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The situation, in brief, is that newtonian physics is incapable of predicting conservation of angular momentum, but no isolated system has yet been encountered experimentally for which
angular momentum is not conserved. We conclude that conservation of angular momentum is an independent physical law, and until a contradiction is observed, our physical understanding must be guided by it.

Dan Kleppner

Why Is newtonian physics incapable of predicting it? Isn't it a consequence of Newton's laws of dynamics?

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You are quoting out of context.

The quote you present is from page 307, the following is from page 306

The situation shown in figure (a) corresponds to the case of central forces, and we conclude that the conservation of angular momnetum following from Newton's laws in the case of central force motion. However, Newton's laws do not explicitly require forces to be central. We must conclude that Newton's laws have no direct bearing on whether or not angular momentum of an isolates system is conserved, since these laws do not in themselves exclude the situation shown in figure (b)

Newtonian mechanics comes with the implicit assumption that the effect of any force is independent of spatial orientation. If you grant that assumption then conservation of angular momentum is a logical consequence.

Obviously, if you decline granting one of the fundamental assumptions of newtonian mechanics then you have moved to a realm where newtonian mechanics may not apply.

As I understand it, Kleppner is arguing that we should allow for the possibility that the Newtonian assumption that implies conservation of angular momentum may not hold good.

Kleppner does assert that within its realm of applicability newtonian mechanics implies conservation of angular momentum.

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