On Santilli's methods in Birkhoffian inverse problem

The order reduction method of the relativistic Birkhoffian equations is studied. For a relativistic autonomous Birkhoffian system, if the conservative law of the Birkhoffian holds, the conservative quantity can be called the generalized energy integral. Through the generalized energy integral, the order of the system can be reduced. If the relativistic Birkhoffian system has a generalized energy integral, then the Birkhoffian equations can be reduced by at least two degrees and the Birkhoffian form can be kept. The relations among the relativistic Birkhoffian mechanics, the relativistic Hamiltonian mechanics and the relativistic Lagrangian mechanics are discussed, and the Whittaker order reduction method of the relativistic Lagrangian system is obtained. And an example is given to illustrate the application of the result.

We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Λ-symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence of first integrals, we recall the definition and the properties of Λ-symmetries. We show that in the presence of a Λ-symmetry for the Hamiltonian equations, one can introduce the notion of "Λ-constant of motion". The presence of a Λ-symmetry leads also to a nice and useful reduction of the form of the equations. We then consider the case in which the Hamiltonian problem is deduced from a Λ-invariant Lagrangian. We illustrate how the Lagrangian Λ-invariance is transferred into the Hamiltonian context and show that the Hamiltonian equations are Λ-symmetric. We also compare the "partial" (Lagrangian) reduction of the Euler–Lagrange equations with the reduction which can be obtained for the Hamiltonian equations. Several examples illustrate and clarify the various situations.

The method of integrating factors is used to study the conservation laws of the Herglotz type Birkhoffian systems in this paper. Firstly, the definition of the integrating factors of the Herglotz type Birkhoffian systems is given. Secondly, the relationship between the integrating factors and conservation laws is studied, and the conservation theorems of Herglotz type Birkhoff's equations and their inverse theorems are established. Thirdly, two types of generalized Killing equations for calculating integrating factors are given. Finally, as an example, a linear damped oscillator is taken. This example can be transformed into a Herglotz type Birkhoffian system. The resulting conservation theorems are used to find the conserved quantities for this example.

The fractional generalization of Hamiltonian mechanics is constructed by using the Lagrangian involving fractional derivatives. In this paper the equation of projectile motion with air friction using fractional Hamiltonian mechanics and equation for current loop involving electric source, a resistor, an inductor and a capacitor has been obtained. Furthermore, fractional optics has been introduced.

In the optimal control theory, the Hamiltonian formalism is a famous one to find an optimal solution. However, when the performance index is complicated or for a degenerate case with a non-convexity of the Hamiltonian function with respect to the control force the Hamiltonian method does not work to find the solution. In this paper we will address this important issue via a quite different approach, which uses the optimal control problem of nonlinear Duffing oscillator as a demonstrative example. The optimally controlled vibration problem of nonlinear oscillator is recast into a nonlinear inverse problem by identifying the unknown heat source in a nonlinear parabolic partial differential equation (PDE). Then through a semi-discretization of the resultant PDE, the inverse problem is further reformulated to be a system of n-dimensional ODEs with n unknown pointwise sources, which allows a Lie-group adaptive method (LGAM) to recover the point-wise sources. The present method has three-fold advantages: it can easily minimize a complicated performance index to find an optimal control force of the nonlinear vibration system, it is effective for highly nonlinear optimal control problem, and it does not resort on the classical Hamiltonian formulation, which provides only a necessary condition, but not a sufficient condition, for the optimality of the control law. Numerical examples show that the LGAM may find a better performance than the classical one.

The formulas and constructive methods for solving inverse problems of source determination correctness of certain inverse problems numerical solution of direct and inverse problems. (Part contents).

A cross section of the rod is taken as object of investigation. The freedom of the section in free or constraint case is analyzed and the definition of virtual displacement of the section is given, which can be expressed by a variational operation. Assuming the variational and partial differential operations has commutativity, based on the hypothesis about surface constraint subjected to the rod, the freedom of the section on constraint surface is discussed and the equations satisfied by virtual displacements of the section are given. Combining D'Alembert principle and the principle of virtual work, D'Alembert-Lagrange principle is established. When constitutive equation of material of the rod is linear, the principle can be transformed to Euler-Lagrange form. From the principle, a dynamical equation in various forms such as Kirchhoff, Lagrange, Nielsen and Appell equation can be derived. For the case when a rod is subjected to a surface or a nonholonomic constraint, Lagrange equation with undetermined multipliers is obtained. Integral variational principle of dynamics of a super-thin elastic rod is also established, from which Hamilton principle formulation is obtained when the material of the rod is linear. Finally, canonical variables to describe the state of the section and Hamilton function are defined, and Hamilton canonical equation is derived. The analytical methods of dynamical modeling of a super-thin elastic rod have been constructed, which can serve as a theoretical framework of analytical dynamics of a super-thin elastic rod with two independent variables.

This paper presents an inverse problem in analytical dynamics. The inverse problem is to construct the Lagrangian when the integrals of a system are given. Firstly, the differential equations are obtained by using the time derivative of the integrals. Secondly, the differential equations can be written in the Lagrange equations under certain conditions and the Lagrangian can be obtained. Finally, two examples are given to illustrate the application of the result.

This study investigates the backward problem of a time-fractional diffusion-wave equation (TFDWE) within a cylindrical domain, confirming the inherent ill-posedness and conditional stability of this inverse problem. To address these challenges, we introduce an iterative weighted Tikhonov regularization technique (IWTRT), which stabilizes the solution process by incorporating constraints and weights into the framework. We rigorously derive the convergence rates for the regularized solution using both a-priori and a-posteriori strategies for selecting the regularization parameter, ensuring a comprehensive understanding of the method’s performance in various scenarios. Numerical experiments validate the method’s effectiveness, demonstrating its ability to accurately reconstruct the initial condition from noisy data. The a-posteriori strategy of the IWTRT, in particular, outperforms the a-priori approach in terms of accuracy and noise resilience, as evidenced by the experimental results. Our findings contribute to the field of fractional calculus, offering a robust framework for addressing inverse problems characterized by fractional dynamics. This work also provides a systematic approach for parameter selection, paving the way for future research and practical applications in engineering, physics, and applied mathematics.

Noether's theorems for nonshifted dynamic systems on time scales

Lagrangian formulation of some q-deformed systems

This work is primarily concerned with various aspects of Lagrangian and Hamiltonian mechanics. These different aspects are related in a somewhat complicated way, which is clarified by the introduction of the unifying concept of a jet bundle. Here, bundles are first considered in some generality and then jet bundles are introduced and their bundle structure investigated especially with a view to formulating Lagrangian and Hamiltonian mechanics invariantly. The second chapter is devoted to a discussion of the two Schouten brackets and it is explained why each is of importance in mechanics. The symmetric Schouten bracket enables the notion of a killing tensor to be defined on a Riemannian or pseudo-Riemannian manifold. A theorem is proved which gives an upper bound on the dimension of the space of killing tensors of a fixed rank. In chapter 3 Hamilton-Jacobi theory and a version of Noether's theorem are presented from a m o d e m viewpoint. Then conditions are obtained which entail the existence of a constant of motion which is polynomial in momenta. These conditions are used to construct several classical Hamiltonians with hidden symmetries - the usage of the latter term is briefly justified. Most importantly, all systems with two degrees of freedom which have a quadratic integral independent of the Hamiltonian are characterized. In chapter 4 the geometrical properties of TM and the closely related space J1 (IR,M) are investigated. It is shown that J1(IR,M) has an intrinsically defined 1-1 tensor in analogy to TM. The behaviour of these tensors under diffeomorphisms is investigated. The chapter concludes with a discussion of the various notions of symmetry in Lagrangian theory. Chapter 5 begins with a review of symplectic geometry and continues with the definition and examination of contact manifolds in the same spirit as symplectic geometry. In particular, contact diffeomorphisms are described on the space J1(IR,M). Finally, two theorems are given which endeavour to explain the general principle enabling the Lie algebra of a subalgebra of vector fields to be transferred to a collection of forms.

A broad class of partitioned differential equations with possible algebraic constraints is considered, including Hamiltonian and mechanical systems with holonomic constraints. For mechanical systems a formulation eliminating the Coriolis forces and closely related to the Euler--Lagrange equations is presented. A new class of integrators is defined: the super partitioned additive Runge--Kutta (SPARK) methods. This class is based on the partitioning of the system into different variables and on the splitting of the differential equations into different terms. A linear stability and convergence analysis of these methods is given. SPARK methods allowing the direct preservation of certain properties are characterized. Different structures and invariants are considered: the manifold of constraints, symplecticness, reversibility, contractivity, dilatation, energy, momentum, and quadratic invariants. With respect to linear stability and structure-preservation, the class of s-stage Lobatto IIIA-B-C-C* SPARK methods is of special interest. Controllable numerical damping can be introduced by the use of additional parameters. Some issues related to the implementation of a reversible variable stepsize strategy are discussed.

The problem of reconstructing an unknown electric conductivity from boundary measurements has applications in medical imaging, geophysics, and nondestructive testing. A. Nachman [Ann. of Math. (2), 143 (1996), pp. 71--96.] proved global uniqueness for the two-dimensional inverse conductivity problem using a constructive method of proof. Based on this proof, Siltanen, Mueller, and Isaacson [Inverse Problems, 16 (2000), pp. 681--699] presented a new numerical reconstruction method that solves the nonlinear problem directly without iteration. The method was verified with nonnoisy rotationally symmetric examples. In this paper the method is extended by introducing a new regularization scheme, which is analyzed theoretically and tested on symmetric and nonsymmetric numerical examples containing computer simulated noise.

The main modern construction methods of geological medium depth-velocity models are based on different solutions to the inverse kinematic problem of seismic prospecting. Unfortunately, solutions to the inverse problems are ill-posed. There are many algorithms for approximate solutions to the inverse problems. Its application and results comparison show the nonexistence of uniqueness and existence of structural instability of solutions to the inverse problems. The article presents an original seismic locating technology designed to computer modeling of seismic images with high-resolution and precision. The technology provides an accurate and consistent design of a computer depth-velocity medium model without inverse problems solutions. To eliminate obstacles inherent to traditional seismic images construction approaches, the technology applies radio locating methods ideas. The technology consequently solves three construction problems of computer models of qualitative seismic images. It begins with qualitative primary processing of field materials obtained with the reflected waves method, and continues with receiving computer images of the processing results. Next, it builds a high-resolution and precision model of the reflected waves hodograph series image. The technology completes with computer modeling of the configuration of the interface between the layers of the medium, linked in space to the observation profile. The concept of accurate and high-quality design of a computer image of the interface between two layers of the geological environment model is to find a line passing through an optimally selected set of wave reflection points. The seismic location technology uses algorithmic means providing an optimal location and estimation of the best reflecting point where the propagating wave has a minimal travel time. The virtual locating grid provides the spatial binding to the observation profile. Precision of the binding is achieved by a stochastic modification operation and the use of grid parameters variation.