## On Hölder's transformation

J. Math. Sci. Univ. Tokyo
Vol. 1 (1994), No. 1, Page 1--21.

Hildebrandt, Stefan
On Hölder's transformation
This paper investigates the main properties of Hölder's transformation $\Hk_F$ of the contact space $\hat M$, generated by some function $F:\hat M\to\oR$, and its relation to Legendre's transformation $\Lk_H$. The commuting diagram $\Lk_H\circ\Hk_F=\Hk_W\circ\Lk_F$ and the related global properties of $\Hk$ and $\Lk$ are of particular interest. Hölder's transformation can be used to transform Hamiltonian systems into Lie systems and Euler-Lagrange equations into Herglotz equations, this way establishing four equivalent pictures of the one-dimensional calculus of variations.