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Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
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Velocity in generalized coordinates
Consider the expression of velocity in generalized coordinates, $\mathbf v = \frac {d \mathbf x}{dt}$, where $\mathbf x = \mathbf x (\mathbf q(t), t)$.
We end up with a total derivative, i.e $$\...
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