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2. Examples from the calculus of variations III. Legendre and Jacobi conditions.

Creator:
Chrastina, Jan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Legendre condition, Jacobi condition, Poincaré-Cartan form, Lagrange problem, and degenerate variational integral
Language:
English
Description:
We will deal with a new geometrical interpretation of the classical Legendre and Jacobi conditions: they are represented by the rate and the magnitude of rotation of certain linear subspaces of the tangent space around the tangents to the extremals. (The linear subspaces can be replaced by conical subsets of the tangent space.) This interpretation can be carried over to nondegenerate Lagrange problems but applies also to the degenerate variational integrals mentioned in the preceding Part II.
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3. Examples from the calculus of variations. I. Nondegenerate problems

Creator:
Chrastina, Jan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
variational integral, critical curve, adjoint module, initial form, Poincaré-Cartan form, Lagrange problem, Mayer field, Weierstrass function, and diffiety
Language:
English
Description:
The criteria of extremality for classical variational integrals depending on several functions of one independent variable and their derivatives of arbitrary orders for constrained, isoperimetrical, degenerate, degenerate constrained, and so on, cases are investigated by means of adapted Poincare-Cartan forms. Without ambitions on a noble generalizing theory, the main part of the article consists of simple illustrative examples within a somewhat naive point of view in order to obtain results resembling the common Euler-Lagrange, Legendre, Jacobi, and Hilbert-Weierstrass conditions whenever possible and to discuss some modifications necessary in the degenerate case. The inverse and the realization problems are mentioned, too.
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4. Examples from the calculus of variations. VI.. Concluding review

Creator:
Chrastina, Jan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Lagrange problem, Poincaré-Cartan form, Hamiltonian-Jacobi equation, and Weierstrass-Hilbert method
Language:
English
Description:
Variational integrals containing several functions of one independent variable subjected moreover to an underdetermined system of ordinary differential equations (the Lagrange problem) are investigated within a survey of examples. More systematical discussion of two crucial examples from Part I with help of the methods of Parts II and III is performed not excluding certain instructive subcases to manifest the significant role of generalized Poincaré-Cartan forms without undetermined multipliers. The classical Weierstrass-Hilbert theory is simulated to obtain sufficient extremality conditions. Unlike the previous parts, this article is adapted to the category of continuous objects and mappings without any substantial references to the general principles, which makes the exposition self-contained.
Rights:
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5. Examples from the calculus of variations: II. a degenerate problem

Creator:
Chrastina, Jan
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Poincaré-Cartan form, degenerate variational integral, and realization problem
Language:
English
Description:
Continuing the previous Part I, the degenerate first order variational integrals depending on two functions of one independent variable are investigated.
Rights:
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6. The symmetry reduction of variational integrals

Creator:
Tryhuk, Václav and Chrastinová, Veronika
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Routh reduction, Lagrange variational problem, Poincaré-Cartan form, diffiety, standard basis, controllability, and variation
Language:
English
Description:
The Routh reduction of cyclic variables in the Lagrange function and the Jacobi-Maupertuis principle of constant energy systems are generalized. The article deals with one-dimensional variational integral subject to differential constraints, the Lagrange variational problem, that admits the Lie group of symmetries. Reduction to the orbit space is investigated in the absolute sense relieved of all accidental structures. In particular, the widest possible coordinate-free approach to the underdetermined systems of ordinary differential equations, Poincaré-Cartan forms, variations and extremals is involved for the preparation of the main task. The self-contained exposition differs from the common actual theories and rests only on the most fundamental tools of classical mathematical analysis, however, they are applied in infinite-dimensional spaces. The article may be of a certain interest for nonspecialists since all concepts of the calculus of variations undergo a deep reconstruction.
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7. The symmetry reduction of variational integrals, complement

Creator:
Chrastinová, Veronika and Tryhuk, Václav
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Lagrange variational problem, Poincaré-Cartan form, and symmetry reduction
Language:
English
Description:
Some open problems appearing in the primary article on the symmetry reduction are solved. A new and quite simple coordinate-free definition of Poincaré-Cartan forms and the substance of divergence symmetries (quasisymmetries) are clarified. The unbeliavable uniqueness and therefore the global existence of Poincaré-Cartan forms without any uncertain multipliers for the Lagrange variational problems are worth extra mentioning.
Rights:
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