# Definition:Transversality Conditions

## Definition

Let $\map {\mathbf y} x$ be a differentiable vector-valued function.

Let $J \sqbrk {\mathbf y}$ be a functional of the following form:

$\displaystyle J \sqbrk {\mathbf y} = \int_{P_1}^{P_2} \map F {x, \mathbf y, \mathbf y', \ldots} \rd x$

where $P_1$, $P_2$ are points on given differentiable manifolds $M_1$ and $M_2$.

Suppose we are looking for $\mathbf y$ extremizing $J$.

The system of equations to be solved consists of differential Euler equations and algebraic equations at both endpoints.

Then the set of all algebraic equations at both endpoints are called transversality conditions.