https://doi.org/10.1007/BF01463873
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DOI: https://doi.org/10.1007/BF01463873