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).