
10
Variational equations of motion
We encountered variational forms of equations of motion in prior chapters, for
example, when solving the brachistocrone problem in Section 1.4.2. Several
dynamic equations of motion will be derived from variational principles in this
chapter.
10.1 Legendre’s dual transformation
This transformation invented by Legendre is of fundamental importance. Let
us consider the function of n variables
f = f (u
1
,u
2
, ..., u
n
).
Legendre proposed to introduce a new set of variables by the transformation of
v
i
=
∂f
∂u
i
,i=1, 2, ..., n.
The Hessian matrix of this transformation is
H(f)=
⎡
⎢
⎢
⎢
⎢
⎣
∂
2
f
∂u
2
1
∂
2
f
∂u
1
∂u
2
...
∂
2
f
∂u
1
∂u
n
∂
2
f
∂u
2
∂u
1
∂
2
f
∂u
2
2
...