Search

In this paper, thermodynamics of a moving medium is considered. Main goal of this paper is to describe thermodynamics state equations and equations of co-existence manifolds of media on two- and three-dimensional manifolds. To this end the phase space of a medium is extended with the deformation tensor and the stress tensor.

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and solve the equivalence problem for actions of algebraic Lie pseudogroups. As a running example of application of the methods, we use the pseudogroup of local symplectomorphisms.

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of order $k\ge 2$ on a smooth manifold of dimension $n\ge 2$ and show their application to the equivalence problem of such operators.

We give a description of the field of rational natural differential invariants for a class of nonlinear differential operators of the third order on a two dimensional manifold and show their application to the equivalence problem of such operators., Comment: 14 pages

We consider the Euler system describing a one-dimensional inviscid flows in space along curves of a certain class. Using differential invariants for the Euler system, we obtain its quotient equation. The solutions of the quotient equation that are constant along characteristic vector field provide some solutions of the Euler system. We discuss solving the quotient using asymptotic expansions of unknown functions and virial expansion of thermodynamic state equations. Thus the quotient is reduced to a series of ODE systems.

A Navier--Stokes system on a curve is discussed. The quotient equation for this system is found. The quotient is used to find some solutions of Navier--Stokes system. Using virial expansion of the Planck potential, we reduce the quotient equation to a series of systems of ordinary differential equations.

In these lectures, we discuss two approaches to studying orbit spaces of algebraic Lie groups. Due to algebraic approach orbit space, or quotient, is an algebraic manifold, while from the differential viewpoint a quotient is a differential equation. The main goal of these lectures is to show that the differential approach gives us a better understanding of structure of invariants and orbit spaces. We illustrate this on classical equivalence problems, such as $\mathrm{SL}$ - classification of binary and ternary forms, and affine classification of algebraic plane curves., Comment: 27 pages

In this paper, we analyze various types of critical phenomena in one-dimensional gas flows described by Euler equations. We give a geometrical interpretation of thermodynamics with a special emphasis on phase transitions. We use ideas from the geometrical theory of PDEs, in particular, symmetries and differential constraints to find solutions to the Euler system. Solutions obtained are multivalued, have singularities of projection to the plane of independent variables. We analyze the propagation of the shock wave front along with phase transitions., Comment: 12 pages, 4 figures

We present the development of the approach to thermodynamics based on measurement. First of all, we recall that considering classical thermodynamics as a theory of measurement of extensive variables one gets the description of thermodynamic states as Legendrian or Lagrangian manifolds representing the average of measurable quantities and extremal measures. Secondly, the variance of random vectors induces the Riemannian structures on the corresponding manifolds. Computing higher order central moments one drives to the corresponding higher order structures, namely, the cubic and the fourth order forms. The cubic form is responsible for the skewness of the extremal distribution. The condition for it to be zero gives us so-called symmetric processes. The positivity of the fourth order structure gives us an additional requirement to thermodynamic state., Comment: 8 pages, 4 figures; the Discussion section has been added, typos and figures corrected

Symmetries and differential invariants of viscid flows with viscosity depending on temperature on a space curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given., Comment: arXiv admin note: text overlap with arXiv:2006.14932, arXiv:2004.11906

We consider Cosserat media as SO(3)-structures over a domain D in R3. Motions of such media are given by automorphisms of the SO(3)-bundle. We present Euler-type equations for such media and discuss their structure., Comment: 12 pages

In this paper, flows of a viscid fluids on curves are considered. Symmetry algebras and the corresponding fields of differential invariants are found. We study their dependence on thermodynamic states of media, and provide classification of thermodynamic states., Comment: arXiv admin note: text overlap with arXiv:2004.11906

In this paper we classify the symbols of the linear differential operators of order $k$, which act from the module $C^\infty(\xi)$ to the module $C^\infty(\xi^t)$, where $\xi\colon E(\xi)\to M$ is vector bundle over the smooth manifold $M$, bundle $\xi^t$ is either $\xi^*$ with fiber $E^*:=\mathrm{Hom}(E,\mathbb{C})$ or $\xi^\flat$ with fiber $E^\flat:=\mathrm{Hom}(E, \Lambda^n T^*)$ and $C^\infty(\xi)$, $C^\infty(\xi^t)$ are the modules of their smooth sections. To find invariants of the symbols we associate with every non-degenerated symbol the tuple of linear operators acting on space $E$ and reduce our problem to the classification of such tuples with respect to some orthogonal transformations. Using the results of C. Procesi, we find generators for the field of rational invariants of the symbols and in terms of these invariants provide a criterion of equivalence of non-degenerated symbols.

We propose a geometrical approach to the mechanics of continuous media equipped with inner structures and give the basic (mass conservation, Navier-Stokes and energy conservation) equations of their motion.

In this paper, we study scalar the forth order linear differential operators over an oriented 2-dimensional manifold. We investigate differential invariants of these operators and show their application to the equivalence problem., Comment: 13 pages. arXiv admin note: text overlap with arXiv:1906.03254

Symmetries and the corresponding fields of differential invariants of the inviscid flows on a curve are given. Their dependence on thermodynamic states of media is studied, and a classification of thermodynamic states is given., Comment: 6 figures. arXiv admin note: text overlap with arXiv:2004.01567

The equivalence problem for linear differential operators of the second order, acting in vector bundles, is discussed. The field of rational invariants of symbols is described and connections, naturally accosiated with differential operators, are found. These geometrical structures are used to solve the problems of local as well as global equivalency of differential operators., Comment: arXiv admin note: text overlap with arXiv:2003.12734

Non-stationary Euler flows of gases are studied. The system of differential equations describing such flows can be represented by means of 2-forms on zero-jet space and we get some exact solutions by means of such a representation. Solutions obtained are multivalued and we provide a method of finding caustics, as well as wave front displacement. The method can be applied to any model of thermodynamic state as well as to any thermodynamic process. We illustrate the method on adiabatic ideal gas flows., Comment: 8 pages, 2 figures

This paper surveys results found by the authors in the previous papers (see for example, A. Duyunova, V. Lychagin, S. Tychkov, Differential invariants for spherical layer flows of a viscid fluid, Journal of Geometry and Physics, 130, 288-292 (2018)) concerning calculation and classification of symmetry Lie algebra and differential invariants for a variety of fundamental systems in fluid mechanics, including the Euler and Navier-Stokes equations on a plane, three-dimensional space, a sphere, and a spherical layer. In each case, the symmetry Lie algebra is found, followed by a list of generating differential invariants and invariant derivations for the group action. The main extension is a detailed analysis of thermodynamic states, symmetries, and differential invariants. This analysis is based on consideration of Riemannian structure naturally associated with Lagrangian manifolds that represent thermodynamic states. This approach radically changes the description of the thermodynamic part of the symmetry algebra as well as the field of differential invariants.

The problem of equivalency for linear differential operators of the first order is discussed., Comment: 14 pages

In this paper, we study Schr\"{o}dinger equations on elliptic curves called generalized Lam\'{e} equations. We suggest a method of finding integrable potentials for Schr\"{o}dinger type equations. We apply this method to the Lam\'{e} equations and provide a sequence of integrable potentials for which the eigenvalue problem is solved explicitly., Comment: 13 pages, 4 figures

In this paper, we consider an optimal control problem in equilibrium thermodynamics of gases. Thermodynamic state of the gas is given by a Legendrian submanifold in a contact thermodynamic space. Using Pontryagin's maximum principle we find a thermodynamic process on this submanifold such that the gas maximizes the work functional. For ideal gases, this problem is shown to be integrable in Liouville's sense and its solution is given by means of action-angle variables. For real gases considered as a perturbation of ideal ones, the integrals are given asymptotically., Comment: 13 pages

In this paper, we study non-stationary filtration of real gases in porous media. Thermodynamic state of the medium is given by van der Waals state equations. Solutions for non-stationary filtration equation are obtained by means of finite dimensional dynamics. The analysis of phase transitions along the flow in case of isentropic and isenthalpic processes is presented as well as singular properties of solutions obtained are discussed. Domains in the jet space where the dynamics found is an attractor are shown., Comment: 13 pages, 7 figures

A non-stationary isentropic filtration of gases in porous media is considered. Thermodynamics in terms of contact and symplectic geometries is briefly discussed. Algebra of symmetries for the PDE system is found, and the classification of media with respect to admissible symmetries is given. Solution for one class of media is found and the phase transitions for this solution are studied. Keywords: gas filtration, porous media, symmetry algebra, phase

We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles over smooth manifolds with respect to groups of authomorphisms.

Stationary adiabatic flows of real gases issued from a source of given intensity are studied. Thermodynamic states of gases are described by Legendrian or Lagrangian manifolds. Solutions of Euler equations are given implicitly for any equation of state and the behavior of solutions of the Navier-Stokes equations with the viscosity considered as a small parameter is discussed. For different intensities of the source we introduce a small parameter into the Navier-Stokes equation and construct corresponding asymptotic expansions. We consider the most popular model of real gases --- the van der Waals model, and ideal gases as well., Comment: 14 pages, 8 figures

We study differential invariants of the third order linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles on two dimensional manifolds with respect to groups of authomorphisms.

Filtration of real gases described by Peng-Robinson equations of state in 3-dimensional space is studied. Thermodynamic states are considered as either Legendrian submanifolds in contact space, or Lagrangian submanifolds in symplectic space. The correspondence between singularities of their projection on the plane of intensives and phase transitions is shown, and coexistence curves in various coordinates are constructed. A method of finding explicit solutions of the Dirichlet boundary problem is provided and the case of a number of sources is discussed in details. The domains corresponding to different phases are shown., Comment: 10 pages, 6 figures

In this paper we study a 3-dimensional filtration of real gases described by Redlich-Kwong equations of state. Thermodynamical states are considered as Legendrian (Lagrangian) submanifolds in contact (symplectic) space. Connection between singularities of their projection on the space of intensive variables and phase transitions is shown. Explicit formulae for the Dirichlet boundary problem are given and the distribution of phases in space is shown., Comment: 10 pages, 8 figures

Steady adiabatic filtration of real gases is studied. Thermodynamical states of real gases are presented by Legendrian surfaces in 5-dimensional thermodynamical contact space. The relation between phase transitions and singularities of projection of the Legendrian surfaces on the plane of intensive variables is shown. The constructive method of finding solutions of the Dirichlet filtration problem together with analysis of critical phenomena is presented. Cases of van der Waals and Peng-Robinson gases are discussed in details., Comment: 27 pages, 10 figures

We describe quantizations on monoidal categories of modules over finite groups. They are given by quantizers which are elements of a group algebra. Over the complex numbers we find these explicitly. For modules over S3 and A4 we are given explicit forms for all quantizations.