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The ergodic hypothesis assumes that a system can explore its whole phase space in the course of time.

Holonomic constraints make the system integrable and reduce its degrees of freedom, which should if I understand it correctly also break any ergodicity that would be present without these constraints.

Is non-holonomicity a necessary and sufficient condition for the ergodicity hypothesis to be true?

Can anything else break ergodicity apart from holonomic constraints?

Holonomic constraints make a system not necessarily integrable.

Symmetries also break ergodicity since they imply additional conservation laws by Noether's theorem.

Even on the manifold defined by fixed values of all conserved quantities, ergodicity is not the rule but a restrictive condition. Systems close to integrability are not ergodic for low energies, due to the KAM theorem.

We had some recent discussion on ergodicity here: http://www.physicsoverflow.org/38287

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