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In my experience it is often said that the Lagrangian formulation of mechanics can be much much more convenient because the form of the (E-L) equations remains the same whatever coordinates we choose, allowing us to pick as convenience dictates. Often this is exemplified by an example featuring some system constrained to a circle or a sphere, or something.

It is true that the Second Law $\mathbf{F} = m\ddot{\mathbf{x}}$ requires us to express the system in special "inertial" coordinates, but this is fixed easily by introducing a connection and writing $\mathbf{F} = \nabla_{\dot{\mathbf{x}}}\dot{\mathbf{x}}$. Of course, introducing a connection is conceptually identical with declaring certain curves to be geodesics which is conceptually identical with the original speak about "inertial frames." However, it is coordinate-invariant, and as far as I'm concerned coordinate-invariant is coordinate-invariant is coordinate-invariant.

Qmechanic
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DPatt
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Well, both work in any coordinate system, but the form of Newton's law can and does change, while the form of the Euler-Lagrange equations do not change. For a very basic example, consider Newton's second law vs the Euler-Lagrange equations in polar coordinates. Newton's second law changes significantly, and becomes $$ \begin{align} F_r&=m\left(\ddot{r}-r\dot{\phi}\right)\\ F_\phi &= m\left(r\ddot{\phi}+2\dot{r}\dot{\phi}\right) \end{align} $$ This is far different then the simple Cartesian form: $$ \begin{align} F_x &= m\ddot{x}\\ F_y &= m\ddot{y} \end{align} $$ Meanwhile, the Euler-Lagrange equations stay precisely the same in form: $$ \begin{align} \frac{\partial L}{\partial r}&=\frac{d}{dt}\frac{\partial L}{\partial \dot{r}}\\ \frac{\partial L}{\partial \phi}&=\frac{d}{dt}\frac{\partial L}{\partial \dot{\phi}} \end{align} $$ compared to $$ \begin{align} \frac{\partial L}{\partial x}&=\frac{d}{dt}\frac{\partial L}{\partial \dot{x}}\\ \frac{\partial L}{\partial y}&=\frac{d}{dt}\frac{\partial L}{\partial \dot{y}} \end{align} $$ So, you'll get the same results no matter what (both are equivalent), but the Euler-Lagrange equations are a whole lot easier to use then Newton's law in certain coordinate systems.

John Dumancic
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