The principles of geometric mechanics are extended to the physical elements of mechanics, including space and time, rigid bodies, constraints, forces, and dynamics. What is arrived at is a comprehensive and rigorous presentation of basic mechanics, starting with precise formulations of the physical axioms. A few components of the presentation are novel. One is a mathematical presentation of force and torque, providing certain well-known, but seldom clearly exposited, fundamental theorems about force and torque. The classical principles of Virtual Work and Lagrange-d'Alembert are also given clear mathematical statements in various guises and contexts. Another novel facet of the presentation is its derivation of the Euler-Lagrange equations. Standard derivations of the Euler-Lagrange equations from the equations of motion for Newtonian mechanics are typically done for interconnections of particles. Here this is carried out in a coordinate-free rmner for rigid bodies, giving for the first time a direct geometric path from the Newton-Euler equations to the Euler-Lagrange equations in the rigid body setting.

https://www.aimsciences.org/article/exportPdf?id=5522bb4e-fd68-47c9-a3ea-d85aaa285e92

The physical foundations of geometric mechanics

There are various ways for introducing the concept of the instantaneous angular velocity vector. In this paper we propose a method based on introducing of this concept by construction of the solution for the system of kinematic equations. These equations connect the function vectors defining the motion of the basis, and their derivatives. Necessary and sufficient conditions for the existence and uniqueness of the solution of this system are established. The instantaneous angular velocity vector is a solution of the algebraic system of equations. It is built explicitly. The derived formulas for the angular velocity vector generalize the earlier results, both for a basis of an affine oblique coordinate system and for an orthonormal basis.

The generalized formula for angular velocity vector of the moving coordinate system

AbstractIn this paper, we consider a Cartesian moving reference frame. Its angular velocity vector is introduced as a solution to a system of kinematic equations of basis vectors. These equations connect the position of the basis vectors with their velocity. The construction of a formula for the angular velocity vector of an orthonormal basis is described. It is shown that the angular velocity vector in the found form is a solution to the system of the equations. Using transformations of the constructed solution, four more representation forms of the angular velocity vector are derived. It is shown that all the obtained forms define the same angular velocity vector of the moving space, though they contain different elements. All of the forms are also solutions of the system of kinematic equations. Presented results can be applied both to a solid body and to any rigid system. 

Representation Forms of the Angular Velocity Vector for an Orthonormal Basis of a Moving Frame

Recent articles in this journal have proposed that certain rather specialised mathematical formalisms (quaternions, or alternatively dyads) should be employed in the derivation of the fundamental equations of rigid-body kinematics. Here, to the contrary, the merits are argued of a description based simply on the properties of rotation matrices. One particular advantage of such a mainstream approach is that the construction of the algebra of angular velocity vectors from the group of rotation matrices is a basic example of a general construction, that of a Lie algebra from a Lie group. It therefore offers the possibility of making the acquaintance, in a familiar context, of ideas which are well known to be of major importance in several branches of theoretical physics. The distinctive features of the rotation group case, as well as the pointers it gives to general theory, are identified in the course of a full, but it is hoped not too technical, discussion of angular velocity in this vein.

http://iopscience.iop.org/article/10.1088/0143-0807/7/4/014/pdf

On the concept of angular velocity

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On the concept of angular velocity