# 60799

## Автор(ов):

1

Параметры публикации

Глава в книге

## Название:

Symmetry and integrals

## ISBN/ISSN:

978-3-030-63252-6

## DOI:

10.1007/978-3-030-63253-3_5

## Наименование источника:

• Differential Geometry, Differential Equations, and Mathematical Physics

## Город:

• Basel, Switzerland

## Издательство:

• 2019 Springer Nature Switzerland AG. Part of Springer Nature.

2021

## Страницы:

73-121
Аннотация
In these lectures, I want to illustrate an application of symmetry ideas to integration of differential equations. Basically, we will consider only differential equations of finite type, i.e. equations with finite-dimensional space Sol of (local) solutions. Ordinary differential equations make up one of the main examples of such equations. The symmetry Lie algebra Sym induces an action on manifold Sol. In the case when this action is transitive, we expect to get more detailed information on solutions. Here, we are going to realize this expectation; namely, we will show that in the case when the Lie algebra Sym is solvable, integration of the differential equation can be done by quadratures due to the Lie–Bianchi theorem (see, for example, [4] or [6]). In the case when the Lie algebra Sym contains simple subalgebras, integration shall use quadratures (for radical of the Lie algebra) and integration of some differential equations, which we will call model equations [6, 10]. The model equations depend on the type of the simple Lie subalgebras and are natural generalizations of the wellknown Riccati differential equations. They possess nonlinear Lie superposition, and all their solutions could be obtained by nonlinear superposition of a finite set of solutions (the so-called fundamental solutions). Once more, the form of this superposition and the number of fundamental solutions are dictated by the symmetry Lie algebra. In order to give a more “practical” reader a feeling of the power of the geometrical approach to differential equations, we included in these lectures a number of examples on the formula level.

## Библиографическая ссылка:

Лычагин В.В. Symmetry and integrals / Differential Geometry, Differential Equations, and Mathematical Physics. Basel, Switzerland: 2019 Springer Nature Switzerland AG. Part of Springer Nature., 2021. С. 73-121.