The General Problem of The Motion of Coupled Rigid Bodies About a Fixed Point

This book focuses on the way in which the problem of the motion of bodies has been viewed and approached over the course of human history. It is not another traditional history of mechanics but rather aims to enable the reader to fully understand the deeper ideas that inspired men, first in attempting to understand the mechanisms of motion and then in formulating theories with predictive as well as explanatory value.

Newton interprets 2D layers of the current composition as rigid bodies in a physical world. These bodies can collide with each other, slide, bounce, and react to gravity. Bodies can also be connected to each other using joints. Newton can deal with various types of bodies: some types allow bodies to be animated in After Effects while reacting to physical forces, others are entirely controlled by the solver. Newton also provides a simple, clean, and easy to use interface, offering fast OpenGL preview and intuitive controls. Once simulation is completed, Newton export the animation to After Effects using standard keyframes.

The aim of the present book is to give a systematic treatment of the inverse problem of the calculus of variations, i.e. how to recognize whether a system of differential equations can be treated as a system for extremals of a variational functional (the Euler-Lagrange equations), using contemporary geometric methods. Selected applications in geometry, physics, optimal control, and general relativity are also considered. The book includes the following chapters: - Helmholtz conditions and the method of controlled Lagrangians (Bloch, Krupka, Zenkov) - The Sonin-Douglas's problem (Krupka) - Inverse variational problem and symmetry in action: The Ostrogradskyj relativistic third order dynamics (Matsyuk.) - Source forms and their variational completion (Voicu) - First-order variational sequences and the inverse problem of the calculus of variations (Urban, Volna) - The inverse problem of the calculus of variations on Grassmann fibrations (Urban).

The shadowing function is an important element in the simulation and calculation of how electromagnetic waves scatter at randomly rough surfaces. Its derivation and use is an active, interdisciplinary area of research with practical applications in fields such as radar, optics, acoustics, geoscience, computer graphics and remote sensing This book addresses the general problem of the derivation of the shadowing function from randomly rough surfaces.