For a viscoelastic material the work
W
(
e
)
W\left ( e \right )
needed to produce a given strain
e
0
{e_0}
in a given time
T
T
depends on the strain path
e
(
t
)
,
0
≤
t
≤
T
e\left ( t \right ), 0 \le t \le T
, connecting the unstrained state with
e
0
{e_0}
. We here ask the question: Of all strain paths of this type, is there one which is optimal,
1
^{1}
that is, one which renders
W
W
a minimum? In answer to this question we show that: (i) There is no smooth optimal strain path. (ii) There exists a unique optimal path in
L
2
(
0
,
T
)
{L_2}\left ( {0,T} \right )
; this path is smooth on the open interval
(
0
,
T
)
\left ( {0,T} \right )
, but suffers jump discontinuities
2
^{2}
at the end points 0 and
T
(
i
.
e
.
,
e
(
0
+
)
≠
0
,
e
(
T
−
)
≠
e
0
)
T\left ( {i.e.,e\left ( {{0^ + }} \right ) \ne 0, \\ e\left ( {{T^ - }} \right ) \ne {e_0}} \right )
. (iii) For a Maxwell material the optimal path is linear on
(
0
,
T
)
\left ( {0,T} \right )
.