The paper deals with the local theory of internal symmetries of underdetermined systems of ordinary differential equations in full generality. The symmetries need not preserve the choice of the independent variable, the hierarchy of dependent variables, and the order of derivatives. Internal approach to the symmetries of one-dimensional constrained variational integrals is moreover proposed without the use of multipliers.
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.
The article treats the geometrical theory of partial differential equations in the absolute sense, i.e., without any additional structures and especially without any preferred choice of independent and dependent variables. The equations are subject to arbitrary transformations of variables in the widest possible sense. In this preparatory Part 1, the involutivity and the related standard bases are investigated as a technical tool within the framework of commutative algebra. The particular case of ordinary differential equations is briefly mentioned in order to demonstrate the strength of this approach in the study of the structure, symmetries and constrained variational integrals under the simplifying condition of one independent variable. In full generality, these topics will be investigated in subsequent Parts of this article.
The paper deals with local symmetries of the infinite-order jet space of C ∞ -smooth n-dimensional submanifolds in R m n . Transformations under consideration are the most general possible. They need not preserve the distinction between dependent, and independent variables, the order of derivatives and the hierarchy of finite-order jet spaces.
The article concerns the geometrical theory of general systems Ω of partial differential equations in the absolute sense, i.e., without any additional structure and subject to arbitrary change of variables in the widest possible meaning. The main result describes the composition series Ω 0 ⊂ Ω 1 ⊂ · · · ⊂ Ω where Ω k is the maximal system of differential equations "induced" by Ω such that the solution of Ω k depends on arbitrary functions of k independent variables (on constants if k = 0). This is a well-known result for the particular case of underdetermined systems of ordinary differential equations. Then Ω = Ω 1 and we have the composition series Ω 0 ⊂ Ω 1 = Ω where Ω 0 involves all first integrals of Ω, therefore Ω 0 is trivial (absent) in the controllable case. The general composition series Ω 0 ⊂ Ω 1 ⊂ · · · ⊂ Ω may be regarded as a "multidimensional" controllability structure for the partial differential equations.Though the result is conceptually clear, it cannot be included into the common jet theory framework of differential equations. Quite other and genuinely coordinate-free approach is introduced.2010 Mathematics Subject Classification. 58A17, 58J99, 35A30.
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