Foundations of Analytical Mechanics

In this article, it is aimed to introduce the Euler-Lagrange equations using a three-dimensional space for mechanical systems. In addition to, the geometrical-physical results related to three-dimensional space for mechanical systems are also given.

This paper presents a non-minimal order dynamics model for many analysis, simulation, and control problems of constrained mechanical systems passing through singular configurations during their motion by making use of linear projection operator. The distinct features of this model describing dynamics of the dependent coordinates are: i) The mass matrix M (q) is always positive definite even at singular configurations; ii) matrix  (d/dt) Mc −2  Cc is skew symmetric, where all nonlinear terms are lumped into vector  Cc(q,  dot q) dot q after elimination of constraint forces. Eigenvalue analysis shows that the condition number of the constraint mass matrix can be minimized upon adequate selection of a scalar parameter called " virtual mass " thereby reducing the sensitivity to round-off errors in numerical computation. It follows by derivation of two oblique projection matrices for computation of constraint forces and actuation forces. It is shown that projection-based model allows feedback control of dependent coordinates which, unlike reduced-order dependent coordinates, uniquely define spatial configuration of constrained systems.

The main purpose of the present paper is to study almost complex structures Euler-Lagrangian Equations
on Walker 4-manifold with Walker metric. In this study, routes of bodies moving in space will be modeled
mathematically Euler-Lagrangian Equations on Walker 4-manifold with Walker metric that these are time-
dependent partial di¤erential equations. Here we present complex analogues of Euler-Lagrangian Equations
on Walker 4-manifold with Walker metric. Also, the geometrical-physical results related to complex mechan-
ical systems are also discussed for Euler-Lagrangian Equations on Walker 4-manifold with Walker metric
for dynamical systems and solution of the motion equations using symbolic computational program will be
made.

Espindola, Maria L.
Equações Diferenciais Parciais de Primeira Ordem - São Carlos, SP :
SBMAC, 2014, 68 p., 21.5 cm - (Notas em Matemática Aplicada; v. 72)

e-ISBN 978-85-8215-058-0
1. EDPs de primeira ordem
2. EDPs de primeira ordem não ineares 
3. Solução Geral de EDPs de primeira ordem
4. Transformada de Legendre
5. Formas diferenciais Pfaffianas
6. Hamiltonização Alternativa

I. Espindola, Maria L. IV. Título. V. Série

A new procedure named direct Hamiltonization gives another foundations to Analytical Mechanics, since in this formalism 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 are established as a consequence of a canonical description of the
mechanical system. The direct Hamiltonization is a generalization of the alternative one, where the usual Hamiltonization and momenta is recovered whenever it exists. Also this procedure assures the existence of a Hamiltonian function without any constraints for any mechanical system,
therefore the usual quantization is always allowed. This procedure can be applied to non Lagrangian, Nambu, non holonomic and dynamical systems since there are no restriction in this formalism as, for example, the number of equations of motion.

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.

The paper aims to introduce Lagrangian and Hamiltonian formalism for mechanical systems using para/pseudo-K¨ahler manifolds, representing an interesting multidisciplinary field of research. Moreover, the geometrical, relativistical, mechanical and physical results related to para/pseudo-K¨ahler mechanical systems are given, too.

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.