Let me give some context. I was just doing some math, calculating the energy of a pendulum and I've arrived to the following: $$ E = \frac{1}{2}m \dot{\theta}^2 l ^2 + mgl(1 - \cos{\theta}) \\ \frac{\mathrm d E}{\mathrm d t} = 0\\ $$ Establishing this conditions: $$ \frac{\mathrm d E}{\mathrm d t} = \frac{1}{2}m l^2 \left(\ddot{\theta}\dot{\theta} + \dot{\theta}\ddot{\theta} \right) + mgl \dot{\theta} \sin{\theta} \\ \frac{\mathrm d E}{\mathrm d t} = m l^2 \ddot{\theta}\dot{\theta} + mgl \dot{\theta} \sin{\theta} = 0 \\ $$ Arriving to the equation of motion of the pendulum: $$ \ddot{\theta} + \frac{g}{l} \sin{\theta} = 0 \\ $$ Not only it gives me this particular equation, but for example the spring-mass oscillator equation and Newton's Laws.
But why this happends? I tought that you could only arrive to equations of motion, using the energy of the system, by using Lagrangian Mechanics or Hamiltonian Mechanics. Is this method similar? It could give me the equations of motion of any system, providing I know the energy?
And, finally, why you can obtain the equations of motion of a system or particle form its energy?
