24 R.N. Coppolino FD dU du C Fe (3.6) Thus the variational integral becomes t2 Z t1 .•T •UC•W/ dt D t2 Z t1 .•LC•W/ dt D0 (3.7) whereLDT – U is known as the Lagrangian function. The postulate-theorem relationship may be reversed by starting with Hamilton’s principle as a postulate. Two consequences follow from this starting point in mechanics of distributed as well as single degree of freedom systems, namely, (a) Lagrange’s equations [4], and (b) Newton’s second law; both may be viewed as theorems resulting from the starting postulate. 3.3 Mathematical Physics and Hamilton’s Principle Much like the above discussion on dynamics of a particle, the partial differential equations of mathematical physics (in particular mechanics) may be derived on the basis of dynamic equilibrium (Newton’s laws and free body diagrams) or on the basis of Hamilton’s principle (energy and virtual work). Results of both approaches produce the same description. Application of Hamilton’s principle to a dynamic system described as a continuum yields a volume integral of the type t2 Z t1 Z V .•TV •UV C•WV/ dV dt D0; (3.8) where TV, UV, and •WV are the kinetic energy, potential (or strain) energy, and virtual work functions per unit volume, respectively. Analysis of any particular dynamic system, described in terms of displacement variables, u(x, y, z, t), which may be vectors, results in the following type of functionals t2 Z t1 Z V .P:D:E/ •u dV dt C t2 Z t1 Z S .N:B:C/ •u dS dt D0 (3.9) “P.D.E” represents the particular partial differential equation(s) within the system’s volume. “N.B.C” represents the natural boundary conditions, which are mathematically and physically admissible on the system’s boundary surface(s). The general process for derivation of a system’s partial differential equations and natural boundary conditions has provided a consistent basis for the development of technical structural theories for prismatic bars, beams, rings, plates and shells [5, 6]. 3.4 The Contributions of Ritz, Galerkin, and Trefftz Three outstanding contributions that led to the development of approximate analysis techniques date back to the early part of the twentieth century. The methods bearing the names of Ritz, Galerkin and Trefftz are all consequences of Hamilton’s principle and the assumption of approximate solution functions. 3.4.1 The Ritz Method A monumental contribution to approximate analysis was introduced by Ritz [7], who described the displacement field in variable separable terms
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