# Weierstrass-Erdmann corner conditions

Necessary conditions for an extremum, additional to the Euler equation, specified at points at which the extremal has a corner. Let

$$J ( x) = \int\limits L ( t, x, \dot{x} ) dt$$

be a functional of the classical calculus of variations (cf. Variational calculus), and let the extremal $x _ {0} ( t)$ be continuously differentiable in a neighbourhood of the point $\tau$ except at the point $\tau$ itself, at which it has a corner. In this situation, for $x _ {0} ( t)$ to be at least a weak local extremum for the functional $J( x)$, it is necessary that the equations

$$p( \tau - 0) = p ( \tau + 0),$$

$$H( \tau - 0) = H ( \tau + 0),$$

where

$$p( t) = \ \frac{\partial L( t , x _ {0} ( t), \dot{x} _ {0} ( t)) }{\partial \dot{x} }$$

and

$$H( t) = \ ( \dot{x} _ {0} ( t), p( t))- L ( t, x _ {0} ( t), \dot{x} _ {0} ( t)),$$

be satisfied at the corner point $\tau$. These equations are known as the corner conditions of K. Weierstrass (1865) and G. Erdmann (1877) [1].

The meaning of the Weierstrass–Erdmann corner conditions is that the canonical variables and the Hamiltonian are continuous at a corner point of the extremal; their meaning in classical mechanics is the continuity of momentum and of energy at a corner point.

In regular problems, when $L$ is a strictly convex function of $\dot{x}$, the extremals cannot have corner points. Corner points appear if $L( t, x, \dot{x} )$ and consequently the Weierstrass ${\mathcal E}$- function, contains segments of $\dot{x}$. For the Lagrange problem with conditions $\phi _ {i} ( t, x, \dot{x} ) = 0$ and Lagrange multipliers $\lambda _ {i} ( t)$, the $L$ in the Weierstrass–Erdmann corner conditions is replaced by $\widetilde{L} = L+ \sum _ {i} \lambda _ {i} \phi _ {i}$.

#### References

 [1] G. Erdmann, "Ueber die unstetige Lösungen in der Variationsrechnung" J. Reine Angew. Math. , 82 (1877) pp. 21–30 [2] O. Bolza, "Lectures on the calculus of variations" , Chelsea, reprint (1960) (Translated from German) [3] N.I. Akhiezer, "The calculus of variations" , Blaisdell (1962) (Translated from Russian)