I Morse Theory & Lagrangian Mechanics: Is There a Connection?

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I read somewhere that Morse originally applied his theory to the calculus of variations. I'm wondering, is this application useful in physics and mechanics, like maybe it sheds light on lagrangian mechanics? Does anyone know?
 

Thread 'About the maximal extension of local charts on a manifold'

I'm a bit confused by the conditions on the existence of coordinate basis given by Frobenius's theorem. Namely, let's take a n-dimensional smooth manifold and a set of n smooth vector fields defined on it. Suppose they are pointwise linearly independent and do commute each other (i.e. zero commutator/Lie bracket). That means they span the entire tangent space at any point and since commute, they define a local coordinate basis. What does this mean? Well, starting from any point on the...
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