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In 2 dimention dynamics, if angular momentum is conserved: mr^2(theta dot)=constant, does that mean degree of freedom is reduced from 2 to 1?

I think it should since r and theta(although theta dot is written in angular momentum conservation) is related.

Also if it's right, shouldn't we add this as a constaint in lagragian equation?

My questions arised from derivng angular momentum conservation in central force part in marion classical dynamics book

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If you write down the lagrangian for a particle in polar coordinates you will find that the conservation of momentum rolls out of the Euler-Lagrange equation for your lagrangian.

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