Consider Figure 10.12, in which a point in 0 must move to reach point p = [4 2.5]^T, crossing three layers of fluids having different densities.

In the first layer, the point can travel at a maximum speed ν₁, while in the second layer and third layers it may travel at a lower maximum speeds, respectively ν² = ν1/η₂, and ν₃ = ν₁/η₃, with η₂, η₃ > 1. Assume ν₁ = 1, η₂ = 1.5, η₃ = 1.2. You have to determine what is the fastest (i.e., minimum time) path from 0 to p. use path leg lengths ℓ₁, ℓ₂, ℓ₃ as variables, and observe that, in this problem, equality constraints of the type ℓ_i = “something” can be equivalently substituted by inequality constraints ℓ_i  ≥ ”something”.

1 2 0 2.5- br P1 l2 P2 l3 P V1 22 V3