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Classical Mechanics and Field Theory
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
i.e., the wave equation for the field q(x, t). The invariance of this Lagrangian density under space-time translations δya=ka for constant ka was discussed already in Example 8.26, where we found the conserved current () Ttt=ρℓ2(qt2+c2qx2),Ttx=−ρℓc2qtqx
Lie symmetry analysis, power series solutions and conservation laws of the time-fractional breaking soliton equation
Published in Waves in Random and Complex Media, 2022
Zhi-Yong Zhang, Hui-Min Zhu, Jia Zheng
We use the idea of nonlinear self-adjointness for the fractional case to construct conservation laws of Equation (1) since it does not admit a Lagrangian. A conservation law of Equation (1) is defined as for all solutions of Equation (1), where and are three functions of t, x, y, u, integer-order derivatives of u and fractional integrals and derivatives of u, and is called a conserved current. It should be noted that, since the conserved current for Equation (1) involves fractional derivatives or fractional integrals, thus the total derivatives and are extended to contain the terms of the partial derivatives of fractional derivatives or fractional integrals, for example, A conservation law is trivial for Equation (1) if the components and of the conserved current vanish identically on the solution of Equation (1), or holds identically for any .
Implications of causality for quantum biology – I: topology change
Published in Molecular Physics, 2018
(Consequences of current density functional): The extremisation of the Hodge norm of the gauge curvature scalar, , implies the existence of three globally defined Noether fields: (1) a conserved symmetric gauge curvature stress-energy tensor, τκεym, (2) a conserved angular-momentum tensor, Mβγλym and (3) a conserved current tensor, Jαym, such that ∂αJαym = 0 where:
Reduction of the classical electromagnetism to a two-dimensional curved surface
Published in Journal of Modern Optics, 2019
The electromagnetic fields can be consistently coupled only to a conserved current, i.e. satisfying the continuity equation which is the obvious consequence of equations (3a) and (3c).