Hello,

I've a problem to calculate the Position of a pendulum as a function of theta.

For example: $\theta (t)$ is a function of time which returns the angle made by the pendulum at a particular instant wrt it's equilibrium Position.

So,

$$

T = \dfrac 12 m l^2 \dot \theta^2

$$$$

U = - mgl \cos \theta

$$

$$

L(\theta, \dot \theta) = \dfrac 12 m l^2 \dot \theta^2 + mgl \cos \theta

$$

Using, the Euler - Lagrangian Formula,

$$

\dfrac d{dt} \left ( \dfrac{\partial L}{\partial \dot \theta}\right) - \dfrac{\partial L}{\partial \theta} = 0

$$

We get,

$$

\boxed{\ddot \theta =- \dfrac gl \sin \theta}

$$

which is the equation of motion.

But, most of the derivations, I've seen/read go this way:

$$

\ddot \theta = \dfrac gl \theta \quad \dots \quad (\text{As, } \sin \theta \approx \theta, \theta \rightarrow 0) \tag{*}

$$

$$

\theta (t) = \cos \left ( \sqrt{\dfrac gl} t \right)

$$

Because it satisfies $(*)$

So, I've 2 questions here.

Other possible solutions of the Second Order Differential Equations exist like $\theta (t) = e^{\left( \sqrt{\dfrac gl}t \right)}$. So,

why we choose that only one? One would argue that the sine function

oscillates similar to the pendulum, so this makes sense to accept the

sine one. But, in general case, when we solve the Lagrangian and get

the equation of motion in differential form, then there are tons of

complex situation possible, How can you determine which kind is

needed?How can we solve the Second Order Differential Equation $\ddot \theta = - \dfrac gl \sin \theta$ and get an exact formula for that?

Thanks :)