  Gyrokinetic theory is arguably the most important tool for numerical studies
of transport physics in magnetized plasmas. However, exact local
energy-momentum conservation law for the electromagnetic gyrokinetic system has
not been found despite continuous effort. Without such a local conservation
law, energy-momentum can be instantaneously transported across spacetime, which
is unphysical and casts doubt on the validity of numerical simulations based on
the gyrokinetic theory. Standard Noether's 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 developed 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 law, for the electromagnetic gyrokinetic system. A weak
Euler-Lagrange equation is established to replace the standard Euler-Lagrange
equation for the particles. It is discovered that an induced weak
Euler-Lagrange current enters the local conservation laws. And it is the new
physics captured by the high-order field theory on heterogeneous manifolds.
