Journal article
Geometric conditions for the positive definiteness of the second variation in one-dimensional problems
- Abstract:
- Given a functional for a one-dimensional physical system, a classical problem is to minimize it by finding stationary solutions and then checking the positive definiteness of the second variation. Establishing the positive definiteness is, in general, analytically untractable. However, we show here that a global geometric analysis of the phase-plane trajectories associated with the stationary solutions leads to generic conditions for minimality. These results provide a straightforward and direct proof of positive definiteness, or lack thereof, in many important cases. In particular, when applied to mechanical systems, the stability or instability of entire classes of solutions can be obtained effortlessly from their geometry in phase-plane, as illustrated on a problem of a mass hanging from an elastic rod with intrinsic curvature.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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- Files:
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(Preview, Accepted manuscript, pdf, 3.8MB, Terms of use)
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- Publisher copy:
- 10.1088/1361-6544/aa6448
Authors
- Publisher:
- IOP Publishing
- Journal:
- Nonlinearity More from this journal
- Volume:
- 30
- Issue:
- 5
- Pages:
- 2023–2062
- Publication date:
- 2017-04-01
- Acceptance date:
- 2017-03-02
- DOI:
- EISSN:
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1361-6544
- ISSN:
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0951-7715
Terms of use
- Copyright holder:
- IOP Publishing Ltd and London Mathematical Society
- Copyright date:
- 2017
- Notes:
- © 2017 IOP Publishing Ltd and London Mathematical Society. This is the accepted manuscript version of the article. The final version is available online from the Institute of Physics at: http://dx.doi.org/10.1088/1361-6544/aa6448
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