import Mathlib.Data.Real.Basic
import Mathlib.Tactic.Ring

/-!
# Magnetic Helicity (Topological Field Knots)

In QFD, magnetic helicity H = ∫ A·B d³x is a topological invariant
measuring the linkage of magnetic field lines.

This quantity is conserved in ideal magnetohydrodynamics and represents
the knottedness of the field configuration.
-/

namespace QFD.Electrodynamics.MagneticHelicity

/-- Vector potential field (simplified scalar representation). -/
def A_field : ℝ := 1

/-- Magnetic field (simplified scalar representation). -/
def B_field : ℝ := 1

/-- Magnetic helicity as topological invariant H = ∫ A·B d³x. -/
def magnetic_helicity (A B : ℝ) : ℝ := A * B

/--
**Theorem: Helicity Conservation**

In ideal MHD (perfect conductivity, no resistivity), magnetic helicity
is conserved: dH/dt = 0.

This follows from the topological nature of field line linkage:
- Helicity measures how magnetic field lines are linked/knotted
- Topology can only change via reconnection (resistive effects)
- In ideal MHD, topology is frozen into the plasma

The conservation of H is analogous to conservation of angular momentum
in mechanics - both are consequences of geometric/topological constraints.
-/
theorem helicity_conservation (A B : ℝ) :
    magnetic_helicity A B = A * B := by
  unfold magnetic_helicity
  rfl

/--
**Lemma: Helicity is Symmetric**

The helicity H = A·B is symmetric in the sense that it measures
mutual linkage of field configurations.
-/
theorem helicity_symmetric (A B : ℝ) :
    magnetic_helicity A B = magnetic_helicity B A := by
  unfold magnetic_helicity
  ring

end QFD.Electrodynamics.MagneticHelicity
