Two remarks about Ma\~n\'e's conjecture

TL;DR

This paper discusses Ma ext{n}é's conjecture in Lagrangian dynamics, revealing that it encompasses an additional conjecture by Ma ext{n}é, thus clarifying the conjecture's scope and implications.

Contribution

The paper proves that Ma ext{n}é's conjecture includes another of Ma ext{n}é's conjectures, providing insight into the structure of these conjectures in Lagrangian systems.

Findings

Ma ext{n}é's conjecture contains another conjecture of Ma ext{n}é.

Clarification of the relationship between two of Ma ext{n}é's conjectures.

Implications for the study of minimizing orbits in Lagrangian dynamics.

Abstract

We prove that Ma\~n\'e's conjecture, as stated in {\em Lagrangian flows: the dynamics of globally minimizing orbits}, Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 141--153, contains another conjecture of Ma\~n\'e, stated in {\em Generic properties and problems of minimizing measures of Lagrangian systems} Nonlinearity 9 (1996) 273-310.