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I have read that in his Mecanique Analytique [1788], Lagrange sought a “coordinate invariant expression for mass times acceleration”.

The discussion regarding this is given in 'Marsden and Ratiu [15, Page 231]', but it includes an explanation that is too technical for me right now since it uses symplectic geomtery to explain this.

What exactly does this mean? What remains invariant here?

$F=ma$ is anyways coordinate invariant so surely that is not what is being referred to here or is it?

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$\vec{F} = m\vec{a}$ is Newton's second law and it is a vector equation. That means, in 3D space, there are three components for which you have to solve for. Let us consider a simpler case, of a free particle in 2 dimensions.

Now we begin to see what coordinate invariance means. If you study the system using Cartesian coordinates, the equations of motion then are $m \ddot{x} = 0$ and $m \ddot{y} = 0$. This is fairly simple.

However, if you solve the same system in plane polar coordinates the equations of motion are NOT $m \ddot{r} = 0$ and $m \ddot{\theta} = 0$. Further, the two coordinates in the plane polar system $r, \theta$ do not even have the same dimensions. So simply replacing the cartesian coordinates with the polar coordinates in the equations of motion cannot work.

The Euler Lagrange equations of motion are invariant in the sense that as long as I have a set of coordinates {${q}$} and velocities {${\dot q}$}, you can always have equations of motion of the form:

$ \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot q} = 0 $

where L is the Lagrangian of the system. So the q here could refer to coordinates in the Cartesian system or in the polar system. It is in this sense that they are coordinate invariant.

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