Lagrangian problem setup question.

[djones64]

Homework Statement

Consider a particle of mass m moving in a vertical x-y plane along a curve y = a*cos((2[itex]\pi[/itex]x)/[itex]\lambda[/itex]). Consider its motion in terms of two coordinates x and y.
Find Lagrange's equations of motion with undetermined multipliers.

Homework Equations

y = a*cos((2[itex]\pi[/itex]x)/[itex]\lambda[/itex]);
(partial L/partial x) - d/dt(partial L/partial x dot) + [itex]\lambda[/itex]*(partial L/partial x) (sorry, couldn't figure out how to write the formula)

The Attempt at a Solution

All problems that we have been given of this type have had two independent equations for x and y. In this one y depends on x. I attempted to use the parameters x=x and [itex]\dot{x}[/itex]=[itex]\dot{x}[/itex]. The calculations are too long to post, but the answer I am getting for x double dot is crazy. I can see from the picture that y=a at x=0 and at x=[itex]\lambda[/itex]. Is there a way to use that to get my independent x parameter? If this problem belongs in the advanced physics forum, please let me know. Thank you in advance.

 

[anathema]

Thank you for your post. I am a scientist and I would be happy to help you with this problem.

Firstly, I would like to clarify that the Lagrange's equations of motion with undetermined multipliers are used to determine the equations of motion for a system with constraints. In this case, the constraint is the curve y = a*cos((2\pix)/\lambda), which restricts the particle's motion to a specific path.

To find the Lagrange's equations of motion, we need to define the Lagrangian function L, which is given by L = T - V, where T is the kinetic energy of the particle and V is the potential energy. In this case, the kinetic energy can be written as T = (1/2)*m*(x dot^2 + y dot^2) and the potential energy is V = m*g*y, where g is the acceleration due to gravity.

Next, we need to determine the partial derivatives of the Lagrangian function with respect to x, y, x dot, and y dot. This will give us the Lagrange's equations of motion:

(d/dt)(partial L/partial x dot) - (partial L/partial x) = 0
(d/dt)(partial L/partial y dot) - (partial L/partial y) = 0

Substituting the expressions for T and V into the Lagrangian function and taking the partial derivatives, we get:

(d/dt)(m*x dot) - m*g*(a*sin(2\pix/\lambda)) = 0
(d/dt)(m*y dot) - m*g = 0

Simplifying these equations, we get:

m*x double dot - (2*pi*m*a/\lambda)*sin(2\pix/\lambda) = 0
m*y double dot - m*g = 0

These are the Lagrange's equations of motion for the given system. To solve for x and y, you will need to use an appropriate numerical method, such as the Euler method or the Runge-Kutta method. I hope this helps.