The major goals of this paper are to address the need of development and application of predictive high fidelity fundamental concepts to analyze, model, and simulate nanoelectromechanical structures at the atomic level. The solution of these problems is critical for designing new materials with improved desired performance and required effects (developing advanced communication, electro-optical, memory, optoelectronic, processing, and storage materials and nanostructures), understanding, prediction and control the reactivity and energy flow in molecules, designing high-energy-density molecules, structural optimization and stability analysis of molecular systems, prediction and modeling of the atomic interactions, control of deposition and growth, calculation of properties of bulk materials from atomistic considerations, etc. Our emphasis is to combine engineering, science, and technology to develop the fundamental aspects of nanoelectomechanics for nanoscale structures. This will form the basis of a nanoelectromechanical theory to achieve high-fidelity modeling and analysis, as well as design, optimization, and synthesis of nanoelectromechanical systems (NEMS). The ability to find mathematical models which adequately describe nanostructure properties, phenomena and effects, is a key problem in modeling, analysis, synthesis, optimization, control, fabrication, and manufacturing of NEMS. In this paper, using electromagnetics, classical and quantum mechanics, we develop nanoelectromechanical theory to model, analyze, and simulate nanoscale structures. The reported fundamental results allow the designer to attack a broad spectrum of problems which cannot be solved applying currently existing methods. The reported theoretical and applied results are verified to demonstrated.
To perform the comprehensive modeling, analysis, and simulation of nanostructures in the time domain, there is a critical need to develop and apply advanced theories using fundamental physical laws. Due to the analytical and numerical complexity, conventional mechanics was widely used to study nanostructures. However, there is a critical need to model basic phenomena, and quantum effects must be integrated. The quantum mechanics gives the system evolution in the form of the Schrödinger equations. Classical and quantum mechanics are widely used, and this paper illustrates that the Schrödinger equation can be derived using Hamilton's concept (it is well known that the Euler-Lagrange equations, given in terms of the generalized coordinates and forces, can be straightforwardly derived applying the variational principle). Furthermore, it is shown that the solution of the Schrödinger equation can be found using the optimal cost function rather than the wavefunction. This avenue leads to meaningful analytical and numerical advantages because the complexity of the Schrödinger equation does not allow one to solve a wide array of problems even for simple atomic structures. In addition, the bridge is built between conventional and quantum mechanics. This paper illustrates the applicability of the Hamilton concept to solve a wide range of important problems for nanoscale structures mapping all essential features and quantum phenomena.