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## Homework Statement

An object of mass m, and constrained to the x-y plane, travels frictionlessly along a curve f(x), while experiencing a gravitational force, m*g. Starting with the Lagrangian for the system and using the method of Lagrange multipliers, derive the equations of motion for the bead, and then derive the condition under which the bead would lose contact with the surface

## Homework Equations

L=T-V, as well as several other

## The Attempt at a Solution

T=.5*m*v^2

=>.5*m*((dx/dt)^2)+((dy/dt)^2)

V=m*g*y

=>m*g*f(x) (When the object is on the curve)

L=.5*m*((dx/dt)^2)+((dy/dt)^2)+m*g*f(x)

I think I'm correct up to this point, but I'm not sure how to apply Lagrange multipliers to the Lagrangian and reduce to the equations of motion. Also, I know intuitively that the object would lose contact with the surface when dy/dt is greater (less negative) than df(x)/dt, but I have no idea how I would find this condition using Lagrangian mechanics. If someone could help, I would be really greatful.