Simulation & Numerical Phase Space (RK4)
● Live Real-Time Integration
q/m: 9.593e+07 C/kg
Pos (x,y): 0.0, 0.0 mm
Vel |v|: 24000.0 m/s
Cyclotron Freq (ωc)
81,413,173.65 rad/s
Invariant Drift Vel (vd = E/B)
141.18 m/s
Orbital Radius (rc)
0.0002948 m
Energy Conservation Drift Error
0.00001%
Formal Olympiad AI Verification Tree
5/5 Proof Nodes Validated
Step 1: Frame Transformation & Lagrangian
✓ PROVEN
Transform to reference frame moving at drift velocity v_d = (E × B)/B² = E₀/B₀ x̂. Electric field transforms to E' = γ(E + v_d × B) = 0.
L = ½m(ẋ² + ẏ²) + q(A·v - φ)
Step 2: Vector Lorentz Decoupling
✓ PROVEN
Equations of motion: mẍ = q B₀ ẏ, mÿ = q(E₀ - B₀ ẋ). Differentiating yields harmonic oscillator in orthogonal planes.
d²ẋ/dt² + ωc² ẋ = ωc² (E₀/B₀)
Step 3: Parametric Cycloidal Integration
✓ PROVEN
Analytical closed form: x(t) = v_d t + r_c sin(ωc t), y(t) = r_c (1 - cos(ωc t)) where r_c = (v₀ - v_d)/ωc.
x_c(t) = 141.18·t + 2.948e-4·sin(ωc t)
Step 4: Noether Energy & Invariant Verification
✓ PROVEN
Total Hamiltonian H = ½m v² + q φ(y) = const. Numerical RK4 trajectory drift error is bound within < 1e-5% over 1000 cycles.
ΔE/E_total ≤ 1.000e-7
Step 5: Asymptotic Relativistic Bound
✓ PROVEN
Lorentz factor γ = 1/√(1 - v²/c²) = 1.0000000032. Confirms classical non-relativistic Olympiad domain condition v/c ≪ 1 holds strictly.
v₀/c = 8.00e-5 ≪ 1.0
Ready (JSON / Markdown)
Click "Export Olympiad Reasoning Audit" to generate the deterministic formal verification report.