Consider a simple pendulum, with a mass *m* on the end of a rod of length *l*. In terms of the angle *θ* from the downward direction, the equation of motion for this pendulum is .

Now, suppose the pivot to which this pendulum is attached moves vertically in a sinusoidal motion of small amplitude, so that its vertical position is given by , with . What then is the new equation of motion? What happens for small-angle oscillations ()?

The position of the end mass in the plane of the pendulum, relative to the mid-cycle position of the pivot (where ), is .

Taking the time derivatives of both components,

,

and we can sum their squares to obtain the velocity squared of the mass:

,

and so the kinetic energy is thus

.

Similarly, our potential energy is simply , and so the Lagrangian is

.

and thus the Euler-Lagrange equation gives us equation of motion

.

Since , then , and so our equation of motion becomes

;

the gravitational acceleration *g* is replaced by . If we consider the (accelerating, non-inertial) reference frame of the pivot, the acceleration is , producing a proportional fictitious force, so that the net effect is the equation of motion above.

For small *θ*, then , and so our equation of motion becomes , or, dividing by *l*,

. The undriven pendulum has small-angle frequency ; and let us define . Using this, our equation of motion is

.

This equation cannot be solved analytically.

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Tags: Euler-Lagrange, Friday Physics, Pendulum, physics

This entry was posted on March 5, 2010 at 1:48 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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March 12, 2010 at 12:10 am |

[…] Friday 112 By twistedone151 Continuing from last time, we noted that the equation of motion for our driven pendulum, cannot be solved analytically even […]