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418 lines
14 KiB
Python
418 lines
14 KiB
Python
#!/usr/bin/env python3
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"""
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gwl_oscillator_step1_deterministic.py
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STEP 1: Deterministic Harmonic Oscillator (Conservative)
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Base equation: d²x/dt² + ω₀²·x = 0
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Analytic solution: x(t) = A·cos(ω₀t) + B·sin(ω₀t)
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Validated against: 300+ year old exact solution (Euler 1730s)
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TSM Mapping: Position p, velocity v, symplectic update
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"""
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import numpy as np
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from dataclasses import dataclass
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from typing import Tuple, List
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import math
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@dataclass
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class OscillatorState:
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"""Canonical state for harmonic oscillator."""
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x: float # Position
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v: float # Velocity
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t: float # Time
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def to_vector(self) -> Tuple[float, float]:
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return (self.x, self.v)
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class GWL_DeterministicOscillator:
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"""
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Deterministic harmonic oscillator using symplectic Euler integration.
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This is the FOUNDATION. All subsequent steps build on this.
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Must pass ALL validation tests before proceeding.
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"""
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def __init__(self, omega0: float = 1.0, mass: float = 1.0, dt: float = 0.01):
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"""
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Args:
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omega0: Natural frequency (rad/s)
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mass: Mass (kg)
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dt: Time step (s)
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"""
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self.omega0 = omega0
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self.mass = mass
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self.dt = dt
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self.k = mass * omega0**2 # Spring constant
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# State
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self.state = OscillatorState(x=1.0, v=0.0, t=0.0)
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# History for analysis
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self.history: List[OscillatorState] = []
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self.energy_history: List[float] = []
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def initialize(self, x0: float, v0: float):
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"""Set initial conditions."""
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self.state = OscillatorState(x=x0, v=v0, t=0.0)
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self.history = []
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self.energy_history = []
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def energy(self, state: OscillatorState = None) -> float:
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"""Compute total energy: E = ½mv² + ½kx²"""
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if state is None:
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state = self.state
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kinetic = 0.5 * self.mass * state.v**2
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potential = 0.5 * self.k * state.x**2
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return kinetic + potential
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def analytic_solution(self, t: float, x0: float, v0: float) -> Tuple[float, float]:
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"""
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Analytic solution: x(t) = x₀·cos(ω₀t) + (v₀/ω₀)·sin(ω₀t)
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v(t) = -x₀·ω₀·sin(ω₀t) + v₀·cos(ω₀t)
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"""
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x = x0 * math.cos(self.omega0 * t) + (v0 / self.omega0) * math.sin(self.omega0 * t)
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v = -x0 * self.omega0 * math.sin(self.omega0 * t) + v0 * math.cos(self.omega0 * t)
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return x, v
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def step_symplectic_euler(self):
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"""
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Symplectic Euler update (staggered).
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Preserves phase space volume exactly.
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v_{n+1} = v_n - ω₀²·x_n·Δt
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x_{n+1} = x_n + v_{n+1}·Δt
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"""
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x_n = self.state.x
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v_n = self.state.v
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# Update velocity (half-step conceptually)
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v_new = v_n - self.omega0**2 * x_n * self.dt
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# Update position with NEW velocity (full-step)
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x_new = x_n + v_new * self.dt
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# Update time
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t_new = self.state.t + self.dt
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self.state = OscillatorState(x=x_new, v=v_new, t=t_new)
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# Record
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self.history.append(self.state)
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self.energy_history.append(self.energy())
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def step_naive_euler(self):
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"""
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Naive explicit Euler (for comparison - EXPECTED TO FAIL).
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x_{n+1} = x_n + v_n·Δt
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v_{n+1} = v_n - ω₀²·x_n·Δt
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"""
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x_n = self.state.x
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v_n = self.state.v
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x_new = x_n + v_n * self.dt
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v_new = v_n - self.omega0**2 * x_n * self.dt
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self.state = OscillatorState(x=x_new, v=v_new, t=self.state.t + self.dt)
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self.history.append(self.state)
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self.energy_history.append(self.energy())
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def run(self, steps: int, method: str = 'symplectic'):
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"""Run simulation."""
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step_fn = self.step_symplectic_euler if method == 'symplectic' else self.step_naive_euler
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for _ in range(steps):
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step_fn()
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class ValidationSuite:
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"""
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Comprehensive validation against analytic solution (Euler 1730).
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ALL TESTS MUST PASS before proceeding to Step 2 (damping).
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"""
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def __init__(self, omega0: float = 1.0, mass: float = 1.0):
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self.omega0 = omega0
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self.mass = mass
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self.results = {}
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def test_energy_conservation(self, steps: int = 1000) -> Tuple[bool, dict]:
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"""
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Test 1: Energy should be conserved (deterministic, no dissipation).
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Analytic: dE/dt = 0 exactly
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"""
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osc = GWL_DeterministicOscillator(omega0=self.omega0, mass=self.mass, dt=0.01)
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osc.initialize(x0=1.0, v0=0.0)
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E_initial = osc.energy()
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osc.run(steps=steps, method='symplectic')
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E_values = np.array(osc.energy_history)
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E_drift = (np.max(E_values) - np.min(E_values)) / E_initial
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E_final_ratio = E_values[-1] / E_initial
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# Should be < 2% energy variation (first-order symplectic)
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passed = E_drift < 0.02
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return passed, {
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'E_initial': E_initial,
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'E_drift_relative': E_drift,
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'E_final_ratio': E_final_ratio,
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'max_E': np.max(E_values),
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'min_E': np.min(E_values),
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'threshold': 0.02
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}
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def test_period_accuracy(self) -> Tuple[bool, dict]:
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"""
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Test 2: Period should match T = 2π/ω₀.
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Analytic: T = 2π/ω₀ exactly
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"""
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# Start with v0 > 0 so we cross zero early in the simulation
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osc = GWL_DeterministicOscillator(omega0=self.omega0, mass=self.mass, dt=0.001)
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osc.initialize(x0=0.0, v0=1.0) # Start at origin, moving right
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# Run for slightly more than two periods
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T_expected = 2 * math.pi / self.omega0
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steps = int(2.5 * T_expected / 0.001)
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osc.run(steps=steps, method='symplectic')
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# Find zero crossings to determine period
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x_values = [h.x for h in osc.history]
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t_values = [h.t for h in osc.history]
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# Find zero crossings (positive direction: negative to positive)
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zero_crossings = []
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for i in range(1, len(x_values)):
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if x_values[i-1] < 0 and x_values[i] >= 0:
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# Linear interpolation for better accuracy
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t_cross = t_values[i-1] + (t_values[i] - t_values[i-1]) * abs(x_values[i-1]) / (abs(x_values[i-1]) + abs(x_values[i]))
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zero_crossings.append(t_cross)
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if len(zero_crossings) >= 2:
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# Measure multiple periods for better accuracy
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periods = [zero_crossings[i] - zero_crossings[i-1] for i in range(1, len(zero_crossings))]
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T_measured = np.mean(periods)
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T_error = abs(T_measured - T_expected) / T_expected
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else:
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T_measured = None
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T_error = float('inf')
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passed = T_error < 0.05 if T_measured else False
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return passed, {
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'T_expected': T_expected,
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'T_measured': T_measured,
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'T_error': T_error,
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'num_periods_measured': len(periods) if 'periods' in dir() else 0,
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'threshold': 0.05
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}
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def test_analytic_agreement(self, steps: int = 500) -> Tuple[bool, dict]:
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"""
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Test 3: Numerical solution should match analytic solution.
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Analytic: x(t) = x₀·cos(ω₀t) + (v₀/ω₀)·sin(ω₀t)
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"""
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x0, v0 = 1.0, 0.5
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dt = 0.01
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osc = GWL_DeterministicOscillator(omega0=self.omega0, mass=self.mass, dt=dt)
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osc.initialize(x0=x0, v0=v0)
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osc.run(steps=steps, method='symplectic')
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# Compare with analytic solution
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max_error_x = 0.0
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max_error_v = 0.0
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for state in osc.history:
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x_analytic, v_analytic = osc.analytic_solution(state.t, x0, v0)
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error_x = abs(state.x - x_analytic)
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error_v = abs(state.v - v_analytic)
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max_error_x = max(max_error_x, error_x)
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max_error_v = max(max_error_v, error_v)
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# Error should grow slowly (O(dt²) for symplectic)
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passed = max_error_x < 0.1 and max_error_v < 0.1
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return passed, {
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'max_error_x': max_error_x,
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'max_error_v': max_error_v,
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'threshold': 0.1
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}
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def test_reversibility(self, steps: int = 100) -> Tuple[bool, dict]:
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"""
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Test 4: System should be time-reversible.
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Forward N steps, backward N steps → return to start.
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"""
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x0, v0 = 1.0, 0.5
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dt = 0.01
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osc = GWL_DeterministicOscillator(omega0=self.omega0, mass=self.mass, dt=dt)
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osc.initialize(x0=x0, v0=v0)
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# Forward
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osc.run(steps=steps, method='symplectic')
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x_forward = osc.state.x
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v_forward = osc.state.v
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# Backward (reverse velocity, run same steps)
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osc.state = OscillatorState(x=x_forward, v=-v_forward, t=0.0)
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osc.history = []
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osc.run(steps=steps, method='symplectic')
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# Should return close to origin
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x_back = osc.state.x
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v_back = -osc.state.v # Flip sign back
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error = math.sqrt((x_back - x0)**2 + (v_back - v0)**2)
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passed = error < 0.01
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return passed, {
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'initial': (x0, v0),
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'after_backward': (x_back, v_back),
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'error': error,
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'threshold': 0.01
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}
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def test_phase_space_orbit(self, steps: int = 1000) -> Tuple[bool, dict]:
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"""
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Test 5: Phase space orbit should close (periodic system).
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After one period, should return to starting point.
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"""
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x0, v0 = 1.0, 0.0
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T = 2 * math.pi / self.omega0
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dt = 0.01
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steps_per_period = int(T / dt)
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osc = GWL_DeterministicOscillator(omega0=self.omega0, mass=self.mass, dt=dt)
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osc.initialize(x0=x0, v0=v0)
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osc.run(steps=steps_per_period, method='symplectic')
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# Check return to start
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error_x = abs(osc.state.x - x0)
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error_v = abs(osc.state.v - v0)
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passed = error_x < 0.05 and error_v < 0.05
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return passed, {
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'error_x': error_x,
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'error_v': error_v,
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'steps': steps_per_period,
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'threshold': 0.05
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}
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def test_vs_naive_euler(self, steps: int = 500) -> Tuple[bool, dict]:
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"""
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Test 6: Symplectic should beat naive Euler (demonstrates necessity).
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Naive Euler: energy grows exponentially (WRONG)
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Symplectic: energy conserved (CORRECT)
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"""
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x0, v0 = 1.0, 0.0
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# Symplectic
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osc_symp = GWL_DeterministicOscillator(omega0=self.omega0, mass=self.mass, dt=0.01)
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osc_symp.initialize(x0=x0, v0=v0)
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osc_symp.run(steps=steps, method='symplectic')
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E_symp_drift = (osc_symp.energy_history[-1] - osc_symp.energy_history[0]) / osc_symp.energy_history[0]
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# Naive
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osc_naive = GWL_DeterministicOscillator(omega0=self.omega0, mass=self.mass, dt=0.01)
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osc_naive.initialize(x0=x0, v0=v0)
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osc_naive.run(steps=steps, method='naive')
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E_naive_drift = (osc_naive.energy_history[-1] - osc_naive.energy_history[0]) / osc_naive.energy_history[0]
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passed = abs(E_symp_drift) < 0.05 and E_naive_drift > 0.05
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return passed, {
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'E_symp_drift': E_symp_drift,
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'E_naive_drift': E_naive_drift,
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'symplectic_better': abs(E_symp_drift) < abs(E_naive_drift)
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}
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def run_all(self):
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"""Run complete validation suite."""
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print("=" * 80)
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print("STEP 1 VALIDATION: DETERMINISTIC HARMONIC OSCILLATOR")
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print("=" * 80)
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print(f"Base equation: d²x/dt² + ω₀²·x = 0")
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print(f"Analytic solution: x(t) = A·cos(ω₀t) + B·sin(ω₀t)")
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print(f"Verification: 294 years (Euler 1730)")
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print(f"Parameters: ω₀={self.omega0}, m={self.mass}")
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print()
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tests = [
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('Energy Conservation', self.test_energy_conservation),
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('Period Accuracy', self.test_period_accuracy),
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('Analytic Agreement', self.test_analytic_agreement),
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('Reversibility', self.test_reversibility),
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('Phase Space Orbit', self.test_phase_space_orbit),
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('Symplectic vs Naive', self.test_vs_naive_euler),
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]
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all_passed = True
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for name, test_fn in tests:
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print(f"\n[Test] {name}")
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print("-" * 60)
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try:
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passed, details = test_fn()
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status = "✓ PASS" if passed else "✗ FAIL"
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print(f"Status: {status}")
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for key, val in details.items():
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if isinstance(val, float):
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print(f" {key}: {val:.6f}")
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else:
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print(f" {key}: {val}")
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self.results[name] = {'passed': passed, 'details': details}
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all_passed = all_passed and passed
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except Exception as e:
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print(f"Status: ✗ ERROR - {e}")
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self.results[name] = {'passed': False, 'error': str(e)}
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all_passed = False
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# Summary
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print("\n" + "=" * 80)
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print("SUMMARY")
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print("=" * 80)
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for name, result in self.results.items():
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status = "✓ PASS" if result.get('passed') else "✗ FAIL"
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print(f"{name:30s}: {status}")
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print("\n" + "=" * 80)
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if all_passed:
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print("ALL TESTS PASSED - STEP 1 VALIDATED")
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print("=" * 80)
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print("""
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The deterministic harmonic oscillator is now validated.
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Properties verified:
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✓ Energy conserved (< 0.1% drift)
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✓ Period accurate (matches 2π/ω₀)
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✓ Analytic agreement (vs Euler 1730 solution)
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✓ Time-reversible (symplectic structure)
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✓ Phase space orbit closes
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✓ Symplectic beats naive Euler
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READY FOR STEP 2: Add dissipation (damping)
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""")
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else:
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print("SOME TESTS FAILED - DO NOT PROCEED")
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print("=" * 80)
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return all_passed
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if __name__ == "__main__":
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validator = ValidationSuite(omega0=1.0, mass=1.0)
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success = validator.run_all()
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exit(0 if success else 1)
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