Lagrange Pendulum Equation of Motion

[Andrew Deleonardis]

Hi, I've derived the equation of motion for a regular single pendulum, but do not know how to solve the differential equation.
I have the following:
r²θ''²=mg(cosθ-rsinθ)

 

[Dr.D]

Before you worry too much about a solution, I suggest that you check your derivation. That equation does not appear to be correct. If you want to continue the discussion, please give us a diagram with proper labels.

 

[mpresic]

unless r is dimensionless (probably not) your equation above cannot be correct. You are adding r sin theta to cosine theta for part of the term on the right. In general the right hand term (apparently) contains a mass, but the term on the left involves solely geometric quantities.

In addition, most of these type problems involve a small amplitude assumption. Is this the case here.

 

[Dr.D]

A small amplitude assumption is not essential. The large amplitude pendulum can be formulated just fine, although it is much more difficult to solve than with the small amplitude assumption.

 

[Andrew Deleonardis]

Before you worry too much about a solution, I suggest that you check your derivation. That equation does not appear to be correct. If you want to continue the discussion, please give us a diagram with proper labels.

You're right, I completely messed up. My mistake was a mistake correcting a mistake. I had accidentally noticed that I wasn't supposed to take the derivative of the lagrangian with respect to the r because it was a constant, and mistakenly removed only half of it from my working out, thus the extra cos.

θ''=-g/r*sin(θ)

 

[mpresic]

Yes, the small angle assumption is not necessary, and the problem is solvable using elliptic integrals. Solving the problem with the small angle assumption is commonly treated at the undergraduate level. Even at the graduate level, the small angle assumption is common, and the problems become more sophisticated because they treat many degrees of freedom (e.g. coupled equations and normal modes).