# Equations of motion for double spherical pendulum simply?

I am attempting to simulate a double spherical pendulum, i.e. a combination of the spherical pendulum and the double pendulum.

I understand that the equations of motion can be derived via the Lagrangian and the Euler-Lagrange equations. However this method rapidly becomes very messy.

Is there an alternate method that could be used to simplify the calculations required?

-UPDATE-

This is an update to specify the Lagrangian of the system as requested in the comments.

We have two spherical pendulums, denoted $$1$$ and $$2$$. The lengths of each pendulum $$l_1=l_2=1$$ whilst the masses $$m_1=m_2=1$$.

There are two parameters that describe the location of the pendulum mass $$\theta$$, the angle from the vertical, and $$\phi$$ the azimuthal angle about the vertical axis. For a diagram see here.

The locations of the masses $$r_1 , r_2$$ are therefore given by (for a vertical $$z$$ axis): $$x_1 = sin(\theta_1)cos(\phi_1)$$ $$y_1 = sin(\theta_1)sin(\phi_1)$$ $$z_1 = -cos(\theta_1)$$

and

$$x_2 = x_1+sin(\theta_2)cos(\phi_2)$$ $$y_2 = y_1+sin(\theta_2)sin(\phi_2)$$ $$z_2 = z_1-cos(\theta_2)$$

The Lagrangian is given by the difference between the kinetic and potential energies $$L = E_k -E_p$$

$$E_k = \frac{1}{2} \left( \dot{r}_1^2 +\dot{r}_2^2 \right)$$ $$E_p = g (z_1 +z_2)$$

The Lagrangian then follows simply from taking the time derivatives of $$r_1$$ and $$r_2$$. (Note that this leads me to a very long and complicated form the Lagrangian)

• I am afraid that this problem actually IS very messy. Traditional way is to use the small amplitude approximation. – Victor Pira Nov 20 '15 at 10:54
• Hi Tom. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. – Qmechanic Nov 20 '15 at 11:10
• ...this method rapidly becomes very messy. welcome to physics ;) – Kyle Kanos Nov 20 '15 at 12:30
• Thanks for all comments. Qmechanic have read the policy now and so accept your edit :). I am happy to accept these comments as an answer. What is the protocol in this situation? Thanks all – Tom Nov 20 '15 at 13:24
• @Tom If you can't be bothered to go through the derivation of the e.o.m. yourself (it's not that tedious), you can do it easily in a program like Mathematica. – JamalS Nov 20 '15 at 16:02

## 1 Answer

Not that I know of. However if you're fine with considering only small oscillations, then you can replace $\sin \theta$ by $\theta$ and $\cos \theta$ by $1-\frac{\theta^2}{2}$. This might make things simpler although the solution you get will be acceptable for small angles.

• This is explained here – Leander Sep 30 '19 at 9:12