A fish swims in the ocean along a straight line with speed $V(t)$. The fish starts its journey from rest (zero velocity at $t=0$ ) and, during a given time $T$, swims subject to the constraint that the total distance travelled is $L$. The energy cost for swimming is $a V^{2}+b \dot{V}^{2}$ per unit time, where $a, b \geqslant 0$ are known and $a^{2}+b^{2} \neq 0$.

(a) Derive the Euler-Lagrange condition on $V(t)$ for the journey to have minimum energetic cost.

(b) In the case $a \neq 0, b \neq 0$ solve for $V(t)$ assuming that the fish starts at $t=0$ with zero acceleration (in addition to zero velocity).

(c) In the case $a=0$, the fish can decide between three different boundary conditions for its journey. In addition to starting with zero velocity, it can:

(1) start at $t=0$ with zero acceleration;

(2) end at $t=T$ with zero velocity; or

(3) end at $t=T$ with zero acceleration.

Which of $(1),(2)$ or (3) is the best minimal-energy cost strategy?