Calculus of variations: 1 [The Lagrangian formalism]
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2004
|
| Edition: | Corr. 2. print. |
| Series: | Grundlehren der mathematischen Wissenschaften
310 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIX, 474 S. Ill., graph. Darst. |
| ISBN: | 354050625X |
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MARC
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| 245 | 1 | 0 | |a Calculus of variations |n 1 |p [The Lagrangian formalism] |c Mariano Giaquinta ; Stefan Hildebrandt |
| 250 | |a Corr. 2. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2004 | |
| 300 | |a XXIX, 474 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 310 | |
| 490 | 0 | |a Grundlehren der mathematischen Wissenschaften |v ... | |
| 650 | 7 | |a Calculo de variações |2 larpcal | |
| 650 | 4 | |a Calculus of variations | |
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| adam_text | Contents
The Lagrangian Formalism
Part I. The First Variation and Necessary Conditions
Chapter
1.
(Necessary conditions for local
variation.)
2.
2.1.
(Linear and nonlinear variations. Extremals and weak extremals.)
2.2.
Euler s Equations, and the Euler Operator LF
(ґ
Area functional, and linear combinations of area and volume. Lagrangians of
the type F{x, p) and F(u, p); conservation of energy. Minimal surfaces of
revolution: catenaries and
2.3.
(Properties of mollifiers. Smooth functions are dense in Lebesgue spaces L ,
1 <.
lemma.)
2.4.
(Dirichlet integral. Area functional. Neumann s boundary conditions.)
3.
3.1.
A Regularity Theorem
for One-Dimensional Variational Problems
(Euler s paradox. Lipschitz extremals. The integral form of Euler s equations:
DuBois-Reymonďs
3.2.
(Weierstrass s example. Surfaces of prescribed mean curvature. Capillary
surfaces. Obstacle problems.)
3.3.
(Weierstrass-Erdmann corner conditions. Inner variations. Conservation of
energy for Lipschitz minimizers.)
4.
4.1.
(Null Lagrangians and invariant integrals. Cauchy s integral
theorem.)
XVIII Contents
4.2.
(Structure of null Lagrangians. Exactly the Lagrangians of divergence form
are null Lagrangians. The divergence and the Jacobian of a vector field as
null Lagrangians.)
5.
(Euler equations. Equilibrium of thin plates. Gauss curvature. Gauss-Bonnet theorem.
Curvature integrals for planar curves. Rotation number of a planar curve. Euler s
area problem.)
6.
Chapter
1.
(The classical isoperimetric problem. The multiplier rule for isoperimetric problems.
Eigenvalues of the vibrating string and of the vibrating membrane. Hypersurfaces of
constant mean curvature. Catenaries.)
2.
(The multiplier rule for holonomic constraints. Harmonic mappings into hypersurfaces
of Rw+1. Shortest connection of two points on a surface in R3.
theorem. Geodesies on a sphere. Hamiltons s principle and holonomic constraints.
Pendulum equation.)
3.
(Normal and abnormal extremals. The multiplier rule for one-dimensional problems
with nonholonomic constraints. The heavy thread on a surface. Lagrange s
formulation of Maupertuis s least action principle. Solenoidal vector fields.)
4.
(Shortest distance in an
integral.
5.
Chapter
1.
(Energy-momentum tensor. Noether s equations. Erdmann s equation and conservation
of energy. Parameter invariant integrals: line and double integrals, multiple integrals.
Jacobi s geometric version of the least action principle. Minimal surfaces.)
2.
(Inner extremals of the generalized Dirichlet integral and conformality relations.
H-surfaces.)
3.
(Fluid flow and continuity equation. Stationary, irrotational, isentropic flow of a
compressible fluid.)
4.
(The n-body problem and Newton s law of gravitation. Equilibrium problems in
elasticity. Conservation laws. Hamilton s principle in continuum mechanics. Killing
equations.)
5.
(Generalized Dirichlet integral. Laplace-Beltrami Operator. Harmonic mappings of
Riemanaian manifolds.)
6.
Contents
Part II. The Second Variation and Sufficient Conditions
Chapter
1.
1.1.
(Weak and strong neighbourhoods; weak and strong minimizers; the properties
(Ж)
Scheeffer s example.)
1.2.
and Accessory Lagrangian
(The accessory Lagrangian and the Jacobi operator.)
1.3.
(Necessary condition for weak minimizers. Ellipticity, strong ellipticity, and
superellipticity.)
1.4.
and Weierstrass s Necessary Condition
(Necessary condition for strong minimizers.)
2.
Based on Convexity Arguments
2.1.
of the Second Variation
(Convex integrals.)
2.2.
(Dirichlet integral, area and length, weighted length.)
2.3.
(Line element in polar coordinates.
of the isoperimetric problem.)
2.4.
(Stability via Sobolev s inequality.)
2.5.
(The ff-surface functional.)
2.6.
3.
Chapter
1.
for Weak Minimizers Based on Eigenvalue Criteria
for the Jacobi Operator
1.1.
(Scheeffer s example: Positiveness of the second variation does not imply
minimality.)
1.2.
(The Jacobi operator as linearization of Euler s operator and as
of the accessory integral. Jacobi equation and Jacobi fields.)
XX
1.3.
for Weak Minima
(The role of the first eigenvalue of the Jacobi operator. Strict Legendre-
Hadamard condition. Results from the eigenvalue theory for strongly elliptic
systems. Conjugate values and conjugate points.)
2
in One Unknown Function
2.1.
(A sufficient condition for weak minimizers.)
2.2.
(Jacobi s function A(x,
sufficient conditions expressed in terms of Jacobi fields and conjugate
points.)
2.3.
(Envelope of families of extremals. Fields of extremals and conjugate points.
Embedding of a given extremal into a field of extremals. Conjugate points and
complete solutions of Euler s equation.)
2.4.
(Quadratic integrals. Sturm s comparison theorem. Conjugate points
of geodesies. Parabolic orbits and Galileo s law. Minimal surfaces of
revolution.)
3.
Chapter
and Strong Minimizers
1.
1.1.
and Mayer Bundles, Stigmatic Ray Bundles
(Definitions. The modified Euler equations. Mayer fields and their eikonals.
Characterization of Mayer fields by
Beltrami form.
bundles.)
1.2.
(Null Lagrangian and
strong minimizers.)
1.3.
Optimal Fields. Kneser s Transversality Theorem
(Sufficient conditions for weak and strong minimizers. Weierstrass fields and
optimal fields. The complete figure generated by a Mayer field: The field lines
and the one-parameter family of transversal surfaces. Stigmatic fields and their
value functions Z{x, e).)
2.
2.1.
(The general case IVzl. Jacobi fields and pairs of conjugate values.
Embedding of extremals by means of stigmatic fields.)
2.2.
(The case
Global embedding of extremals.)
Contents
2.3.
(Field theory for integrals of the kind
to Riemannian metrics ds
Minimal surfaces of revolution.
Brachystochrone.)
2.4.
(Conjugate base of Jacobi fields and its Mayer determinant A(x). The zeros of
A(x) are isolated. Sufficient conditions for minimality of an extremal whose left
endpoint freely varies on a prescribed hypersurface.)
3.
Lichtenstein s Theorem
(Fields for nonparametric hypersurfaces.
integral. Embedding of extremals. Lichtenstein s theorem.)
4.
Supplement. Some Facts from Differential Geometry and Analysis
1.
2.
3.
4.
5.
6.
A List of Examples
Bibliography
Subject Index
|
| any_adam_object | 1 |
| author | Giaquinta, Mariano 1947- Hildebrandt, Stefan 1936-2015 |
| author_GND | (DE-588)111595738 (DE-588)119219050 |
| author_facet | Giaquinta, Mariano 1947- Hildebrandt, Stefan 1936-2015 |
| author_role | aut aut |
| author_sort | Giaquinta, Mariano 1947- |
| author_variant | m g mg s h sh |
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| bvnumber | BV019335956 |
| callnumber-first | Q - Science |
| callnumber-label | QA315 |
| callnumber-raw | QA315 |
| callnumber-search | QA315 |
| callnumber-sort | QA 3315 |
| callnumber-subject | QA - Mathematics |
| ctrlnum | (OCoLC)69172652 (DE-599)BVBBV019335956 |
| dewey-full | 515.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.64 |
| dewey-search | 515.64 |
| dewey-sort | 3515.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Corr. 2. print. |
| format | Book |
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| id | DE-604.BV019335956 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:57:54Z |
| institution | BVB |
| isbn | 354050625X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-012800750 |
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| owner | DE-355 DE-BY-UBR DE-29T DE-83 |
| owner_facet | DE-355 DE-BY-UBR DE-29T DE-83 |
| physical | XXIX, 474 S. Ill., graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Giaquinta, Mariano 1947- Verfasser (DE-588)111595738 aut Calculus of variations 1 [The Lagrangian formalism] Mariano Giaquinta ; Stefan Hildebrandt Corr. 2. print. Berlin [u.a.] Springer 2004 XXIX, 474 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 310 Grundlehren der mathematischen Wissenschaften ... Calculo de variações larpcal Calculus of variations Hildebrandt, Stefan 1936-2015 Verfasser (DE-588)119219050 aut (DE-604)BV010544890 1 Grundlehren der mathematischen Wissenschaften 310 (DE-604)BV000000395 310 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012800750&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Giaquinta, Mariano 1947- Hildebrandt, Stefan 1936-2015 Calculus of variations Grundlehren der mathematischen Wissenschaften Calculo de variações larpcal Calculus of variations |
| title | Calculus of variations |
| title_auth | Calculus of variations |
| title_exact_search | Calculus of variations |
| title_full | Calculus of variations 1 [The Lagrangian formalism] Mariano Giaquinta ; Stefan Hildebrandt |
| title_fullStr | Calculus of variations 1 [The Lagrangian formalism] Mariano Giaquinta ; Stefan Hildebrandt |
| title_full_unstemmed | Calculus of variations 1 [The Lagrangian formalism] Mariano Giaquinta ; Stefan Hildebrandt |
| title_short | Calculus of variations |
| title_sort | calculus of variations the lagrangian formalism |
| topic | Calculo de variações larpcal Calculus of variations |
| topic_facet | Calculo de variações Calculus of variations |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012800750&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010544890 (DE-604)BV000000395 |
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