Discovering exact, gauge-invariant, local energy–momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds

Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas. However, exact local energy–momentum conservation laws for the electromagnetic gyrokinetic system have not been found despite continuous effort. Without such local conservation laws, energy and momentum can be instantaneously transported across spacetime, which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory. The standard Noether procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic field reside on different manifolds. To overcome this difficulty, we develop a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact, local conservation laws, in particular the energy–momentum conservation laws, for the electromagnetic gyrokinetic system. A weak Euler–Lagrange (EL) equation is established to replace the standard EL equation for the particles. It is discovered that an induced weak EL current enters the local conservation laws, and it is the new physics captured by the high-order field theory on heterogeneous manifolds. A recently developed gauge-symmetrization method for high-order electromagnetic field theories using the electromagnetic displacement-potential tensor is applied to render the derived energy–momentum conservation laws electromagnetic gauge-invariant.