## 4 Hamiltonian Variational Formalisms versus Canonical Ones
### Some Remarks on These Variational Principles and Their Numerical Treatments
#### 4.2.1 Advantages and Limitations of the Foregoing Variational Formalisms  
**Limitations**. The multi-primary field principles presented previously are valid beyond the framework of the classical theory of linear elastodynamics under some conditions. In particular, to express the complementary (elastic and kinetic) energy potential densities, it is necessary that the _momentum velocity_ constitutive law and _stress-strain_ constitutive law can be inverted explicitly (even by using the appropriate implicit Legendre-Fenchel transforms in Equations (56) and 72)). When this requirement fails, the foregoing variational formalism can employ as well more multi-primary fields as in the (Chien-)Hu-Washizu (**HW**) and Hamilton-Pontryagin variational formalisms [65; 88; 89; 97; 104; 117; 118; 119; 120; 121; 122; 123; 124; 125]. In such a case, the knowledge gained from the last development and the cited works suggests that the appropriate integral of action would depend on the generalized internal force field \(\overline{\mathbf{F}}^{\rm int}\) (or its impulse \(\overline{\mathbf{F}}^{\rm int}\)), the generalized strain field \(\overline{\mathbf{E}}\), the displacement field \(\mathfrak{q}\), the velocity field \(\mathfrak{v}\sim\dot{\mathfrak{q}}\), and the generalized momentum field \(\boldsymbol{\pi}\). Thus, in order to obtain the canonical equations in Equation (76) from the integral of action of this new multi-field formulation, the generalized internal force field \(\overline{\mathbf{F}}^{\rm int}\) and the generalized momentum \(\boldsymbol{\pi}\) must enter like Lagrange's multipliers [119; 120; 122]; the first field enforces, therefore, the strain-displacement relation in Equation (23), while the second field enforces the velocity-displacement relation \(\mathfrak{q}=\stackrel{{\star}}{{=}}\mathfrak{v}\) (defined in accordance with Equation (86)), wherein the displacement field \(\mathfrak{q}\) is considered as the impulse of the velocity field \(\boldsymbol{v}=\dot{\mathfrak{q}}\). This yields, then, for the integral of action of the Hamilton-Pontryagin-Hu-Washizu type,  
\[\overline{\mathcal{A}}_{\rm HPHW^{\star}}[\mathfrak{q},\overline{ \mathbf{F}}^{\rm int},\boldsymbol{\pi},\boldsymbol{v},\overline{\mathbf{E}},t _{f}] \stackrel{{\rm def}}{{=}} \int_{t_{1}}^{t_{f}}\left\{\overline{\mathcal{Z}}[\mathfrak{v}, \overline{\mathbf{E}},\boldsymbol{v}]\right.\] \[\left.+\int_{\overline{\mathcal{S}}}\left[\left(\dot{\mathfrak{q}}- \boldsymbol{v}\right)^{\rm T}\boldsymbol{\pi}-\left(\overline{\overline{ \mathbf{B}}}\mathfrak{q}-\overline{\mathbf{E}}\right)^{\rm T}\overline{ \mathbf{F}}^{\rm int}\right]d\overline{\mathcal{S}}\right.\] \[\left.+\oint_{\overline{\mathcal{C}}}\left[\left(\dot{\mathfrak{q}}- \boldsymbol{v}\right)^{\rm T}\boldsymbol{\pi}\right]d\bar{s}_{\ell}\right\}d \bar{t} \tag{103}\]or \(\overline{\mathcal{A}}_{\rm HPHW}[\mathfrak{q},\overline{\mathbf{F}}^{\rm int}, \boldsymbol{\pi},\boldsymbol{v},\overline{\mathbf{E}},t_{f}]\sim\overline{ \mathcal{A}}_{\rm HPHW^{\ast}}[\mathfrak{q},\overline{\mathbf{F}}^{\rm int}, \boldsymbol{\pi},\boldsymbol{v},\overline{\mathbf{E}},t_{f}]\) (with the adequate variational conditions at \(\{t_{i},t_{f}\}\)). These integrals of action involve the Lagrangian \(\overline{\mathcal{L}}[\dot{\boldsymbol{v}},\overline{\mathbf{E}},\boldsymbol{ v}]\stackrel{{\rm def}}{{=}}\overline{\mathcal{K}}[\dot{ \boldsymbol{v}},\boldsymbol{v}]-\overline{\mathcal{E}}^{\rm cas}[\dot{ \boldsymbol{v}},\overline{\mathbf{E}}]\) introduced in Equation (74); as before, the potential energy \(\overline{\mathcal{E}}^{\rm elas}[\dot{\boldsymbol{v}},\overline{\mathbf{E}}]\) and the kinetic energy \(\overline{\mathcal{K}}[\dot{\boldsymbol{v}},\boldsymbol{v}]\) are, respectively, defined according to Equations (51) and (68) in the linear elastodynamic case. Nevertheless, as mentioned in [117], this type of Hamilton-Pontryagin-Hu-Washizu integral of action is also suitable for nonlinear elastic theory with finite displacements.