Contact Dynamics: Legendrian and Lagrangian Submanifolds

We are proposing Tulczyjew’s triple for contact dynamics. The most important ingredients of the triple, namely symplectic diffeomorphisms, special symplectic manifolds, and Morse families, are generalized to the contact framework. These geometries permit us to determine so-called generating family (obtained by merging a special contact manifold and a Morse family) for a Legendrian submanifold. Contact Hamiltonian and Lagrangian Dynamics are recast as Legendrian submanifolds of the tangent contact manifold. In this picture, the Legendre transformation is determined to be a passage between two different generators of the same Legendrian submanifold. A variant of contact Tulczyjew’s triple is constructed for evolution contact dynamics.

Boundary conditions in topological AdS4/CFT3

We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional $$ \mathcal{N} $$ N = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.

The Lagrangian and Hamiltonian for RLC Circuit: Simple Case

The Lagrangian and Hamiltonian for series RLC circuit has been formulated. We use the analogical concept of classical mechanics with electrical quantity. The analogy is as follow mass, position, spring constant, velocity, and damping constant corresponding with inductance, charge, the reciprocal of capacitance, electric current, and resistance respectively. We find the Lagrangian for the LC, RL, RC, and RLC circuit by using the analogy and find the kinetic and potential energy. First, we formulate the Lagrangian of the system. Second, we construct the Hamiltonian of the system by using the Legendre transformation of the Lagrangian. The results indicate that the Hamiltonian is the total energy of the system which means the equation of constraints is time independent. In addition, the Hamiltonian of overdamping and critical damping oscillation is distinguished by a certain factor.

Idempotent Analog of the Legendre Transformation and lts Application in Digital Processing of Signals

In recent years, a new area of mathematics — idempotent or “tropical” mathematics — has been intensively developed within the framework of the Sofus Lee international center, which is reflected in the works of V.P. Maslov, G.L. Litvinov, and A.N. Sobolevsky. The Legendre transformation plays an important role in theoretical physics, classical and statistical mechanics, and thermodynamics. In mathematics and its applications, the Legendre transformation is based on the concept of duality of vector spaces and duality theory for convex functions and subsets of a vector space. The purpose of this paper is to go beyond linear vector spaces using similar notions of duality in conformally flat Riemannian geometry and in idempotent algebra.An abstract idempotent analog of the Legendre transformation is constructed in a way similar to the polar transformation of the conformally flat Riemannian metric introduced in the works of E.D. Rodionov and V.V. Slavsky. Its capabilities for digital processing of signals and images are being investigated

On the connections of sub-Finslerian geometry

A sub-Finslerian manifold is, roughly speaking, a manifold endowed with a Finsler type metric which is defined on a [Formula: see text]-dimensional smooth distribution only, not on the whole tangent manifold. Our purpose is to construct a generalized nonlinear connection for a sub-Finslerian manifold, called [Formula: see text]-connection by the Legendre transformation which characterizes normal extremals of a sub-Finsler structure as geodesics of this connection. We also wish to investigate some of its properties like normal, adapted, partial and metrical.

QUSI-LAGRANGIAN INTEGRATION OF TWO-DIMENSIONAL PRANDTL AND BOUSSINESQ EQUATIONS IN THE INVISCID LIMIT

For the hydrodynamic type systems locally describing incompressible twodimensional fluid flows without dissipation we suggest quasi-Lagrangian approach for their integration. This method is based on application of incomplete Legendre transformation when the independent variables become a Lagrangian invariant (i.e. the constant quantity along the fluid particle trajectory), instead of one of the spatial coordinate, and the rest ones which are another spatial coordinate and time. Thus, this method is based on the inverse transform of one of the spatial coordinate and by this reason differs from the the complete Legendre transformation. The classical example of the complete Legendre transformation is the Hodograph transformation applying to solve the equations for one-dimensional isoentropycal gas flows. In this paper it is shown that equation for the Lagrangian invariant after applying the incomplete Legendre transformation and introducing the stream function transforms into the linear eqation can be resolved by means of the generating function introduction. This method is turned to be effective for solving the inviscid twodimensional Prandtl equation (this equation describes the boundary layer behavior) that allows one to integrate this equation completely. In the case of the constant pressure along the boundary the parallel velocity component represents the Lagrangian invariant. The obtained solution is written through the initial data and satisfies the non-penetrate boundary condition. Analysis of this solution shows the formation of the singularity for the velocity gradient on the wall. This singularity appears as the result of breaking. At the breaking point the velocity gradient tends to infinity according to the power ~(t0–t)-1 where t0 is the singular time. This solution describes the appearance of the folding type singularity. It is shown also that the Prandtl equation admits complete integration for arbitrary dependence of pressure on the longitudinal coordinate. The simplest solution is written for the case of the constant pressure gradient. For the Boussinesq system the equation for density can be resolved by this method that reduces to one equation for the generating function. This work was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

Higher Order Hamiltonian Systems with Generalized Legendre Transformation

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition.