This is not a homework equation at all, however I have devised my own example problem in order to convey my misunderstanding. (My question is at the end of the problem) Question that I had come up with: A particle's motion is described in the x direction by the equation x = x(t). The particle's motion in the y direction is given by the constraint equation, y = (x(t))2. This particle's motion is on earth and is acted on by the force, -mg. Find the equation of motion. Attempted solution: because of the constraint equation for y, I did not write the lagrangian in terms of either x-dx/dt-y, but with just the coordinates x-dx/dt. Because of this logic, my Lagrangian was: http://www.sciweavers.org/upload/Tex2Img_1428780064/render.png [Broken] Plugging into the Euler-Lagrange, I obtained the Equation of Motion to be: http://www.sciweavers.org/upload/Tex2Img_1428780484/render.png [Broken] This is undoubtedly wrong as the equation of motion is one dimensional while the actual motion of the particle is in two dimensions. What did I do wrong? And more importantly, how would you guys solve this type of problem?