Page 202 - Intro to Tensor Calculus

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Figure 2.2-5. Motion along curves c and c

where
i
i
L = T − V = L(t, x , ˙ x )
is the Lagrangian evaluated along the curve c. We ask the question, “What conditions must be satisfied by
the curve c in order that the integral I( ) have an extremum value when is zero?”If the integral I( )has
a minimum value when is zero it follows that its derivative with respect to will be zero at this value and
we will have
dI( )
=0.
d
=0
Employing the definition

dI I( ) − I(0)
0
= lim = I (0) = 0
d →0
=0
we expand the Lagrangian in equation (2.2.30) in a series about the point =0. Substituting the expansion

∂L i ∂L i 2
i
i
i
i
i
i
L(t, x + η , ˙x + ˙η )= L(t, x , ˙x )+ η + ˙ η + [] + ···
∂x i ∂ ˙x i
into equation (2.2.30) we calculate the derivative
Z t 1
I( ) − I(0) ∂L i ∂L i
I (0) = lim = lim η (t)+ ˙ η (t) dt + [] + ··· =0,
0
→0 →0 t 0 ∂x i ∂ ˙x i
where we have neglected higher order powers of since is approaching zero. Analysis of this equation
informs us that the integral I has a minimum value at = 0 provided that the integral
Z t 1 ∂L ∂L
i
i
δI = η (t)+ ˙ η (t) dt =0 (2.2.31)
∂x i ∂ ˙x i
t 0