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One of the fundamental problems in the calculus of variations (cf. [[Variational calculus|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|Bolza problem]].

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