Symmetries and Related Physical Balances for Discontinuous Flow Phenomena within the Framework of Lagrange Formalism

By rigorous analysis, it is proven that from discontinuous Lagrangians, which are invariant with respect to the Galilean group, Rankine–Hugoniot conditions for propagating discontinuities can be derived via a straight forward procedure that can be considered an extension of Noether’s theorem. The use of this general procedure is demonstrated in particular for a Lagrangian for viscous flow, reproducing the well known Rankine–Hugoniot conditions for shock waves.

Bateman Oscillators: Caldirola-Kanai and Null Lagrangians and Gauge Functions

The Lagrange formalism is developed for Bateman oscillators, which includes both damped and amplified systems, and a novel method to derive the Caldirola-Kanai and null Lagrangians is presented. For the null Lagrangians, the corresponding gauge functions are obtained. It is shown that the gauge functions can be used to convert the undriven Bateman oscillators into the driven ones. Applications of the obtained results to quantizatation of the Bateman oscillators are briefly discussed.

Lagrange formalism in field theory

This chapter is devoted to a general discussion of classical field theory. It presents the minimum information required about classical fields for the subsequent treatment of quantum theory in the rest of the book. The Lagrange formalism for the fields is introduced, based on the least action principle. Global symmetries are described, and the proof of Noether's theorem given. In addition, the energy-momentum tensor for a field system is constructed as an example.

The Relativistic Harmonic Oscillator in a Uniform Gravitational Field

We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach based on the Euler–Lagrange formalism. Due to the conceptual differences of mass in the framework of special relativity compared with the classical model, the correct treatment of the relativistic gravitational potential requires special attention. It is proved that the corresponding relativistic equation of motion has unique periodic solutions. Some approximate analytical results including the next-to-leading-order term in the non-relativistic limit are also examined. The discussion is rounded up with a numerical simulation of the full relativistic results in the case of a strong gravity field. Finally, the dynamics of the model is further explored by investigating phase space and its quantitative relativistic features.

Robotic device for automation of the control of the status of cables in the mining industry

In this work, we propose a design of a robotic device designed to automate the process of nondestructive testing of wire rope equipment. A schematic hydraulic diagram of a robotic device with a description of the logic of work of the system is given. A mathematical model of the dynamics of the system is obtained using the Lagrange formalism. An algorithm for evaluating the control moments necessary for the implementation of programmed motion has been developed. Numerical simulation of the motion corresponding to the mode of approaching the magnetic flaw detector heads to the steel rope has been carried out. The graphs of the traction-speed characteristics and load diagrams of the drives providing the programmed movement of the links of the pantograph mechanism are presented. The results presented in this paper can be used to calculate the parameters of the power section of the electro-hydraulic drives included in the robotic device.

Qualitative Investigation of Hamiltonian Systems by Application of Skew-Symmetric Differential Forms

In the present paper, a role of Hamiltonian systems in mathematical and physical formalisms is considered with the help of skew-symmetric differential forms. In classical mechanics the Hamiltonian system is realized from the Euler–Lagrange equation as the integrability condition of the Euler-Lagrange equation and discloses specific features of Lagrange formalism. In the theory of differential equations, the Hamiltonian systems reveals canonical relations that define the integrability conditions of differential equations. The Hamiltonian systems, as a self-independent equations, are an example of dynamic systems that describe a behavior of dynamical systems in phase space. The connection of the Hamiltonian systems with differential equations and dynamical systems point to the fact that dynamical systems can be generated by differential equations. Under the investigation of Hamiltonian systems, in addition to exterior skew-symmetric differential forms it is suggested to use the skew-symmetric differential forms that are defined on a nonintegrable manifolds and possess a nontraditional mathematical apparatus, such as degenerate transformations and transitions from nonintegrable manifold to integral structures.

Design of the state feedback-based feed-forward controller asymptotically stabilizing the double-pendulum-type overhead cranes with time-varying hoisting rope length

In this paper we focus our attention on the design of the feedback-based feed-forward controller asymptotically stabilizing the double-pendulum-type (D-P-T) crane system with the time-varying rope length in the desired end position of payload (the origin of the coordinate system). In principle, two cases are considered, in the first case, the sway angle of payload is uncontrolled and second case, when the sway angle of payload is controlled by an external force. Precise mathematical modeling in the framework of Lagrange formalism without the traditional neglect of the important structural parameters of the D-P-T crane system and numerical simulation in the Matlab environment indicate the substantial reduction of the transportation time to the desired end position.