Foundations of Analytical Mechanics

This paper presents a method for formation control of marine surface craft inspired by Lagrangian mechanics. The desired formation configuration and response of the marine sur- face craft are given as a set of constraint functions. The functions are treated according to constraints in analytical mechanics. Thus, constraint forces arise and feedback from the constraint functions keeps the formation assembled. Since the constraint functions are designed for a desired effect, the forces can be interpreted as con- trol laws. Examples of constraint functions that maintain a forma- tion are presented. Furthermore, the same approach has been ap- plied with no major modification to position control purposes for a single vessel. An extension to underactuated vessels is given. Sim- ulations with nonlinear models of tugboats illustrate the proposed method versatility and its robustness with respect to environmental disturbances and time delays.

A new procedure named direct Hamiltonization gives another foundations to Hamiltonian Analytical Mechanics, since in this formalism the Hamiltonian function can be obtained for any mechanical system. The main change proposed in this procedure is that the conjugate momenta cannot be defined a priori, but instead of this, they are determinate as a consequence of a canonical description of the mechanical system. As the direct Hamiltonization contains the alternative one, then the usual Hamiltonization and momenta is recovered while the envelope solution is selected. Also this procedure assures the existence of a Hamiltonian function without any constraints whatsoever mechanical system is considered, therefore the usual quantization is always allowed.

Historical, physical, and geometrical relations between two different momenta, characterized here as Cartesian and Lagrangian, are explored. Cartesian momentum is determined by the mass tensor, and gives rise to a kinematical geometry. Lagrangian momentum, which is more general, is given by the fiber derivative, and produces a dynamical geometry. This differs from the kinematical in the presence of a velocity-dependent potential. The relation between trajectories and level surfaces in Hamilton-Jacobi theory can also be Cartesian and kinematical or, more generally, Lagrangian and dynamical.

A field-theory Hamiltonization procedure valid for both linear (on \phi_t,) and nonlinear Lagrangians is proposed. It is shown that the usual Hamiltonian formulation is based on an envelope solution of a partíal differential equation. The choice of the complete solution of the same equation leads to an alternative Hamiltonian description of the same field and is the only open possibility for linear Lagrangian systems.

Kinematical transformations are expressed as time independent and time dependent functions of work and energy to be employed in motions of mechanical systems. Relations between the kinematical parameters of moving mechanical systems and energy transformations occurring in them are also considered.

A new procedure named direct Hamiltonization gives another foundations to Analytical Mechanics, since in this formalism of the Hamiltonian Mechanics the Hamiltonian function can be obtained for all mechanical systems. The principal change proposed in this procedure is that the conjugate momenta cannot be defined a priori, but instead of this, they are determinate as a consequence of a canonical description of the mechanical system. As the direct Hamiltonization contains the alternative one, then the usual Hamiltonization and momenta is recovered while the envelope solution is selected. Also this procedure assures the existence of a Hamiltonian function without any constraints for any mechanical system therefore the usual quantization is always allowed.