{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:28:05Z","timestamp":1787232485831,"version":"build-2736575974"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler--Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values $\\sigma \\le 1$ at which a second ODE (the Jacobi equation) has a solution vanishing at 0 and $\\sigma$.<\/jats:p>\n                  <jats:p>Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter $\\lambda$, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of $\\lambda$ of interest.<\/jats:p>\n                  <jats:p>Here we present an alternative approach that takes advantage of the presence of a parameter $\\lambda$. Rather than search for conjugate points $\\sigma \\le 1$ at a fixed value of $\\lambda$, we search for a set of special parameter values $\\lambda_m$ (with corresponding Jacobi solution $\\bfzeta^m$) for which $\\sigma=1$ is a conjugate point. We show that, under appropriate assumptions, the index of an equilibrium at any $\\lambda$ equals the number of these $\\bfzeta^m$ for which $\\langle \\bfzeta^m, \\Op \\bfzeta^m \\rangle &lt; 0$, where $\\Op$ is the Jacobi differential operator at $\\lambda$. This computation is particularly simple when $\\lambda$ appears linearly in $\\Op$.<\/jats:p>\n                  <jats:p>We apply this approach to the elastic strut, in which the force appears linearly in $\\Op$, and, as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute two-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE.<\/jats:p>","DOI":"10.1137\/s1111111101396622","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"115-145","source":"Crossref","is-referenced-by-count":17,"title":["Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut"],"prefix":"10.1137","volume":"1","author":[{"given":"Kathleen A.","family":"Hoffman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Robert S.","family":"Manning","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Randy C.","family":"Paffenroth","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,8,7]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4147-6"},{"key":"R2","unstructured":"O. Bolza,\n                      Lectures on the Calculus of Variations\n                      , 2nd ed., Chelsea, New York, 1961."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1996.0132"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/BF00379680"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000397"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127491000555"},{"key":"R7","volume-title":"Calculus of variations with applications","author":"Ewing George","year":"1969"},{"key":"R8","volume-title":"Foundations of modern analysis","author":"Friedman Avner","year":"1970"},{"key":"R9","volume-title":"Calculus of variations","author":"Gelfand I.","year":"1963"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/s003320010009"},{"key":"R11","unstructured":"A. G. Greenhill, Proc. Inst. Mech. Eng. (1883), p. 182."},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-2918-3"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0983-6"},{"key":"R14","volume-title":"Calculus of variations and optimal control theory","author":"Hestenes Magnus","year":"1966"},{"key":"R15","unstructured":"H. B. Keller,\n                      Numerical solution of bifurcation and nonlinear eigenvalue problems\n                      , in Applications of Bifurcation Theory, P. H. Rabinowitz, ed., Academic Press, New York, 1977, pp. 359\u2013384."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1859.56.285"},{"key":"R17","volume-title":"A treatise on the Mathematical Theory of Elasticity","author":"Love A.","year":"1944"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127497001333"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1998.0291"},{"key":"R20","volume-title":"Introduction to analysis in the large","author":"Morse Marston","year":"1951"},{"key":"R21","unstructured":"S. Neukirch and M. E. Henderson,\n                      Classification of the Spatial Clamped Elastica\n                      , I\n                      and\n                      II, preprint, available online from http:\/\/lcvmsun9.epfl.ch\/\u02dcneukirch\/publi.html."},{"key":"R22","unstructured":"R. C. Paffenroth,\n                      Mathematical Visualization, Parameter Continuation, and Steered Computations\n                      , Ph.D. thesis, University of Maryland, College Park, MD, 1999."},{"key":"R23","volume-title":"An introduction to partial differential equations","author":"Renardy Michael","year":"1993"},{"key":"R24","volume-title":"Introduction to the calculus of variations","author":"Sagan Hans","year":"1992"},{"key":"R25","first-page":"439","volume":"41","author":"Shuster Malcolm","year":"1993","journal-title":"J. Astronaut. Sci."},{"key":"R26","unstructured":"G. H. M. van der Heijden, S. Neukirch, V. G. A. Goss, and J. M. T. Thompson,\n                      Instability and Self\u2010Contact Phenomena in the Writhing of Clamped Rods\n                      , preprint, available online from http:\/\/lcvmsun9.epfl.ch\/\u02dcneukirch\/publi.htm."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(97)00211-X"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1111111101396622","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:58:35Z","timestamp":1787230715000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1111111101396622"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1]]},"references-count":27,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2002,1]]}},"alternative-id":["10.1137\/S1111111101396622"],"URL":"https:\/\/doi.org\/10.1137\/s1111111101396622","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1]]}}}