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## Elastic pendulum - Lagrangian approach

1. The problem statement, all variables and given/known data

A spring of rest length $$L_0$$ (no tension) is connected to a support at one end and has a mass $$M$$ attached at the other. Neglect the mass of the spring, the dimension of the mass $$M$$, and assume that the motion is confined to a vertical plane. Also, assume that the spring only stretches without bending but it can swing in the plane.

(a) Using the angular displacement of the mass from the vertical and the length that the spring has stretched from its rest length (hanging with the mass $$M$$), solve Lagrange's equations for small stretching and angular displacement.

(b) Solve Lagrange's equations to the next order in both stretching and angular displacement. This is still amenable to hand calculations. Using some reasonable assumptions about the spring constant, the mass, and the rest length, discuss the motion. Is resonance likely under the assumptions stated?

2. Relevant equations

Formula for the Lagrangian; Euler-Lagrange equations.

3. The attempt at a solution

The Lagrangian is:

$$\mathcal{L} = T - V = \frac{M}{2}(\dot{r}^2 + r^2\dot{\phi}^2) - (\frac{k}{2} x^2 - Mgr\cos\phi)$$

where $$r = L_0 + x$$ (i.e. $$r$$ is the stretched length of the spring, and $$x$$ is the deviation from the rest length). $$\phi$$ is the angular displacement where zero is hanging straight down.

The Euler-Lagrange equations in $$x$$ and $$\phi$$ are:

$$\frac{\partial \mathcal{L}}{\partial x} - \frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{\partial \mathcal{L}}{\partial \dot{x}}\right) = M(L_0 + x)\dot{\phi}^2 - (kx - Mg\cos\phi) - M\ddot{x} = 0$$

$$\frac{\partial \mathcal{L}}{\partial \phi} - \frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{\partial \mathcal{L}}{\partial \dot{\phi}}\right) = - Mg(L_0 + x)\sin\phi - M((L_0 + x)^2\ddot{\phi} + 2(L_0 + x)\dot{x}\dot{\phi}) = 0$$

My guess at approximating for small angular displacements is to set $$\sin\phi = \phi$$ and $$\cos\phi = 1$$, but I don't know whether this should be before or after differentiating the Lagrangian, and I don't know the right way to approximate for small stretching.
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