Direct Energy, 2018a

11 CALCULUS OF VARIATIONS 249
11.4 Derivation of the Euler-Lagrange Equation
In this section, we use the Principle ofLeast Action to derive a dierential
relationship for the path, and the result is the Euler-Lagrange equation.
This derivation closely follows [163, p. 23-33], so see that reference for
a more rigorous derivation. Assume that we know the Lagrangian which
describes the dierence between two forms of energy, and we know the
action. We want to nd a dierential relationship for the path y(t) which
minimizes the action. This path has the smallest integral over t ofthe
dierence between the two forms of energy.
Suppose that the path y(t) minimizes the action and is the path found
in nature. Consider a path ỹ(t) which is very close to the path y(t). Path
ỹ(t) is equal to path y(t) plus a small dierence.
ỹ = y + εη (11.9)
In Eq. 11.9, ε is a small parameter, and η = η(t) is a function of t. Wecan
evaluate the Lagrangian at this nearby path.
(
L t, ỹ, dỹ ) (
= L t, y + εη, ẏ + ε dη )
(11.10)
dt
dt
The Lagrangian ofthe nearby path ỹ(t) can be related to the Lagrangian
ofthe path y(t).
(
L t, ỹ, dỹ )
(
= L (t, y, ẏ)+ε η ∂L
dt
∂y + dη
dt
)
∂L
+ O(ε 2 ) (11.11)
∂ẏ
Equation 11.11 is written as an expansion in the small parameter ε. The
lowest order terms are shown, and O(ε 2 ) indicates that all additional terms
are multiplied by ε 2 or higher powers ofthis small parameter.
We can also express the dierence in the action for paths ỹ and y as an
expansion in the small parameter ε.
[ˆ t1
S(ŷ) − S(y) =ε η ∂L
t 0
∂y + dη ]
∂L
dt ∂ẏ dt + O(ε 2 ) (11.12)
The term in brackets is called the rst variation ofthe action, and it is
denoted by the symbol δ.
δS(η, y) =
ˆ t1
t 0
η ∂L
∂y + dη
dt
∂L
dt (11.13)
∂ẏ