Geodesic
Second-Order Evolution Equations Associated with Convex Hamiltonians(1)
Canadian mathematical bulletin, Tome 23 (1980) no. 1, pp. 81-94

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Many problems in mathematical physics can be formulated as differential equations of second order in time: with V a convex functional. This is the Euler equation for the Lagrangian which is convex with respect to x, and concave with respect to x. 
Aubin, J. P.; Ekeland, I. Second-Order Evolution Equations Associated with Convex Hamiltonians(1). Canadian mathematical bulletin, Tome 23 (1980) no. 1, pp. 81-94. doi: 10.4153/CMB-1980-011-x
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