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ContinuousSymmetriesandConservationLaws. Noether’sTheorem

Nature always creates the best of all (384 BC 322 BC)8 Continuous Symmetries and conservation s TheoremIn many physical systems, the action is invariant under some continuous set oftransformations. In such systems, there exist local and globalconservation lawsanalogous to current and charge conservation in electrodynamics. The analogs ofthe charges can be used to generate the symmetry transformation, from which theywere derived, with the help of Poisson brackets, or after quantization, with the helpof Point MechanicsConsider a simple mechanical system with a generic actionA=ZtbtadtL(q(t), q(t),t).

While the conservation law of energy follows from the symmetry of the action under time translations, conservation laws ofmomentum and angular momentum are found if the action is invariant under translations and rotations. Consider a Lagrangian of a point particle in a euclidean space L= L(xi(t),x˙i(t),t). (8.35)

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