# Mayer problem

One of the fundamental problems in the calculus of variations (cf. Variational calculus) on a conditional extremum. The Mayer problem is the following: Find a minimum of the functional

$$J ( y) = g ( x _ {1} , y ( x _ {1} ) , x _ {2} , y ( x _ {2} ) ) ,\ \ g: \mathbf R \times \mathbf R ^ {n} \times \mathbf R \times \mathbf R ^ {n} \rightarrow \mathbf R ,$$

in the presence of differential constraints of the type

$$\phi ( x , y , y ^ \prime ) = 0 ,\ \ \phi : \mathbf R \times \mathbf R ^ {n} \times \mathbf R ^ {n} \rightarrow \mathbf R ^ {m} ,\ \ m < n ,$$

and boundary conditions

$$\psi ( x _ {1} , y ( x _ {1} ) , x _ {2} , y ( x _ {2} ) ) = 0 ,\ \ \psi : \mathbf R \times \mathbf R ^ {n} \times \mathbf R \times \mathbf R ^ {n} \rightarrow \ \mathbf R ^ {p} ,$$

$$p < 2 n + 2 .$$

For details see Bolza problem.

The Mayer problem is named after A. Mayer, who studied necessary conditions for its solution (at the end of the 19th century).

How to Cite This Entry:
Mayer problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mayer_problem&oldid=47810
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article