- #1

Pythagorean

Gold Member

- 4,292

- 278

## Homework Statement

Standard double pendulum setup. A string with mass, connected to a string with a mass, mounted to the ceiling. Given is m1,m2,l1,l2

a) choose a suitable set of coordinates and write a lagrangian function, assuming it swings in a single vertical plane (I did this, using L = T - U)

b)write out lagrange's equations and show that they reduce to the equations for a pair of coupled harmonic oscillators. (here's where my problem arises)

## Homework Equations

The Lagrangian

d/dt[dL/(dq/dt)] - dL/dq = 0

[tex]\frac {d}{dt} \frac {\partial L}{\partial d \theta_k[/tex]

## The Attempt at a Solution

My issue is really a implicit/explicit differentiation problem.

I come up with a term under the d/dt (first term) of the lagrangian that involves three variables (all degrees) in this form:

(dx1/dt)*sin(x1 - x2)

when I take the time derivative of this, how do I handle the x1 and x2 (which are actually angles theta in my written notation)

I realize the first term (by the product rule) would be:

(d^2x1/dt^2)*sin(x1 - x2)

but how do I handle the two angles under the sin term that have no explicit time dependence?

Thank you for your help.

LATEX VERSION BELOW (probably being updated, I'm slow at it)

[tex] \frac {d}{dt} \left \dot{\theta_1}sin(\theta_1 - \theta_2) \right

= \ddot{\theta_1}sin(\theta_1 - \theta_2) + \dot{\theta_1} (?) + \dot{\theta_1} (?)[/tex]

the above equation is what I have, where I don't know what to do for the (?) that involves taking the time derivative of theta (which has no explicit time-dependence)

Last edited: