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is a modern scientific approach to . Its has been developed in the work of the present authors (see [Iva91, ILI95, IS01, IP01b, IP01a, Iva02, Iva04, Iva05, IB05, PI03, PI04]). The dynamics of human motion is extremely complex, multi–dimensional, highly nonlinear and hierarchical. Human skeleton has more than two hundred rigid bones, connected by rotational joints, witch have up to three axes of rotation. Nevertheless, in classical biomechanics the main analytical tool was (see Figure 1.1). The skeleton is driven by a synergistic action of its 640 skeletal muscles. Each of these muscles has its own and dynamics, in which neural action potentials are transformed into muscular force vectors (see [Hat77a, Hat77b, Hat78]).

The core of geometrodynamics is the concept of the , the stage where our , = ,works. To get some dynamical feeling before we dive into more serious geometry, let us consider a simple 3DOF biomechanical system (e.g., a representative point of the center of mass of the human body) determined by three = . There is a unique way to represent this system as a 3D manifold, such that to each point of the manifold there corresponds a definite configuration of the biomechanical system with coordinates ; therefore, we have a geometric representation of the configurations of our biomechanical system. For this reason, the manifold is called the . If the biomechanical system moves in any way, its coordinates are given as the functions of the time. Thus, the motion is given by equations of the form: = (). As varies we observe that the system’s in the configuration manifold describes a and = ()are the equations of this curve.

This Chapter studies various aspects of modern mechanics as is currently used in biomechanics. It includes both Lagrangian and Hamiltonian variations on the central theme of our , = . We start with the basics of Lagrangian and Hamiltonian formalisms. After that we move on to the general variational principles of holonomic mechanics. Next we depart to nonholonomic. At the end, we present the current research in biomechanics given in the framework of Lie–Lagrangian and Lie–Hamiltonian functors.

In this Chapter we develop the basics of algebraic topology as is used in modern biomechanics. It includes both tangent (Lagrangian) and cotangent (Hamiltonian) topological variations on the central theme of our , = .

In this Chapter we develop the basics of nonlinear control theory as is used in modern human–like biomechanics. It includes control variations on the central theme of our , = , and its associated : (see section 2.7 above).

In this Chapter we develop covariant biophysics of electro-muscular stimulation, as an externally induced generator of our , = . The so–called (FES) of human skeletal muscles is used in rehabilitation and in medical orthotics to externally stimulate the muscles with damaged neural control (see, e.g., [VHI87]). However, the repetitive use of electro–muscular stimulation, besides functional, causes also structural changes in the stimulated muscles, giving the physiological effect of muscular training.