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Appendix 3: Derivation of
Hamilton’s Principle from
D’Alembert’s Principle
We consider, for simplicity, a system of N particles of masses m
i
, located at the point r
i
and acted
upon by external forces F
i
. By D’Alembert’s principle, we have
Fmrr
ii
i
N
−⋅=
=
∑
δ
1
. (A3.1)
This is the dynamic condition to be satised by the impressed forces and the internal forces for arbi-
trary
consistent with the constraints. We may interpret the virtual displacements
as the small
variations in the following sense: a particle in its motion will take a certain path described by r=f(t)
under the action of external forces. Varied paths described i ...