Editing Inverted pendulum (section)

=== From Euler-Lagrange equations ===

The generalized forces can be both written as potential energy <math>V_x</math> and <math>V_\theta</math>,

{| class="wikitable"
|-
! Generalized Forces !! Potential Energy
|-
| <math>Q_x=F</math> || <math>V_x=\int_{t_0}^tF\dot{x}\ {\rm d}t</math> 
|-
| <math>Q_\theta=mgl\sin\theta</math> || <math>V_\theta=mgl\cos\theta</math>
|}

According to the [[D'Alembert's principle]], generalized forces and potential energy are connected:

:<math>Q_j = \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial V}{\partial \dot{q}_j} - \frac{\partial V}{\partial q_j} \,,</math>

However, under certain circumstances, the potential energy is not accessible, only generalized forces are available.

After getting the [[Lagrangian mechanics|Lagrangian]] <math>L=T-V</math>, we can also use [[Euler–Lagrange equation]] to solve for equations of motion:

:<math>\frac {\partial L}{\partial x} - \frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial  L}{\partial \dot{x}} \right ) = 0</math>,
:<math>\frac {\partial L}{\partial \theta} - \frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial  L}{\partial \dot{\theta}} \right ) = 0</math>.

The only difference is whether to incorporate the generalized forces into the potential energy <math>V_j</math> or write them explicitly as <math>Q_j</math> on the right side, they all lead to the same equations in the final.