The principle of virtual work and Hamilton's principle on Galilean manifolds (Q2230356)

In the modern theory of the mechanical systems the description based on manifolds is intensively used (see, for instance [\textit{T. Lee} et al., Global formulations of Lagrangian and Hamiltonian dynamics on manifolds. A geometric approach to modeling and analysis. Cham: Springer (2018; Zbl 1381.70005)]). Such geometric formulation is an effective tool for formulation and analyzing the kinematics, dynamics, and their temporal evolution on the configuration manifold. The configuration is often described and analyzed globally and does not require the use of local charts, coordinates or parameters. It often prevents from appearing of some unnecessary singularities or ambiguities during the analysis. This so called coordinate-free approach is used by authors in the paper. They described coordinate-free theory of the time-dependent finite-dimensional mechanical systems with \(n\) degrees of freedom on \(n\)-dimensional configuration manifold and time as an additional dimension, which obviously needs to be included in the case of time-dependent system to obtain the corresponding physical results. Due to difficulties in physical interpretation of the results when time is separate from the base space, as a convenient space for providing suitable analysis the Galilean manifold is chosen. It is already \((n+1)\)-dimensional manifold with an included time structure and a defined metric named Galilean metric, which additionally allows to measure the inertia of the mechanical system. As a consequence, the action differential two-form can be defined in a natural way. It opens the door to formulate Lagrange's and Hamiltonian theory in the frame of this coordinate-free approach. In the present paper the authors continue their previous work [\textit{S. R. Eugster} et al., Math. Mech. Solids 25, No. 11, 2050--2075 (2020; Zbl 1485.70014)] and apply Galilean manifold approach to the principle of virtual work and Hamilton's principle. To do that they introduce three different but related action functionals on corresponding sets of curves and by means of the variational calculus technique state the Hamilton's principle within the Galilean manifold approach. The paper includes introduction with all necessary definitions and concepts, e.i. Galilean manifold, metric, action forms, kinetic and potential energy, forces and so on in the frame of the approach. In subsection 10 the fundamental postulate for the description of the dynamics of time-dependent finite-dimensional mechanical systems, which is not based on the calculus of variations, is formulated. In the next subsections the latter allows to establish the principle of virtual work and to find equations of motion in the Hamilton's form. The paper ended with the study of three different types of virtual displacement fields which led to the formulation of three versions of Hamilton's principle. These three principles are traced back to six versions of the principle of Hamilton found in classical mechanics.