Solving Lagrangian Problem: Find Lagrange's Eqns & Steps

[FluxCapacitator]

A spring of rest length $$L_0$$ (no tension) is connected to a support at one end and has a mass M attached to the other. Neglect the mass of the spring, the dimension of mass M, and assume that the motion is confined to the vertical direction, and that the spring stretches without bending but can swing in the plane.
Find Lagrange's equations and solve them for small stretching and angular displacements.
I'm having trouble just mathematically expressing the kinetic energy and potential functions in an easily solvable form.
So far I have $$T= mv^{2}/2$$ and $$V=mgy+kl^{2}/2$$, where l is the displacement of the string.
At this point I'll confess that I'm trying to learn this by myself, and I could have missed some important concepts in how exactly to solve the equations in the right form.
One key point of confusion I have is whether or not resolving the Lagrangian into components is a valid method to solve a problem. If so, it would probably be easier to solve. If not, I suppose that I'd have to use trigonometry to try to get everything into agreeable terms and then solve.
I suppose what I'm asking for is someone to discuss how to solve the problem step by step sot that I could understand how to do this once and for all.

 

[vanesch]

A spring of rest length $$L_0$$ (no tension) is connected to a support at one end and has a mass M attached to the other. Neglect the mass of the spring, the dimension of mass M, and assume that the motion is confined to the vertical direction, and that the spring stretches without bending but can swing in the plane.
Find Lagrange's equations and solve them for small stretching and angular displacements.
I'm having trouble just mathematically expressing the kinetic energy and potential functions in an easily solvable form.
So far I have $$T= mv^{2}/2$$ and $$V=mgy+kl^{2}/2$$, where l is the displacement of the string.

This looks correct ; you now have to write L = T - V

One key point of confusion I have is whether or not resolving the Lagrangian into components is a valid method to solve a problem.

You cannot "resolve the Lagrangian in components" because it is a scalar quantity, and not a vector !
But you should identify the configuration space variables: I'd say that here, they are x and y. Once you've expressed the lagrangian completely as a function of the configuration space variables and their first time derivatives (v!), you can write down the Euler-Lagrange equation for each configuration space variable (here, there are 2 of them because you have 2 variables x and y).

 

[FluxCapacitator]

Thanks, then I was doing the right thing, but this problem is just a little geometrically involved, so it looked more confusing that it perhaps really was.

 

[lightgrav]

Am I missing some constraint? I see 3 variables.
I'd try to use r, theta, and phi for this;
so modes of spring and pendulums (regular and conical)
have only one variable and two parameters.

 

[vanesch]

Am I missing some constraint? I see 3 variables.
I'd try to use r, theta, and phi for this;
so modes of spring and pendulums (regular and conical)
have only one variable and two parameters.

Maybe I misunderstood the problem description: I made up of it that the motion was confined in a vertical plane...