guidance, flight mechanics and trajectory optimization

NOW, combining these two expressions provides 8R
/lx+, Y’, - f(x, 7, $-- ) + q = 0 (2.4.40)
But from Equation (2.4.38)
aR dR af
-= -
df ajr (xPf) = - 27 (x9 y 5 f6t)
Thus, substituting this equation into (2.4.401, yields the Weierstrass
condition of the Calculus of Variations.
f 4 jf9 (2.4.41)
When the slope Y' in (2.4.37) is computed according to the optimiz-
ing condition of (2.4.38) it follows that
(2.4.42)
Note, that
At points
t/as
(X,#)
developed
for which
from (2.4.38) will be a function of JL and
'C%,@is differentiable, (i.e;, 9 and F'
v*
exist). Eqs. (2.4.42) an 1 2.4.38) can be combined to yie &if a thi& necessary
condition. Taking the total derivative of (2.4.38) with respect
to EL and the partial derivative of (2.4.42) with respect to f yields
(2.4.43)
which is the Euler-Lagrange equation; an equation which must be satisfied
at points (r,Y ) where
the required derivatives
y'is differentiable. Across
do not exist, and (2.4.43)
discontinuities
does not hold.
in y' ,
However,
at such pointsRU,v) is continuous and so is$
assumptions of (2.4.22). Thus, from Eq. (2.4.38)
and the Weierstrass-Erdman corner condition
according to the original
-5 , is also continuous
3
must hold.
70
(2.4.43~)