The local conserved quantities of the closed XXZ chain

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

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Conservation laws

10.1093/oso/9780198786399.003.0007 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

Dynamical System ◽

Angular Momentum ◽

Conservation Laws ◽

Total Energy ◽

Superposition Principle ◽

Momentum Conservation ◽

Conserved Quantities ◽

Angular Momentum Conservation ◽

Third Law ◽

The Law

This chapter defines the conserved quantities associated with an isolated dynamical system, that is, the quantities which remain constant during the motion of the system. The law of momentum conservation follows directly from Newton’s third law. The superposition principle for forces allows Newton’s law of motion for a body Pa acted on by other bodies Pa′ in an inertial Cartesian frame S. The law of angular momentum conservation holds if the forces acting on the elements of the system depend only on the separation of the elements. Finally, the conservation of total energy requires in addition that the forces be derivable from a potential.

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On the accuracy of the binary-collision algorithm in particle-in-cell simulations of magnetically confined fusion plasmas

Journal of Plasma Physics ◽

10.1017/s0022377821000398 ◽

2021 ◽

Vol 87 (2) ◽

Author(s):

Timo P. Kiviniemi ◽

Eero Hirvijoki ◽

Antti J. Virtanen

Keyword(s):

Finite Difference ◽

Electric Fields ◽

Finite Size ◽

Binary Collision ◽

Conserved Quantities ◽

Fusion Plasmas ◽

Fusion Plasma ◽

Particle In Cell ◽

Magnetically Confined ◽

Scale Length

Ideally, binary-collision algorithms conserve kinetic momentum and energy. In practice, the finite size of collision cells and the finite difference in the particle locations affect the conservation properties. In the present work, we investigate numerically how the accuracy of these algorithms is affected when the size of collision cells is large compared with gradient scale length of the background plasma, a parameter essential in full- $f$ fusion plasma simulations. Additionally, we discuss implications for the conserved quantities in drift-kinetic formulations when fluctuating magnetic and electric fields are present: we suggest how the accuracy of the algorithms could potentially be improved with minor modifications.

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Mass and spin for classical strings in dS3

Journal of High Energy Physics ◽

10.1007/jhep05(2021)277 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Klaas Parmentier

Keyword(s):

Black Hole ◽

Evolution Equation ◽

Cosmic Strings ◽

Center Of Mass ◽

Open String ◽

Conserved Quantities ◽

Numerical Example ◽

All Solutions

Abstract We demonstrate that all rigidly rotating strings with center of mass at the origin of the dS3 static patch satisfy the Higuchi bound. This extends the observation of Noumi et al. for the open GKP-like string to all solutions of the Larsen-Sanchez class. We argue that strings violating the bound end up expanding towards the horizon and provide a numerical example. Adding point masses to the open string only increases the mass/spin ratio. For segmented strings, we write the conserved quantities, invariant under Gubser’s algebraic evolution equation, in terms of discrete lightcone coordinates describing kink collisions. Randomly generated strings are found to have a tendency to escape through the horizon that is mostly determined by their energy. For rapidly rotating segmented strings with mass/spin < 1, the kink collisions eventually become causally disconnected. Finally we consider the scenario of cosmic strings captured by a black hole in dS and find that horizon friction can make the strings longer.

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