Geometric conditions for the positive definiteness of the second variation in one-dimensional problems
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...
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| Format: | Journal article |
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IOP Publishing
2017
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| _version_ | 1826287117564641280 |
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| author | Lessinnes, T Goriely, A |
| author_facet | Lessinnes, T Goriely, A |
| author_sort | Lessinnes, T |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T01:53:49Z |
| format | Journal article |
| id | oxford-uuid:9b0afa51-a5ab-4849-9622-f5149c29c829 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T01:53:49Z |
| publishDate | 2017 |
| publisher | IOP Publishing |
| record_format | dspace |
| spelling | oxford-uuid:9b0afa51-a5ab-4849-9622-f5149c29c8292022-03-27T00:25:47ZGeometric conditions for the positive definiteness of the second variation in one-dimensional problemsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:9b0afa51-a5ab-4849-9622-f5149c29c829Symplectic Elements at OxfordIOP Publishing2017Lessinnes, TGoriely, AGiven 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. |
| spellingShingle | Lessinnes, T Goriely, A Geometric conditions for the positive definiteness of the second variation in one-dimensional problems |
| title | Geometric conditions for the positive definiteness of the second variation in one-dimensional problems |
| title_full | Geometric conditions for the positive definiteness of the second variation in one-dimensional problems |
| title_fullStr | Geometric conditions for the positive definiteness of the second variation in one-dimensional problems |
| title_full_unstemmed | Geometric conditions for the positive definiteness of the second variation in one-dimensional problems |
| title_short | Geometric conditions for the positive definiteness of the second variation in one-dimensional problems |
| title_sort | geometric conditions for the positive definiteness of the second variation in one dimensional problems |
| work_keys_str_mv | AT lessinnest geometricconditionsforthepositivedefinitenessofthesecondvariationinonedimensionalproblems AT gorielya geometricconditionsforthepositivedefinitenessofthesecondvariationinonedimensionalproblems |