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Department "Theoretical Mechanics" > Adminstration > V.A. Kuzkin > Conservation of angular momentum

Why angular momentum is not conserved by periodic boundary conditions ?

In a paper [1] the well-known issue with the absence of conservation of angular momentum in classical particle systems with periodic boundary conditions is addressed. It is shown that conventional theory based on Noether's theorem fails to explain the simplest possible example, notably jumps of angular momentum in the case of single particle moving in a periodic cell. It is suggested to consider the periodic cell as an {\it open system}, exchanging mass, momentum, angular momentum, and energy with neighboring cells. Then the behavior of the cell is described by balance laws rather than conservation laws. It is shown using the law of angular momentum balance that the variation of the angular momentum in systems with periodic boundary conditions is a consequence of (i) the non-zero flux of angular momentum through the boundaries and (ii) torque acting on the cell due to interactions between particles in the cell with images in the neighboring cells. Three simple examples demonstrating individual and combined effect of these factors are presented. Thus the paper provides a rational explanation for the absence of angular momentum conservation in particle systems with periodic boundary conditions.

## References

[1] Kuzkin V.A. On angular momentum balance in particle systems with periodic boundary conditions // ZAMM, 2014, DOI: 10.1002/zamm.201400045 (download author's copy)