Further remarks on the law of varying action and the symbol δ

Numerical dissipation in time‐step integration algorithms for structural dynamic analysis

“…Since the initial position and initial velocity are imposed when constructing the approximate solutions, the variations δq and δq̇ are set to zero in the formulation. In this case, the variations in the strict sense of calculus of variations are used to obtain numerical solutions46. Fig.…”

Time Modes and Linear Systems

“…This will provide a consistent comparison between the traditional and universal procedures for implementing HLVA. The reader is referred to reference [1}4, 8,11,12] for details and examples of the traditional procedure from a non-vector-space perspective.…”

“…HLVA is a fundamental law [12,19] and is not a numerical/computational procedure. However, in most practical applications, numerical procedure are used to solve the AEM resulting from HLVA.…”

“…The chain rule may now be utilized in the operationR (as de"ned in the sense of Bailey [12]) as follows…”

“…¹ is the kinetic energy and = is the virtual work expression. The operator is de"ned in the sense of Bailey [12]. HLVA originally being set forth by Hamilton [13,14] in his classic 1834 and 1835 papers presenting a general energy method in dynamics, had not been used successfully to obtain direct solutions to initial value problems until its implementation by Bailey [1] beginning in the 1970s.…”