Differential equation after using Euler-Lagrange equations

[Zaknife]

Homework Statement

Particle is moving along the curve parametrized as below (x,y,z) in uniform gravitational field. Using Euler- Lagrange equations find the motion of the particle.

The Attempt at a Solution

$$\begin{array}{ll} x=a \cos \phi & \dot{x}= -\dot{\phi} a \sin \phi \\ y=a \sin \phi & \dot{y}=\dot{\phi} a \cos \phi \\ z=b \phi & \dot{z}= b \dot{\phi} \\ \end{array}$$
Lagrangean will be :
$$L=T-V=\frac{m}{2}\dot{\phi}^{2}(a^{2}+b^{2})-mgb\phi$$
Using Euler-Lagrange equations we obtain:
$$\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q_{l}}}\right)-\frac{\partial \mathcal{L}}{\partial q_{l}}=0$$
$$m\ddot{\phi}(a^{2}+b^{2})+mgb=0$$

How to deal with such differential equation ?

 

[CompuChip]

So $\phi$ is the only generalised co-ordinate, and all the other letters are constants, right?

In that case, this is a differential equation of the form $$\ddot\phi = c$$ which gives a simple linear solution (which makes sense, because the only freedom you have is how fast you move along the curve).

 

[Zaknife]

So all i need to do is double integration ?

 

[Steely Dan]

So all i need to do is double integration ?

That's correct. Write the equation in the form suggested by CompuChip and then integrate twice with respect to time. Since the right side is a constant is pretty simple to do.

 

[dextercioby]

The particle is moving along a helix (on a circular cylinder) with constant angular velocity. Therefore the angle in the plane xOy depends quadratically on time.