Geodesic
The Erdmann Condition and Hamiltonian Inclusions in Optimal Control and the Calculus of Variations
Canadian journal of mathematics, Tome 32 (1980) no. 2, pp. 494-509

Voir la notice de l'article provenant de la source Cambridge University Press

Consider the basic problem in the calculus of variations, that of minimizing 1.1 over a class of functions x satisfying certain boundary conditions at 0 and 1. One of the classical first order necessary conditions for optimality is the second Erdmann condition, which asserts, in the case in which L is independent mof t, that 1.2 along any local solution x. This formula is the customary basis for solving many of the classical problems, such as the brachistochrone. When it is possible to define via the Legendre transform a Hamiltonian H(t, x, p) corresponding to L, the second Erdmann condition, again in the autonomous case, is the assertion that 1.3 a relation which always evokes classical Hamiltonian mechanics and conservation laws. 
Clarke, Frank H. The Erdmann Condition and Hamiltonian Inclusions in Optimal Control and the Calculus of Variations. Canadian journal of mathematics, Tome 32 (1980) no. 2, pp. 494-509. doi: 10.4153/CJM-1980-039-9
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