Extended General Hamiltons Principle

D Alembert s principle in the Lagrangian version, as derived in Section 3.4.5, uses infinitesimal virtual displacements about the instantaneous system state. For this reason, it is referred to as a differential principle. When infinitesimal virtual deviations from the entire motion of a system between two instants in time are examined, then it is an integral principle like Hamilton s principle, see Goldstein [86], Sokolnikoff [167], Szabb [172] or Morgenstern and Szaho [126]. Here the derivation from the prior to the latter principle will be demonstrated, starting with conversion of the virtual work of the inertia loads included in Eq. (3.59). With Eq. (3.54) and acceleration as derivative of velocity, it 

The last term, with the aid of Schwarz s theorem stating the possibility to interchange the order of taking partial derivatives and velocity as derivative of displacement, can be shown to represent the virtual change of kinetic energy V  

With its substitution into d Alembert s principle in the Lagrangian version of Eq. (3.59), one obtains Eq. (3.73). Integration over the period of time from to to ti, where the virtual displacements are zero by definition at these end points such that 5u (to) = 6u (ti) = 0, leads to the general Hamilton s principle of 

Instead of using d Alembert s principle in the Lagrangian version, Lagrange s central equation, Eq, (3.72), may be substituted into the complete principle of virtual work, Eq. (3.41) with (3.62) and (3.63). After the intermediate step of Eq. (3.75), this finally leads to the general Hamilton s principle with an extension to deformable piezoelectric bodies of Eq. (3.76)  

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The principle of virtual work is suitable for solving a wide range of problems. There are tasks however where different but related formulations might be more useful. Thus, two prominent variational principles will be extended here to take into account materials with electromechanical couplings. This novel approach to Dirichlet s principle of minimum potential energy will be employed later in Section 6.3.2. In comparison to the principle of virtual work, the extended general Hamilton s principle is considered to be equivalent and even more versatile, but only its derivation will be demonstrated here. 

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