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One of the fundamental problems in the classical calculus of variations. It consists in minimizing the functional

$$ 
J ( y)  =  \int\limits _ { x _ {1} } ^ { {x _ 2 } } f
( x , y , y  ^  \prime  )  d x ,\ \ 
f :  \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {n}  \rightarrow  \mathbf R ,
$$

in the presence of differential constraints of equality type:

$$ \tag{1 }
\phi ( x , y , y  ^  \prime  )  =  0 ,\ \ 
\phi :  \mathbf R \times \mathbf R  ^ {n} \times \mathbf R  ^ {n}  \rightarrow \ 
\mathbf R  ^ {m} ,\  m < n ,
$$

and boundary conditions

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

$$ 
p  \leq   2 n + 2 .
$$

The Lagrange problem is usually considered under the condition that the system (1) is regular, that is, the matrix  $  \| \partial  \phi / \partial  y  ^  \prime  \| $
has maximal rank:

$$ 
 \mathop{\rm rank}  \left \| 
\frac{\partial  \phi }{\partial  y  ^  \prime  }
 \right \|  =  m .
$$

Under this condition the system (1) can be solved for part of the variables and, using a different notation ( $  t , x $
instead of  $  x , y $), 
the Lagrange problem can be reduced to the form

$$ \tag{2 }
\left . 
\begin{array}{c}
\int\limits _ { t _ {0} } ^ { {t _ 1 } } F ( t , x , u ) \ 
dt ,\  F :  \mathbf R \times \mathbf R  ^ {n}
\times \mathbf R  ^ {r}  \rightarrow  \mathbf R ,  \\
\dot{x}  =  \Phi ( t , x , u ) ,\  \Phi :  \mathbf R \times
\mathbf R  ^ {n} \times \mathbf R  ^ {r}  \rightarrow  \mathbf R  ^ {n} .  \\
\end{array}

\right \}
$$

The function  $  F $
and the mapping  $  \Phi $
are usually assumed to be continuously differentiable. Problems of optimal control are often specified in the form (2) (the Pontryagin form), and restrictions are, moreover, imposed on the control  $  u \in U $. 
Necessary conditions for a strong extremum for the problem (2) (for simplicity, with fixed left-hand end  $  x _ {0} $
and free right-hand end  $  x _ {1} $) 
have the following form. Let

$$ 
L ( t , x , \dot{x} , u , p ( t) )  = \ 
( p ( t)  \mid   - \dot{x} + \Phi ( t , x , u )) - F ( t , x , u )
$$

be the [[Lagrange function|Lagrange function]]. For a vector function  $  ( x  ^ {*} ( t) , u  ^ {*} ( t) ) $
to be a strong minimum in the Lagrange problem (2) it is necessary that the following relations hold:

$$ \tag{3 }
\left . 
\frac{\partial  L }{\partial  \dot{x} }

\right | _ {( x  ^ {*}  , u  ^ {*} ) } +
\int\limits _ { t _ {0} } ^ { {t _ 1 } }
\left . 
\frac{\partial  L }{d x }

\right | _ {( x  ^ {*}  , u  ^ {*} ) }  d t  =  0 ,
$$

$$ \tag{4 }
p ( t _ {1} )  =  0 ,
$$

$$ \tag{5 }
{\mathcal E}  \equiv  L ( t , x  ^ {*} ( t) , \dot{x} , u , p ( t) ) +
$$

$$ 
- L ( t , x  ^ {*} ( t) , \dot{x}   ^ {*} ( t) , u  ^ {*} ( t) , p ( t) ) +
$$

$$ 
- ( ( \dot{x} - \dot{x}   ^ {*} ( t) )  \mid   L _ {\dot{x} }  ( t , x  ^ {*}
( t ) , \dot{x}   ^ {*} ( t ) , u  ^ {*} ( t ) , p ( t )) ) =
$$

$$ 
= \ 
( p ( t)  \mid   \Phi ( t , x  ^ {*} ( t) , u ) - F ( t , x  ^ {*} ( t) , u )) +
$$

$$ 
- ( p ( t)  \mid   \Phi ( t , x  ^ {*} ( t) , u  ^ {*} ( t) )
+ F ( t , x  ^ {*} ( t) , u  ^ {*} ( t) ))  \leq   0
$$

for all possible admissible values of  $  \dot{x} $
and  $  u $.

If one carries out differentiation in (3) with respect to  $  t $
and uses the notation

$$ 
{\mathcal H} ( t , x , u , p )  =  ( p  \mid   \Phi ) - F ,
$$

then a necessary condition for a strong minimum can be stated in the form of a maximum principle, in which the Euler equation (3), the transversality condition (4) and the Weierstrass condition (5) are combined. For a vector function  $  ( x  ^ {*} , u  ^ {*} ) $
to be a strong minimum in the problem (2) with fixed left-hand end and free right-hand end it is necessary that there is a solution of the system

$$ 
\dot{p} ( t)  =  - 
\frac{\partial  {\mathcal H} ( t , x  ^ {*} , u  ^ {*} , p ) }{\partial  x }
 ,\  p ( t _ {1} )  =  0 ,
$$

for which

$$ 
{\mathcal H} ( t , x  ^ {*} ( t) , u  ^ {*} ( t) , p ( t))  =  \max _
{u \in U }  {\mathcal H} ( t , x  ^ {*} ( t) , u , p ( t) ) .
$$

J.L. Lagrange considered similar problems in connection with studies in mechanics (in the second half of the 18th century).

For references see [[Variational calculus|Variational calculus]].

====Comments====
The notation  $  ( a  \mid   b) $
denotes the [[Inner product|inner product]] of the vectors  $  a $
and  $  b $.