- #1
Zaknife
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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 ?