Varying inclination on plane: Undetermined multipliers

[davidbenari]

Homework Statement

A particle of mass $m$ rests on a smooth plane. The plane is raised to an inclination $\theta$ at constant rate $\alpha$. Find the constraint force.

Homework Equations

The Attempt at a Solution

$L=\frac{1}{2}m(\dot{x}^2+\dot{y}^2)-mgy$ Lagrangian

$f=\frac{y}{x}-\tan\alpha t = 0$ constraint equation

$\partial_y f = \frac{1}{x}$

$\partial_x f = \frac{-y}{x^2}$

$\partial_q L - d_t \partial_\dot{q} L + \lambda \partial_q f = 0$ Method of undetermined multipliers formula.

$\to \boxed{m\ddot{x}+\lambda \frac{y}{x^2} = 0} \quad \boxed{mg+m\ddot{y}=\frac{\lambda}{x}}$

Using tedious manipulation I've gotten to the point where I can say

$\ddot{x}x+\ddot{y}y+gy=0$

And haven't found any other useful formula.

I know I could switch to a polar coordinate basis and find $r(t)$ there and solve $x$ and $y$ and indirectly find constraint forces, but I'm not interested in that. Unless I'm clearly using the Lagrange undetermined multipliers.

 

[Geofleur]

Here's an idea: Write the Lagrangian in terms of polar coordinates, but keep both $r$ and $\phi$ as the generalized coordinates - do not incorporate the constraint into the kinetic or potential energy terms. Rather, incorporate it through the Lagrange multiplier technique.