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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))

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?

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?

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