Calculus of variations: 2 [The Hamiltonian formalism]
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2004
|
| Edition: | Corr. 2. print. |
| Series: | Grundlehren der mathematischen Wissenschaften
311 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIX, 652 S. graph. Darst. |
| ISBN: | 3540579613 |
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| 245 | 1 | 0 | |a Calculus of variations |n 2 |p [The Hamiltonian formalism] |c Mariano Giaquinta ; Stefan Hildebrandt |
| 250 | |a Corr. 2. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2004 | |
| 300 | |a XXIX, 652 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 311 | |
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| 650 | 7 | |a Calculo de variações |2 larpcal | |
| 650 | 4 | |a Calculus of variations | |
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| _version_ | 1804132763409842176 |
|---|---|
| adam_text | Contents
The Hamiltonian Formalism
Part III.
Chapter
Field Theories
1.
1.1.
(Definitions. Involutory character of the Legendre transformation. Conjugate
convex functions. Young s inequality. Support function. Clairaut s differential
equation. Minimal surface equation. Compressible two-dimensional steady
flow. Application of Legendre transformations to quadratic forms and convex
bodies. Partial Legendre transformations.)
1.2.
Euler
{Configuration space, phase space, cophase space, extended configuration
(phase, cophase) space. Momenta. Hamiltonians. Energy-momentum tensor.
Hamiltonian systems of canonical equations. Dual Noether equations. Free
boundary conditions in canonical form. Canonical form of E. Noether s
theorem, of Weierstrass s excess function and of transversality.)
2.
of the One-Dimensional Variational Calculus
2.1.
of Hamilton-Jacobi
(Euierian flows and Hamiltonian flows as prolongations of extremal bundles.
Canonical description of Mayer fields. The
The Hamilton-Jacobi equation as canonical version of Caratheodory s
equations.
2.2.
Regular Mayer Flows and
(The eigentime function of an r-parameter Hamiltonian flow. The Cauchy
representation of the pull-back
r-parameter Hamilton flow
field-like Mayer bundles, and
2.3.
of the Jacobi Equation
(The Legendre transform of the accessory Lagrangian is the accessory
Hamiltonian,
equations describe Jacobi fields. Expressions for the first and second
variations.)
XVIII Contents
2.4.
(Necessary and sufficient conditions for the local solvability of the Cauchy
problem. The Hamilton-Jacobi equation. Extension to discontinuous media:
refracted light bundles and the theorem of
3.
3.1.
(Basic properties of convex sets and convex bodies. Supporting
Convex hull. Lipschitz continuity of convex functions.)
3.2.
(Gauge functions. Distance function and support function. The support
function of a convex body is the distance function of its polar body, and vice
versa. The polarity map. Polar body and Legendre transform.)
3.3.
Fenchel
(Characterization of smooth convex functions. Supporting
differentiability. Regularization of convex functions. Legendre-
transform.)
4.
4.1.
(Null Lagrangians of divergence type as calibrators. Weyl equations. Geodesic
slope fields or Weyl fields, eikonal mappings. Beltrami form. Legendre
transformation. Cattan form. DeDonder s partial differential equation.
Extremals fitting a geodesic slope field. Solution of the local fitting problem.)
4.2.
(Carathéodory s
Transversality.
eikonal maps.
Generalization of Kneser s transversality theorem. Solution of the local fitting
problem for a given extremal)
4.3.
(The general Beltrami form. Lepage s formalism. Geodesic slope fields. Lepage
calibrators.)
4.4.
(Calibrators and pseudonecessary optimality conditions. (I) One-dimensional
variational problems with nonholonomic constraints:
Pontryagin s function, Hamilton function, Pontryagin s maximum principle
and canonical equations. (II) Pontryagin s maximum principle for multi¬
dimensional problems of optimal control.)
5.
Chapter
1.
1.1.
and Weak Extremals
(Parametric Lagrangians. Parameter-invariant integrals. Riemannian metrics.
Finsler metrics. Parametric extremals. Transversality of line elements. Eulerian
covector field and Noether s equation. Gauss s equation. Jacobi s variational
principle for the motion of a point mass in R3.)
Contents
1.2.
and Vice Versa
(Nonparametric restrictions of parametric Lagrangians. Parametric extensions
of nonparametric Lagrangians. Relations between parametric and
nonparametric extremals.)
1.3.
Weierstrass-Erdmann Comer Conditions. Fermat s Principle
and the Law of Refraction
(Weak Z>1- and
Erdmann corner conditions. Regularity theorem for weak D -extremals.
Snellius s law of refraction and Fermat s principle.)
2.
2.1.
and the Canonical Formalism
(The associated quadratic Lagrangian Q of a parametric Lagrangian F.
Elliptic and nonsingular line elements. A natural Hamiltonian and the
corresponding canonical formalism. Parametric form of Hamilton s canonical
equations.)
2.2.
(The conservation of energy and Jacobi s least action principle: a geometric
description of orbits.)
2.3.
and
(The parametric Legendre condition or C-regularity.
formalism.)
2.4.
(Indicatrix, figuratrix and canonical coordinates. Strong and semistrong line
elements. Regularity of broken extremals. Geometric interpretation of the
excess function.)
3.
3.1.
(Parametric fields and their direction fields. Equivalent fields. The parametric
Carathéodory
integral. Weierstrass s representation formula. Kneser s transversality theorem.
The parametric Beltrami form. Normal fields of extremals and Mayer fields,
Weierstrass fields, optimal fields, Mayer bundles of extremals.)
3.2.
(The parametric Cartan form. The parametric Hamilton-Jacobi equation or
eikonal equation. One-parameter families of
3.3.
(F- and Q-minimizers. Regular Q-minimizers are
values and conjugate points of F-extremals. F-extremals without conjugate
points are local minimizers. Stigmatic bundles of quasinormal extremals and
the exponential map of a parametric Lagrangian. F- and
fronts.)
3.4.
(Complete Figures. Duality between light rays and wave fronts. Huygens s
envelope construction of wave fronts. F-distance function. Foliations
by one-parameter families of F-equidistant surfaces and optimal
fields.)
XX
4.
4.1.
(The distance function d(P, F) related to
semicontinuity properties. Existence of global minimizers based on the local
existence theory developed in
4.2.
(Minimizing sequences. An equivalent minimum problem. Compactness of
minimizing sequences. Lower semicontinuity of the variational integral. A
general existence theorem for obstacle problems. Regularity of minimizers.
Existence of minimizing F-extremals. Inclusion principle.)
4.3.
(Comparison of curves with the
ellipse. Comparison of catenaries and
results.)
4.4.
(Existence and regularity of F-extremals
length.)
5.
Part IV. Hamilton-Jacobi Theory
and
Chapter
1.
1.1.
(Trajectories, integral curves, maximal flows.)
1.2.
of Transformations
(Infinitesimal transformations.)
1.3.
(The symbol of a vector field and its transformation law.)
1.4.
(Commuting flows. Lie derivative. Jacobi identity.)
1.5.
(Rectification of nonsingular vector fields.)
1.6.
(Time-dependent and time-independent first integrals. Functionally
independent first integrals. The motion in a central field. Kepler s problem.
The two-body problem.)
1.7.
(Lax pairs.
1.8.
for Matrix-Valued Functions. Variational Equations.
Volume Preserving Flows
(Liouville formula. Liouville theorem. Autonomous Hamiltonian flows are
volume preserving.)
1.9.
(Geodesies on S2.)
Contents
2.
2.1.
Revisited
(Mechanical systems. Action. Hamiltonian systems and Hamilton-Jacobi
equation.)
2.2.
(Principal function and canonical transformations.)
2.3.
(Cyclic variables. Routhian systems.)
2.4.
for Hamiltonian Systems
(The Cartan form and the canonical variational principle.)
3.
3.1.
and Their Symplectic Characterization
(Symplectic matrices. The harmonic oscillator.
Poincaré
3.2.
Hamilton Flows and One-Parameter Groups
of Canonical Transformations
(Elementary canonical transformation. The transformations of
Levi-Civita. Homogeneous canonical transformations.)
3.3.
(Complete solutions. Jacobi s theorem and its geometric interpretation.
Harmonic oscillator. Brachystochrone. Canonical perturbations.)
3.4.
(Arbitrary functions generate canonical mappings.)
3.5.
(Liouville systems. A point mass attracted by two fixed centers. Addition
theorem of Euler. Regularization of the three-body problem.)
3.6.
(Poisson
3.7.
(Symplectic geometry. Darboux theorem. Symplectic maps. Exact symplectic
maps. Lagrangian submanifolds.)
4.
Chapter
and Contact Transformations
1.
1.1.
of Characteristics
(Configuration space, base space, contact space. Contact elements and their
support points and directions. Contact form,
manifolds, characteristic equations, characteristics, null (integral) characteristic,
characteristic curve, characteristic base curve. Cauchy problem and its local
solvability for noncharacteristic initial values: the characteristic flow and its
first integral F, Cauchy s formulas.)
XXII Contents
1.2.
Quasilinear
(Lie s equations. First order linear and quasilinear equations, noncharacteristic
initial values. First integrals of Cauchy s characteristic equations, Mayer
brackets [F,
1.3.
(Homogeneous linear equations, inhomogeneous linear equations, Euler s
equation for homogeneous functions. The reduced Hamilton-Jacobi equation
H(x, ux)
Congruences or ray systems, focal points. Monge cones, Monge lines, and
focal curves, focal strips. Partial differential equations of first order and cone
fields.)
1.4.
for the Hamilton-Jacobi Equation
(A discussion of the method of characteristics for the equation
S,
values.)
2.
2.1.
(Strip equation, strips of maximal dimension
of type Cj, contact transformations, transformation of strips into strips,
characterization of contact transformations. Examples: Contact
transformations of Legendre, Euler, Ampere, dilations, prolongated point
transformations.)
2.2.
and Canonical Mappings
(Contact transformations commuting with translations in z-direction and exact
canonical transformations. Review of various characterizations of canonical
mappings.)
2.3.
(Contact transformations of R2* 1 1 can be prolonged to special contact
transformations of R2*+3, or to homogeneous canonical transformations of
R2 +2. Connection between
contact transformations.)
2.4.
(The directrix equation for contact transformations of first type:
Q(x, z,x,z) =
first type from an arbitrary directrix equation. Contact transformations of type
r and the associated systems of directrix equations. Examples: Legendre s
transformation, transformation by reciprocal
transformation, pedal transformation, dilations, contact transformations
commuting with all dilations, partial Legendre transformations, apsidal
transformation, Fresnel surfaces and conical refraction. Differential equations
and contact transformations of second order. Canonical prolongation of
first-order to second-order contact transformations. Lie s G-K-transformation.)
2.5.
Huygens Flows and Huygens Fields; Vessiot s Equation
(One-parameter flows of contact transformations and their characteristic Lie
functions. Lie equations and Lie flows. Huygens flows are Lie flows generated
by n-strips as initial values. Huygens fields as ray maps of Huygens flows.
Vessiot s equation for the eikonal of a Huygens field.)
Contents
2.6.
(Propagation of wave fronts by Huygens s envelope construction: Huygens s
principle. The indicatrix
Huygens s principle by the Lie equations generated by F.)
3.
3.1.
(Description of Huygens s principle by Herglotz equations generated by the
indicatrix function W. Description of Lie s equations and Herglotz s equations
by variational principles. The characteristic equations Sx
for the eikonal
3.2.
(The generating function
The Holder transform
T
formulas. Connections between Holder s transformation
transformation S£f generated by F: the commuting diagram and Haar s
transformation 3te. Examples.)
3.3.
and Hamiltonian Systems
(Holder s transformation
independent variable generated by
Hamiltonian system x
together with the eigentime transformation
Hamiltonian system into a Lie system. Equivalence of Mayer flows and
Huygens flows, and of Mayer fields and Huygens fields.)
3.4.
and Huygens s Principles
(Under suitable assumptions, the four pictures of rays and waves due to
Euler-Lagrange, Huygens-Lie, Hamilton, and Herglotz are equivalent.
Correspondingly the two principles of
4.
A List of Examples
A Glimpse at the Literature
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 | BV019339822 |
| callnumber-first | Q - Science |
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| callnumber-subject | QA - Mathematics |
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| 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.BV019339822 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:57:59Z |
| institution | BVB |
| isbn | 3540579613 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-012804407 |
| oclc_num | 69172652 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-29T DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-29T DE-11 |
| physical | XXIX, 652 S. 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 2 [The Hamiltonian formalism] Mariano Giaquinta ; Stefan Hildebrandt Corr. 2. print. Berlin [u.a.] Springer 2004 XXIX, 652 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 311 Grundlehren der mathematischen Wissenschaften ... Calculo de variações larpcal Calculus of variations Hildebrandt, Stefan 1936-2015 Verfasser (DE-588)119219050 aut (DE-604)BV010544890 2 Grundlehren der mathematischen Wissenschaften 311 (DE-604)BV000000395 311 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012804407&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 2 [The Hamiltonian formalism] Mariano Giaquinta ; Stefan Hildebrandt |
| title_fullStr | Calculus of variations 2 [The Hamiltonian formalism] Mariano Giaquinta ; Stefan Hildebrandt |
| title_full_unstemmed | Calculus of variations 2 [The Hamiltonian formalism] Mariano Giaquinta ; Stefan Hildebrandt |
| title_short | Calculus of variations |
| title_sort | calculus of variations the hamiltonian 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=012804407&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010544890 (DE-604)BV000000395 |
| work_keys_str_mv | AT giaquintamariano calculusofvariations2 AT hildebrandtstefan calculusofvariations2 |