If there is only one degree of freedom then one talks in singular: Euler-Lagrange equation.
Unlike in the usual classical mechanical systems, in some other problems of calculus of variations one has Lagrangians involving higher time derivatives; the Euler-Lagrange equations have then correspondingly more terms. Similarly, one can have dependence on more parameters than a single time parameter, what can also be easily incorporated.
Robert Bryant, Phillip Griffiths, Daniel Grossman, Exterior differential systems and Euler-Lagrange partial differential equations, Chicago Lectures in Mathematics 2003, 205+xiv pp. math.DG/0207039bookpage