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## Homework Statement

WWe know the model of an ideal pendulum at rest is given by

[tex]L \ddot{\theta} + g sin \theta =0, t\geq0[/tex]

[tex] \dot{\theta}(0)=0 [/tex]

[tex] \theta(0)=\theta_0 [/tex]

where [tex]\theta(t)[/tex] is the pendulum angle at time t, L is the length of the pendulum, and g is gravity.

Now, consider the total energy per unit mass of the pendulum given by

[tex]E(\theta,\dot{\theta}) = \frac{1}{2}L \dot{\theta}^2 - g cos \theta[/tex]

Show that this equation is constant along the previously given solutions (the first 3 equations initially given, lets call them EQ [1] ), [tex]\forall t\geq0[/tex]. Use this result to show that the solution of [1] should satisfy

[tex]\frac{1}{2}L\dot{\theta}^2-g cos \theta + g cos \theta_0 = 0, \forall t\geq0[/tex]

(lets call this EQN [2])

## Homework Equations

we know that if E(theta, thetadot) is a constant, then

[tex]\frac{d E(\theta,\dot{\theta})}{dt} = 0 [/tex]

## The Attempt at a Solution

so, taking the derivative of the energy, we have:

[tex]\frac{d E(\theta,\dot{\theta})}{dt} = \frac{d}{dt}[\frac{1}{2} L \dot{\theta}^2 - g cos \theta][/tex]

[tex] = L \ddot{\theta} +\dot{\theta}g sin\theta[/tex]

however, this does not seem to equal [1], as [tex]\dot{\theta}gsin\theta [/tex] never appears in the solution. what am I doing wrong?

or do we not take the time derivative of the second term in the energy equation (the sin(theta) term), thus giving

[tex]\frac{d E(\theta,\dot{\theta})}{dt} =L \ddot{\theta} +g sin\theta[/tex] which is equal to zero by definition of [1], thus if the derivative of the energy equation is equal to zero, then by definition of constant, energy is constant!

now, i am in confusion as how to use this result to prove EQN [2]?

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