- #1

- 12

- 0

## 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

[tex]\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}

[/tex]

Lagrangean will be :

[tex] L=T-V=\frac{m}{2}\dot{\phi}^{2}(a^{2}+b^{2})-mgb\phi[/tex]

Using Euler-Lagrange equations we obtain:

[tex]\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q_{l}}}\right)-\frac{\partial \mathcal{L}}{\partial q_{l}}=0[/tex]

[tex]m\ddot{\phi}(a^{2}+b^{2})+mgb=0[/tex]

How to deal with such differential equation ?