| Title:
|
On the notion of Jacobi fields in constrained calculus of variations (English) |
| Author:
|
Massa, Enrico |
| Author:
|
Pagani, Enrico |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 |
| Volume:
|
24 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
91-113 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In variational calculus, the minimality of a given functional under arbitrary deformations with fixed end-points is established through an analysis of the so called \emph {second variation}. In this paper, the argument is examined in the context of constrained variational calculus, assuming piecewise differentiable extremals, commonly referred to as \emph {extremaloids}. The approach relies on the existence of a fully covariant representation of the second variation of the action functional, based on a family of \emph {local} gauge transformations of the original Lagrangian and on a set of scalar attributes of the extremaloid, called the corners' \emph {strengths} \cite {mlp}. In discussing the positivity of the second variation, a relevant role is played by the \emph {Jacobi fields}, defined as infinitesimal generators of $1$-parameter groups of diffeomorphisms preserving the extremaloids. Along a piecewise differentiable extremal, these fields are generally discontinuous across the corners. A thorough analysis of this point is presented. An alternative characterization of the Jacobi fields as solutions of a suitable \emph {accessory variational problem} is established. (English) |
| Keyword:
|
constrained variational calculus; second variation; Jacobi fields. |
| MSC:
|
49J-- |
| MSC:
|
53B05 |
| MSC:
|
58F05 |
| MSC:
|
65K10 |
| MSC:
|
70D10 |
| MSC:
|
70Q05 |
| idZBL:
|
Zbl 1364.49022 |
| idMR:
|
MR3590208 |
| . |
| Date available:
|
2017-02-28T16:40:31Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146014 |
| . |
| Reference:
|
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