transversality condition

Natural boundary conditions handle an end that slides freely up and down. The transversality condition handles a richer situation: an endpoint that is not fixed but must lie on a given curve or surface — say the path must end somewhere on a target line, or a ship must reach a moving coastline, with the exact landing point left for the optimization to choose. Transversality is the extra equation that tells you the right angle, so to speak, at which the optimal path should meet that target.

When the endpoint is free to slide along a constraint curve, the boundary term in the first variation no longer simply vanishes; instead the variation of the endpoint is tied to the slope of the constraint curve. Demanding the total first variation be zero produces a relation linking L, dL/dy', and the slope of the target curve at the meeting point. That relation is the transversality condition. In the special case where the endpoint can move purely vertically (the target is a vertical line, i.e. a free value at fixed x), it collapses back to the natural boundary condition dL/dy' = 0; transversality is the general parent, natural boundary conditions the special child.

Transversality is indispensable in optimal control and in problems with free terminal time or a target set rather than a target point — the shortest path from a point to a circle meets the circle perpendicularly (the transversality condition for arc length is literally orthogonality), and a minimum-time trajectory to a moving boundary satisfies its own transversality relation. The honest framing: an interior Euler-Lagrange equation alone never determines a variational problem; you always need the right boundary information, and when ends are free or constrained to curves, transversality is what correctly supplies it. Mislabeling or omitting it is a frequent source of wrong 'optimal' paths.