Euler-Lagrange Equation

Given as

$$\frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q}}-\frac{\partial \mathcal{L}}{\partial q}=Q$$

Or we can write it as

$$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right)-\left(\frac{\partial T}{\partial q}-\frac{\partial V}{\partial q}\right)=Q$$

Where $Q$ is the set of Generalized Forces acting on the system, and $q$ is the set of Generalized Coordinates defined for the system.

Doesn’t matter which one you choose

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This can be further rewritten as

$$\frac{d}{dt}{\nabla }_{\dot{q}}\mathcal{L}-{\nabla }_q\mathcal{L}=0$$

This is allowed since ${\nabla }_{\dot{q}}\mathcal{L}={\frac{\partial \mathcal{L}}{\partial \dot{q}}}^T$. We prefer to work with “standing-vector form”, which is why we do this.