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Global Bifurcations
Published in LM Pismen, Working with Dynamical Systems, 2020
The origin of the Hamiltonian formalism is in classical mechanics, as we discussed in Sect. 1.1.1, but it emerged, rather unexpectedly, in a completely different context in Sect. 1.3.1. Now we encounter it again in the normal forms that can be derived though the formal procedure of the preceding section from any underlying system without special symmetries. At δ = 0, (3.25) is a conservative system. The conserved quantity is “energy” E=12y2+V(x),?V(x)=−∫f(x)dx.
Analysis of Thermal Energy Systems
Published in Steven G. Penoncello, Thermal Energy Systems, 2018
The conservation of mass equation can be developed by making the substitution, m = Ω in Equation 3.2. Mass is a conserved quantity. Therefore, the generation term is zero. This results in the following expression, ∑m˙in−∑m˙out=dmsysdt
Elementary Particles and Interactions — Overview
Published in K Grotz, H V Klapdor, S S Wilson, The Weak Interaction in Nuclear, Particle and Astrophysics, 2020
K Grotz, H V Klapdor, S S Wilson
Every conserved quantity corresponds to an invariance of the equations of motion under a given symmetry operation. We distinguish between external and internal symmetries. The external symmetries are symmetries of the space-time continuum such as translation invariance, rotation invariance, or invariance under spatial reflection through the origin. As a consequence of these invariances we have conservation of momentum, angular momentum and parity. Internal symmetries concern internal parameters of the particle wave functions, e.g. the phase of a wave function. Most quantum numbers are associated with such internal symmetries or invariances, the electric charge (see Chapter 4) is one example.
Role of vortical structures for enstrophy and scalar transport in flows with and without stable stratification
Published in Journal of Turbulence, 2021
M. M. Neamtu-Halic, J.-P. Mollicone, M. van Reeuwijk, M. Holzner
Following Neamtu-Halic et al. [41], we simulated three different flow cases, namely, a wall-jet () and two different gravity currents (). The flow cases differ in the initial Richardson number , where is a conserved quantity in the simulations, whereas the initial bulk Reynolds number is kept constant. Table 1 summarises the parameters of the simulations employed in this study. Note that the label of the flow cases indicates the value of . The results presented here are based on data over six independent xz planes, which are equally spaced in the y-direction, amounting to 280 snapshots over a period of 140, with .
The Discontinuous Asymptotic Telegrapher’s Equation (P1) Approximation
Published in Nuclear Science and Engineering, 2018
Avner P. Cohen, Roy Perry, Shay I. Heizler
Equation (41a) replaces the conservation law [Eq. (5)] and thus does not conserve particles [the conserved quantity is instead], which makes it less favorable. Equation (41b) is identical to Eq. (27), replacing with . Equations (41a) and (41b) yield a discontinuous asymptotic diffusion that does not conserves particles. By recalling that , this diffusion approximation is called the approximation.
Quantization of magnetoelectric fields
Published in Journal of Modern Optics, 2019
Energy, linear momentum, and angular momentum are constants of motion, which constitute the key physical quantities that characterize the EM field configuration. Their conservation can be cast as a continuity equation relating to a density or a flux density, or a current, associated to the conserved quantity (1). Recently, a new constant of motion and a current associated to this conserved quantity have been introduced in electrodynamics. For a circularly polarized EM plane wave, there are the EM-field chirality and the flux density of the EM-field chirality (2,3).