## 1 Introduction
It is important to point out as well that the PHS formulation for other higher-order equation systems was notably given in [17; 28], thus sustaining the promise that, whenever the equations of a theory can be derived from a variational principle, symplectic geometry [4; 5; 29; 30] can clear up and systematize the relations between the quantities entering into the theory. The readers interested in the PHS formalism can also further refer to the books and thesis disserations [1; 8; 9; 31; 32] for a more general overview of the theory and control applications. **The numerical motivation of this work for PHS. ** Our interest in the PHS formulation greatly relies on the robustness (i.e., long-term stability) and accuracy of the Numerical Symplectic Integration (**NSI**) methods [33; 34; 35; 36; 37; 38], which are concurrent with the Variational Numerical Integration (**VNI**) methods [30; 35; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48] within the Geometric Numerical Integration (**GNI**) methods (also often called structure-preserving numerical methods, as they preserve the internal geometric structure of differential equations) [35; 47; 49; 50; 51]. These numerical integrators or algorithms are known to be clearly superior to other methods over long times of simulation. This is generally attributed to the fact that the underlying geometric structures of the dynamic models have a certain importance that influences the nature of their solutions. More precisely, according to [29; 33; 52] (and the references therein), these important features can be the Hamiltonian structure, namely the Poisson bracket, the energy, and the conserved quantities in the long time computation of conservative systems. Standard numerical integration schemes (such as, for instance, the forward Euler integrator and some time-adaptive Runge-Kutta schemes, unless purposely designed for [37; 53]) naively neglect important special features of the system dynamics (such as the energy and momentum conservation and phase-space volume preservation). These advantages partly motivate the present study, although we do not fully address the numerical aspects in this paper and postpone this task with numerical benchmarking illustrations on the NR shells to a separate communication. **Related variational formulations for the elastodynamics of shells and plates. ** To the best of the authors' knowledge, the PHS formulation of the _linear elastodynamics of the NR shell model_ has not yet been given in the literature, at least in an explicit practical form. To fill this gap for the NR shell model, an idea that we promote and intend to outline below is the specific link that exists between the PHS formulation and the existing mixed (i.e., multiple primary field) variational principles. Indeed, in some other research communities in parallel, there is an equally strong effort to reformulate the classical theory of continuum mechanics in the (generalized) frames of the Hamiltonian/Lagrangian variational formalism and/or Hamiltonian canonical equation system [29; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68]. Many of these works were motivated by the availability of GNI [29; 35; 69], i.e., time-stepping methods designed to exactly satisfy conservation laws, symmetries, or symplectic properties of a system of differential equations. In elastodynamics, in particular, various analytical methods and computational approaches using these formalisms for vibration analysis of (often cylindrical or spherical) shells have been identified and highlighted more recently in excellent review papers such as [57; 70; 71]. However, none have formulated the systems of PHS equations proposed here. As regards the aforementioned 2D shell theories, in general, they are based on _a priori_ behavioral assumptions. These are also often seen as generalizing features of flat plate theories [72], which are viewed as special formulation cases for shells having no curvature. The historical equations of these structural theories are richly discussed in [PERSON]'s book [72] or [PERSON]'s book [73].