physics-solver

# Physics Solver

Physics computation and problem solving. Venv: `source /Users/zhangmingda/clawd/.venv/bin/activate`

## Physical Constants

```python
from scipy import constants as const
import numpy as np

# Key constants
c = const.c           # speed of light (m/s)
h = const.h           # Planck's constant (J·s)
hbar = const.hbar     # reduced Planck's constant
k_B = const.k         # Boltzmann constant (J/K)
e = const.e           # elementary charge (C)
m_e = const.m_e       # electron mass (kg)
m_p = const.m_p       # proton mass (kg)
G = const.G           # gravitational constant
N_A = const.N_A       # Avogadro's number
epsilon_0 = const.epsilon_0  # vacuum permittivity
mu_0 = const.mu_0     # vacuum permeability
sigma = const.sigma   # Stefan-Boltzmann constant
```

## Classical Mechanics

```python
from sympy import *

t = symbols('t')
m, g, k, L = symbols('m g k L', positive=True)

# Lagrangian mechanics
# Example: Simple pendulum
theta = Function('theta')(t)
T = Rational(1,2) * m * (L * diff(theta, t))**2  # kinetic energy
V = -m * g * L * cos(theta)                        # potential energy
Lag = T - V

# Euler-Lagrange equation
EL = diff(diff(Lag, diff(theta, t)), t) - diff(Lag, theta)
eq = simplify(EL)
print(f"Equation of motion: {eq} = 0")

# Numerical simulation (projectile, pendulum, etc.)
from scipy.integrate import solve_ivp

def pendulum(t, state, g=9.81, L=1.0):
    theta, omega = state
    return [omega, -g/L * np.sin(theta)]

sol = solve_ivp(pendulum, [0, 10], [np.pi/4, 0], max_step=0.01)
```

## Electromagnetism

```python
# Coulomb's law
def coulomb_force(q1, q2, r):
    """Force between two charges (N)"""
    return const.k * q1 * q2 / r**2  # k = 1/(4πε₀)

# Capacitor energy
def capacitor_energy(C, V):
    return 0.5 * C * V**2

# RC circuit
def rc_discharge(V0, R, C, t):
    tau = R * C
    return V0 * np.exp(-t / tau)

# Electromagnetic wave
def em_wavelength(frequency):
    return const.c / frequency

def photon_energy(wavelength):
    return const.h * const.c / wavelength
```

## Quantum Mechanics

```python
# Particle in a box energy levels
def particle_in_box(n, L, m=const.m_e):
    """Energy of nth level, box length L"""
    return (n**2 * const.h**2) / (8 * m * L**2)

# Hydrogen atom energy levels
def hydrogen_energy(n):
    """Energy in eV"""
    return -13.6 / n**2

# de Broglie wavelength
def de_broglie(p):
    return const.h / p

# Heisenberg uncertainty
# Δx · Δp ≥ ℏ/2
```

## Thermodynamics & Statistical Mechanics

```python
# Ideal gas
def ideal_gas_pressure(n, T, V):
    return n * const.R * T / V

# Carnot efficiency
def carnot_efficiency(T_hot, T_cold):
    return 1 - T_cold / T_hot

# Blackbody radiation (Planck's law)
def planck_spectral_radiance(wavelength, T):
    """W/(m²·sr·m)"""
    return (2 * const.h * const.c**2 / wavelength**5) / \
           (np.exp(const.h * const.c / (wavelength * const.k * T)) - 1)

# Maxwell-Boltzmann speed distribution
def mb_speed_dist(v, T, m):
    return 4 * np.pi * (m / (2 * np.pi * const.k * T))**1.5 * \
           v**2 * np.exp(-m * v**2 / (2 * const.k * T))
```

## Unit Conversion

```python
# scipy.constants has conversion factors
from scipy.constants import eV, atm, calorie, mile, inch

# Common conversions
def eV_to_J(energy_eV): return energy_eV * eV
def J_to_eV(energy_J): return energy_J / eV
def celsius_to_kelvin(T_C): return T_C + 273.15
def atm_to_Pa(P_atm): return P_atm * atm
```

## Problem-Solving Framework

1. **Identify** the physical system and relevant principles
2. **Draw** a diagram (describe it textually)
3. **List** knowns and unknowns
4. **Choose** appropriate equations/laws
5. **Solve** symbolically first (SymPy), then substitute numbers
6. **Check** units, limiting cases, and order of magnitude
7. **Interpret** the result physically

## Tips
- Always carry units through calculations
- Check dimensional consistency
- Verify with limiting cases (e.g., v << c for classical limit)
- Use SymPy for symbolic derivations, SciPy for numerical
- For complex simulations, consider specialized tools (COMSOL, OpenFOAM)