# Definition:Transversality Conditions

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## 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**.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 3.14$: End Points Lying on Two Given Lines or Surfaces