Kinematics, energetics, and design of single and multi-body systems: With applications to non-commutative rotations and protein models

Kinematics, energetics, and design of single and multi-body systems are explored in this thesis through the examples of non-commutative rotations and protein models.; In the first part of the thesis, kinematics of a single body is focused on the analysis and the synthesis of its fully reversed (FR) sequence of rotations. An FR sequence of rotations is defined as a series of rotations of a free rigid body about its body-fixed axes such that the rotations about every axis are fully reversed at the end of the sequence. A feasibility of the FR sequence of rotations is analyzed and methods for its synthesis are developed.; In the second part, the design of multi-body systems found in protein molecules is investigated based on the energetics of their reduced models. Deterministic optimization methods are used for the design of protein namely models based on the continuous modeling of the discrete combinatorial space of sequences. The continuous modeling and the notion of optimization enable the application of efficient continuous optimization algorithms leading to drastic reduction in the computational cost of protein sequence design problem. The new optimal design methodology is first implemented using lattice models of proteins with only two monomer types viz., hydrophobic (H) and polar (P), and later extended to handle multiple monomer types in real proteins. Effective initial guess for the optimization problem are obtained using a spectral graph analysis and a new quadratic programming formulation of the sequence design problem. The latter is capable of determining a lower bound of the energy for a real protein for a given conformation and inter-residue energies.