This text presents some basic notions in symplectic geometry, Poisson geometry, Hamiltonian systems, Lie algebras and Lie groups actions on symplectic or Poisson manifolds, momentum maps and their use for the reduction of Hamiltonian systems. It should be accessible to readers with a general knowledge of basic notions in differential geometry. Full proofs of many results are provided.

We classify, under affine equivalence, the quadratic Hamilton-Poisson systems on the Lie-Poisson space $\mathfrak{so}^*_{-}(3)$. For the simplest strictly inhomogeneous quadratic system, we find explicit expressions for the integral curves in terms of Jacobi elliptic functions.

We classify homogeneous positive semidefinite quadratic Hamilton-Poisson systems on a certain subclass of three-dimensional Lie-Poisson spaces.

Here we use an extension of Rodrigues' vector parameter construction for pseudo-rotations in order to obtain explicit formulae for the generalized Euler decomposition with arbitrary axes for the structure groups in the classical models of hyperbolic geometry. Although the construction is projected from the universal cover $\,\mathsf{SU}(1,1)\simeq\mathsf{SL}(2,\mathbb{R})$, most attention is paid to the 2+1 Minkowski space model, following the close analogy with the Euclidean case, and various decompositions of the restricted Lorentz group $\mathsf{SO}^+(2,1)$ are investigated in detail. At the end we propose some possible applications in special relativity and scattering theory.

Lie symmetries of $K(m,n)$ equations with time-dependent coefficients are classified. Group classification is presented up to widest possible equivalence groups, the usual equivalence group of the whole class for the general case and the conditional equivalence groups for special values of the exponents $m$ and $n$. Examples on reduction of $K(m,n)$ equations (with initial and boundary conditions to nonlinear ordinary differential equations (with initial conditions) are presented.

The notion of the Jacobian group of graph (also known as Picard group, sandpile group, critical group) was independently given by many authors. This is a very important algebraic invariant of a finite graph. In particular, the order of the Jacobian group coincides with the number of spanning trees for a graph. The latter number is known for the simplest families of graphs such as Wheel, Fan, Prism, Ladder and Möbius ladder graphs. At the same time the structure of the Jacobian group is known only in several cases. The aim of this paper is to determine the structure of the Jacobian group of the Möbius ladder and Prism graphs.

The curved Josephson junction is described. In the framework of the Maxwell equations the equation that describes the influence of the curvature on the fluxion motion was obtained. The method of geometrical reduction of the sine-Gordon model from three to lower dimensional manifold was applied to the long Josephson junction. It was argued that the geometrical reduction describes the junctions with slowly varying curvatures.

Approximate solution for the Dirac equation with the $q$-deformed Manning-Rosen potential, under the condition of spin and pseudospin symmetry are obtained. Also the energy spectrum and wave functions are obtained by the Nikiforov-Uvarov (NU) method. The special cases $q=1$, Hulthén potential $ (b\rightarrow0)$ and the nonrelativistic limit are studied for the $q$-deformed Manning-Rosen potential, and then results are compared with the other works.

For a completely integrable system, the way of finding first integrals is not formulated in general. A new characterization for integrable systems using the particular tensor field is investigated which is called a recursion operator. A recursion operator $T$ for a vector field $\Delta$ is a diagonizable $(1, 1)$-type tensor field, invariant under $\Delta$ and has vanishing Nijenhuis torsion. One of the important property of $T$ is that $T$ gives constants of the motion (the sequence of first integrals) for the vector field $\Delta$. The purpose of this paper is to discuss a recursion operator $T$ for the geodesic flow on $S^n$.

Starting from the Moyal product on eight-dimensional canonical Euclidean phase space $T^* \mathbb{R}^4$ with an $S^1$-symplectic action, we construct a non-formal star product, i.e., the deformation parameter is a real number, on the cotangent bundle of three-dimensional Euclidean space except the origin $T^* (\mathbb{R}^3\backslash\{\rm 0\})$ which is the reduced symplectic manifold by the $S^1$-action.

Since the last two decades, several differential operators appeared in connection with the so-called oscillatory geometry. These operators act on sections of infinite rank vector bundles. Definitions of the oscillatory representation, metaplectic structure, oscillatory Dirac operator, as well as some necessary fundamental results in the analysis in $C^*$-Hilbert bundles are recalled here. These results are used for a description of the kernel of a certain second order differential operator arising from oscillatory geometry and the cohomology groups of the de Rham complex of exterior forms with values in the oscillatory representation.

The Kepler problem for planetary motion is a two-body dynamic model with an attractive force obeying the inverse square law, and has a direct analogue in any dimension. While the magnetized Kepler problems were discovered in the late 1960s, it is not clear until recently that their higher dimensional analogues can exist at all. Here we present a possible route leading to the discovery of these high dimensional magnetized models.

Here we discus the diffusion of particles in a curved tube. This kind of transport phenomenon is observed in biological cells and porous media. To solve such a problem, we discuss the three dimensional diffusion equation with a confining wall forming a thinner tube. We find that the curvature appears in a effective diffusion coefficient for such a quasi-one-dimensional system. As an application to higher dimensional case, we discuss the diffusion in a curved surface with thickness. In this case the diffusion coefficient changes to the tensor form depending on the mean and Gaussian curvatures. Then the diffusion flow can be interpreted as usual flow plus anomalous flow. The anomalous flow shows not only the diffusion but also the concentration depending on mean and Gaussian curvatures, and also it includes the flow proportional to the gradient of Gaussian curvature.

Discussed is the problem of the mutual relationship of differentially first-order and second-order field theories and quantum-mechanical concepts. We show that unlike the real history of physics, the theories with algebraically second-order Lagrangians are primary, and in any case more adequate. It is shown that in principle, the primary Schrödinger idea about Lagrangians which are quadratic in derivatives, and leading to second-order differential equations, is not only acceptable, but just it opens some new perspective in field theory. This has to do with using the Lorentz-conformal or rather its universal covering SU(2,2) as a gauge group. This has also some influence on the theory of defects in continua.

In this paper we obtain the ladder operators for the $1D$ and $3D$ Morse potential. Then we show that these operators satisfy SU(2) commutation relation. Finally we obtain the Hamiltonian in terms of the $\frak{su}(2)$ algebra.

The (1,1)-tensor field on symplectic manifold that satisfies some integrability conditions is called a recursion operator. It is known the recursion operator is a characterization for integrable systems, and gives constants of motion for integrable systems. We construct recursion operators for the geodesic flows of some solutions of Einstein equation like Schwarzschild, Reissner-Nordström, Kerr and Kerr-Newman metrics.

Relativistic hyperbolic geometry is a model of the hyperbolic geometry of Lobachevsky and Bolyai in which Einstein addition of relativistically admissible velocities plays the role of vector addition. The adaptation of barycentric coordinates for use in relativistic hyperbolic geometry results in the relativistic barycentric coordinates. The latter are covariant with respect to the Lorentz transformation group just as the former are covariant with respect to the Galilei transformation group. Furthermore, the latter give rise to hyperbolically convex sets just as the former give rise to convex sets in Euclidean geometry. Convexity considerations are important in non-relativistic quantum mechanics where mixed states are positive barycentric combinations of pure states and where barycentric coordinates are interpreted as probabilities. In order to set the stage for its application in the geometry of relativistic quantum states, the notion of the relativistic barycentric coordinates that relativistic hyperbolic geometry admits is studied.

Newton-Lorentz equations describing the motion of charged particles in the equatorial plane of a magnetic dipole field are considered. The parametric equations of the trajectories of the particles are obtained explicitly in terms of Jacobi elliptic functions and elliptic integrals.

We consider the hierarchies of Nonlinear Evolution Equations related to auxiliary problem of Caudrey-Beals-Coifman type. We give a proof that the conservation laws for these equations have local densities based on the theory of the generating operators related to the Caudrey-Beals-Coifman linear problem.

Asymmetric heavenly equation, presented in a two-component form, is known to be 3+1-dimensional bi-Hamiltonian system. We show that symmetry reduction of this equation yields a new two component 2+1-dimensional integrable bi-Hamiltonian system. We prove that this new 2+1-dimensional system admits bi-Hamiltonian structure, so that it is integrable according to Magri's theorem.