---

name: sympy-symbolic description: > SymPy symbolic computation for physics: algebra, calculus, linear algebra, ODE solving, Lagrangian mechanics, quantum physics, and code generation. tags:

• sympy
• symbolic-computation
• physics
• mechanics
• quantum
• code-generation version: "1.0.0" authors:
• name: "awesome-rosetta-skills contributors" github: "@awesome-rosetta-skills" license: "MIT" platforms:
• claude-code
• codex
• gemini-cli
• cursor dependencies:
• sympy>=1.12
• numpy>=1.24.0
• matplotlib>=3.7.0
• scipy>=1.11.0 last_updated: "2026-03-17"

---

SymPy Symbolic Computation for Physics

SymPy is a pure-Python computer algebra system. This skill demonstrates how to use SymPy for symbolic physics: manipulating equations, solving ODEs analytically, deriving equations of motion via Lagrangian mechanics, performing quantum mechanics calculations, and converting symbolic expressions to fast numerical code.

---

1. Symbolic Algebra Fundamentals

import sympy as sp

# Define symbols with assumptions
x, y, z = sp.symbols("x y z", real=True)
t = sp.Symbol("t", positive=True)
n = sp.Symbol("n", integer=True, positive=True)
epsilon, omega, phi = sp.symbols("epsilon omega phi", positive=True)

# Basic algebra
expr = (x + y) ** 4
print("Expanded:", sp.expand(expr))

factored = sp.factor(x**4 - 1)
print("Factored:", factored)

# Simplification
trig_expr = sp.sin(x) ** 2 + sp.cos(x) ** 2
print("Trig simplified:", sp.trigsimp(trig_expr))

# Power series expansion
f = sp.exp(sp.I * x)
series = sp.series(f, x, 0, 6)
print("Euler series:", series)

# Solve algebraic equations
quadratic = sp.Eq(x**2 + 3*x + 2, 0)
solutions = sp.solve(quadratic, x)
print("Quadratic roots:", solutions)

# System of equations
eq1 = sp.Eq(2*x + y, 5)
eq2 = sp.Eq(x - y, 1)
system_sol = sp.solve([eq1, eq2], [x, y])
print("System solution:", system_sol)

---

2. Symbolic Calculus

import sympy as sp

x, t, a, b = sp.symbols("x t a b", real=True)

# Differentiation
f = sp.sin(x**2) * sp.exp(-x)
df = sp.diff(f, x)
d2f = sp.diff(f, x, 2)
print("f(x)   =", f)
print("f'(x)  =", sp.simplify(df))
print("f''(x) =", sp.simplify(d2f))

# Partial derivatives
phi = sp.Function("phi")(x, t)
wave_eq = sp.diff(phi, t, 2) - sp.diff(phi, x, 2)
print("\nWave equation LHS:", wave_eq)

# Definite and indefinite integrals
I1 = sp.integrate(sp.exp(-x**2), (x, -sp.oo, sp.oo))
print("\nGaussian integral:", I1)  # sqrt(pi)

I2 = sp.integrate(sp.sin(x) * sp.cos(x), x)
print("Trig integral:", sp.simplify(I2))

# Integration with parameters
I3 = sp.integrate(sp.exp(-a * x**2), (x, 0, sp.oo))
print("Parameterized Gaussian:", sp.simplify(I3))

# Limits
lim1 = sp.limit(sp.sin(x) / x, x, 0)
print("\nlim_{x->0} sin(x)/x =", lim1)

lim2 = sp.limit((1 + 1/x)**x, x, sp.oo)
print("lim_{x->inf} (1+1/x)^x =", lim2)

# Fourier series (manual via integration)
def fourier_coefficient(f, L, n, var):
    """Compute the nth Fourier coefficient of f on [-L, L]."""
    a_n = (1 / L) * sp.integrate(f * sp.cos(n * sp.pi * var / L), (var, -L, L))
    b_n = (1 / L) * sp.integrate(f * sp.sin(n * sp.pi * var / L), (var, -L, L))
    return sp.simplify(a_n), sp.simplify(b_n)

n = sp.Symbol("n", integer=True, positive=True)
L = sp.pi
# Square wave f(x) = x on [-pi, pi]
a_n, b_n = fourier_coefficient(x, L, n, x)
print("\nFourier coefficients of f(x)=x on [-pi,pi]:")
print("  a_n =", a_n)
print("  b_n =", b_n)

---

3. Symbolic Linear Algebra

import sympy as sp

# Symbolic matrix
a, b, c, d = sp.symbols("a b c d")
M = sp.Matrix([[a, b], [c, d]])
print("Matrix M:\n", M)
print("Determinant:", M.det())
print("Inverse:\n", sp.simplify(M.inv()))

# Eigenvalues and eigenvectors
eigenvals = M.eigenvals()
eigenvects = M.eigenvects()
print("\nEigenvalues:", eigenvals)
for eigenval, mult, vecs in eigenvects:
    print(f"  λ = {eigenval}, multiplicity = {mult}, eigenvector = {vecs[0].T}")

# Pauli matrices (quantum mechanics)
sx = sp.Matrix([[0, 1], [1, 0]])
sy = sp.Matrix([[0, -sp.I], [sp.I, 0]])
sz = sp.Matrix([[1, 0], [0, -1]])

print("\nPauli matrix commutation relations:")
print("[sx, sy] = 2i*sz:", sp.simplify(sx * sy - sy * sx) == 2 * sp.I * sz)
print("[sy, sz] = 2i*sx:", sp.simplify(sy * sz - sz * sy) == 2 * sp.I * sx)
print("[sz, sx] = 2i*sy:", sp.simplify(sz * sx - sx * sz) == 2 * sp.I * sy)

# Rotation matrix and its properties
theta = sp.Symbol("theta", real=True)
R = sp.Matrix([
    [sp.cos(theta), -sp.sin(theta)],
    [sp.sin(theta),  sp.cos(theta)],
])
print("\nRotation matrix R @ R.T:")
print(sp.simplify(R * R.T))  # Should be identity
print("det(R):", sp.simplify(R.det()))

# Solve linear system symbolically
A = sp.Matrix([[2, 1, -1], [-3, -1, 2], [-2, 1, 2]])
b_vec = sp.Matrix([8, -11, -3])
solution = A.solve(b_vec)
print("\nLinear system solution:", solution.T)

---

4. Solving Differential Equations Symbolically

import sympy as sp

t = sp.Symbol("t", positive=True)
x = sp.Function("x")

# --- Simple harmonic oscillator ---
omega = sp.Symbol("omega", positive=True)
ode_sho = sp.Eq(x(t).diff(t, 2) + omega**2 * x(t), 0)
sol_sho = sp.dsolve(ode_sho, x(t))
print("SHO solution:", sol_sho)

# Apply initial conditions x(0)=1, x'(0)=0
C1, C2 = sp.symbols("C1 C2")
ics = {x(0): 1, x(t).diff(t).subs(t, 0): 0}
sol_ics = sp.dsolve(ode_sho, x(t), ics=ics)
print("SHO with ICs:", sol_ics)

# --- Damped oscillator ---
gamma, omega0 = sp.symbols("gamma omega_0", positive=True)
ode_damped = sp.Eq(x(t).diff(t, 2) + 2*gamma*x(t).diff(t) + omega0**2*x(t), 0)
sol_damped = sp.dsolve(ode_damped, x(t))
print("\nDamped oscillator solution:", sp.simplify(sol_damped.rhs))

# --- Driven oscillator ---
F0, omega_d = sp.symbols("F_0 omega_d", positive=True)
ode_driven = sp.Eq(
    x(t).diff(t, 2) + 2*gamma*x(t).diff(t) + omega0**2*x(t),
    F0 * sp.cos(omega_d * t)
)
sol_driven = sp.dsolve(ode_driven, x(t))
print("\nDriven oscillator particular solution:")
print(sp.simplify(sol_driven.rhs))

# --- Heat equation (separation of variables) ---
# Verify that u(x, t) = exp(-alpha * k^2 * t) * sin(k * x) satisfies u_t = alpha * u_xx
x_var = sp.Symbol("x", real=True)
alpha, k = sp.symbols("alpha k", positive=True)
u = sp.exp(-alpha * k**2 * t) * sp.sin(k * x_var)
residual = sp.diff(u, t) - alpha * sp.diff(u, x_var, 2)
print("\nHeat equation residual (should be 0):", sp.simplify(residual))

---

5. Lagrangian Mechanics — Double Pendulum

Derive the equations of motion for a double pendulum using the Lagrangian formalism.

import sympy as sp
from sympy.physics.mechanics import dynamicsymbols, LagrangesMethod, Particle, Point, ReferenceFrame


def derive_double_pendulum():
    """
    Derive equations of motion for a planar double pendulum via Lagrangian mechanics.

    q1 = angle of rod 1 from vertical
    q2 = angle of rod 2 from vertical
    """
    t = sp.Symbol("t")
    m1, m2, l1, l2, g = sp.symbols("m_1 m_2 l_1 l_2 g", positive=True)

    # Generalized coordinates (functions of time)
    q1 = dynamicsymbols("q_1")
    q2 = dynamicsymbols("q_2")
    q1d = dynamicsymbols("q_1", 1)
    q2d = dynamicsymbols("q_2", 1)

    # Cartesian positions
    x1 = l1 * sp.sin(q1)
    y1 = -l1 * sp.cos(q1)
    x2 = x1 + l2 * sp.sin(q2)
    y2 = y1 - l2 * sp.cos(q2)

    # Velocities
    x1d = sp.diff(x1, t)
    y1d = sp.diff(y1, t)
    x2d = sp.diff(x2, t)
    y2d = sp.diff(y2, t)

    # Kinetic and potential energy
    T = sp.Rational(1, 2) * m1 * (x1d**2 + y1d**2) + \
        sp.Rational(1, 2) * m2 * (x2d**2 + y2d**2)
    V = m1 * g * y1 + m2 * g * y2

    L = T - V
    L = sp.simplify(L)
    print("Lagrangian L = T - V:")
    print(sp.simplify(L))

    # Euler-Lagrange equations
    def euler_lagrange(L, q, qd, t):
        """Compute d/dt(dL/dqd) - dL/dq = 0."""
        dL_dqd = sp.diff(L, qd)
        d_dt_dL_dqd = sp.diff(dL_dqd, t)
        dL_dq = sp.diff(L, q)
        return sp.simplify(d_dt_dL_dqd - dL_dq)

    eq1 = euler_lagrange(L, q1, q1d, t)
    eq2 = euler_lagrange(L, q2, q2d, t)

    print("\nEuler-Lagrange equation 1 (q1):")
    print(sp.simplify(eq1))
    print("\nEuler-Lagrange equation 2 (q2):")
    print(sp.simplify(eq2))

    # Linearize for small angles (sin(q) ≈ q, cos(q) ≈ 1)
    def linearize(expr, coords):
        result = expr
        for q in coords:
            result = result.subs(sp.sin(q), q).subs(sp.cos(q), 1)
        return sp.simplify(result)

    eq1_lin = linearize(eq1, [q1, q2])
    eq2_lin = linearize(eq2, [q1, q2])
    print("\nLinearized EOM (small angles):")
    print("EOM1:", eq1_lin)
    print("EOM2:", eq2_lin)

    return eq1, eq2, eq1_lin, eq2_lin


if __name__ == "__main__":
    eqs = derive_double_pendulum()

---

6. Quantum Harmonic Oscillator

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt


def quantum_harmonic_oscillator():
    """
    Symbolic solution of the quantum harmonic oscillator.

    H psi_n = E_n psi_n
    E_n = hbar * omega * (n + 1/2)
    psi_n(x) = N_n * H_n(xi) * exp(-xi^2/2)  where xi = sqrt(m*omega/hbar)*x
    """
    x, xi = sp.symbols("x xi", real=True)
    n = sp.Symbol("n", nonnegative=True, integer=True)
    hbar, omega, m = sp.symbols("hbar omega m", positive=True)

    # Energy eigenvalues
    E_n = hbar * omega * (n + sp.Rational(1, 2))
    print("Energy eigenvalues:")
    for k in range(6):
        print(f"  E_{k} = {E_n.subs(n, k)} * hbar*omega")

    # Hermite polynomials (physicist's convention)
    def hermite(n_val, xi_sym):
        """Generate Hermite polynomial H_n(xi) via recurrence."""
        if n_val == 0:
            return sp.Integer(1)
        elif n_val == 1:
            return 2 * xi_sym
        else:
            H_prev2 = sp.Integer(1)
            H_prev1 = 2 * xi_sym
            for k in range(2, n_val + 1):
                H_curr = 2 * xi_sym * H_prev1 - 2 * (k - 1) * H_prev2
                H_prev2 = H_prev1
                H_prev1 = H_curr
            return sp.expand(H_curr)

    # Normalization constant N_n = 1/sqrt(2^n * n! * sqrt(pi))
    def norm_constant(n_val):
        return 1 / sp.sqrt(2**n_val * sp.factorial(n_val) * sp.sqrt(sp.pi))

    # Wavefunctions
    print("\nWavefunctions psi_n(xi):")
    wavefunctions = []
    for k in range(5):
        Hn = hermite(k, xi)
        Nn = norm_constant(k)
        psi = Nn * Hn * sp.exp(-xi**2 / 2)
        psi_simplified = sp.simplify(psi)
        wavefunctions.append(psi_simplified)
        print(f"  psi_{k}(xi) = {psi_simplified}")

    # Verify orthonormality of psi_0 and psi_1
    inner_00 = sp.integrate(wavefunctions[0]**2, (xi, -sp.oo, sp.oo))
    inner_01 = sp.integrate(wavefunctions[0] * wavefunctions[1], (xi, -sp.oo, sp.oo))
    inner_11 = sp.integrate(wavefunctions[1]**2, (xi, -sp.oo, sp.oo))
    print(f"\nOrthonormality check:")
    print(f"  <0|0> = {sp.simplify(inner_00)}")
    print(f"  <0|1> = {sp.simplify(inner_01)}")
    print(f"  <1|1> = {sp.simplify(inner_11)}")

    # Convert to numpy functions for plotting
    psi_funcs = [sp.lambdify(xi, psi, "numpy") for psi in wavefunctions]

    xi_vals = np.linspace(-4, 4, 500)
    fig, ax = plt.subplots(figsize=(10, 6))
    colors = ["steelblue", "darkorange", "green", "red", "purple"]
    for k, (psi_np, color) in enumerate(zip(psi_funcs, colors)):
        psi_vals = psi_np(xi_vals)
        ax.plot(xi_vals, psi_vals + k + 0.5, label=f"ψ_{k}(ξ)", color=color, lw=1.8)
        ax.axhline(k + 0.5, color=color, ls="--", alpha=0.3)

    # Harmonic potential
    V = xi_vals**2 / 2
    ax.plot(xi_vals, V / 5, "k-", lw=1.5, alpha=0.3, label="V(ξ)/5")
    ax.set_xlim(-4, 4)
    ax.set_ylim(-0.5, 5.5)
    ax.set_xlabel("ξ = x/x₀")
    ax.set_ylabel("ψₙ(ξ) + Eₙ/ℏω")
    ax.set_title("Quantum Harmonic Oscillator Wavefunctions")
    ax.legend(loc="upper right")
    plt.tight_layout()
    plt.show()

    return wavefunctions


if __name__ == "__main__":
    quantum_harmonic_oscillator()

---

7. Code Generation: lambdify and CSE

import sympy as sp
import numpy as np
import time


def code_generation_demo():
    """
    Demonstrate converting symbolic expressions to fast numerical functions.
    lambdify: converts SymPy expression to numpy/scipy callable.
    cse: common subexpression elimination for optimized computation.
    """
    x, y, z = sp.symbols("x y z", real=True)

    # Complex symbolic expression
    expr = (sp.sin(x) * sp.cos(y) + sp.exp(-x**2 - y**2)) / (1 + sp.sqrt(x**2 + y**2 + z**2))

    # lambdify with numpy backend
    f_np = sp.lambdify([x, y, z], expr, modules="numpy")

    # Test on arrays
    x_arr = np.random.randn(100_000)
    y_arr = np.random.randn(100_000)
    z_arr = np.random.randn(100_000)

    t0 = time.perf_counter()
    result = f_np(x_arr, y_arr, z_arr)
    print(f"lambdify evaluation on 100k points: {time.perf_counter()-t0:.4f}s")
    print(f"  Result sample: {result[:5]}")

    # CSE for expensive multi-output expressions
    expr_list = [
        sp.sin(x)**2 + sp.cos(x)*sp.cos(y),
        sp.sin(x)**2 * sp.exp(-y),
        sp.cos(x)**2 - sp.sin(x)*sp.sin(y),
    ]

    replacements, reduced_exprs = sp.cse(expr_list)
    print("\nCSE replacements:")
    for sym, val in replacements:
        print(f"  {sym} = {val}")
    print("Reduced expressions:")
    for e in reduced_exprs:
        print(f"  {e}")

    # Generate Python code from CSE output
    from sympy.printing.pycode import pycode
    print("\nGenerated Python code snippet:")
    for sym, val in replacements:
        print(f"  {sym} = {pycode(val)}")

    # LaTeX output for documentation
    omega, t_sym = sp.symbols("omega t", real=True)
    gamma_sym = sp.Symbol("gamma", positive=True)
    transfer_fn = omega**2 / (omega**2 - t_sym**2 + 2*sp.I*gamma_sym*t_sym)
    print("\nLaTeX representation of transfer function:")
    print(sp.latex(transfer_fn))

    # Numerical integration via lambdify + scipy.integrate
    from scipy.integrate import quad
    integrand_sym = sp.exp(-x**2) * sp.cos(x)
    integrand_np = sp.lambdify(x, integrand_sym, modules="numpy")
    result_int, err = quad(integrand_np, -np.inf, np.inf)
    # Analytical: sqrt(pi) * exp(-1/4)
    exact = np.sqrt(np.pi) * np.exp(-0.25)
    print(f"\nIntegral of exp(-x^2)*cos(x): {result_int:.8f} (exact: {exact:.8f})")


if __name__ == "__main__":
    code_generation_demo()

---

8. Maxwell Equations Symbolic Manipulation

import sympy as sp
from sympy.vector import CoordSys3D, curl, divergence, gradient


def maxwell_equations_demo():
    """
    Demonstrate Maxwell equations using SymPy's vector calculus.

    In Gaussian units:
      div E = 4*pi*rho
      div B = 0
      curl E = -1/c * dB/dt
      curl B = 4*pi/c * J + 1/c * dE/dt
    """
    # Setup coordinate system
    N = CoordSys3D("N")
    x, y, z, t = sp.symbols("x y z t", real=True)
    epsilon0, mu0, c = sp.symbols("epsilon_0 mu_0 c", positive=True)
    omega_k, k_val = sp.symbols("omega k", positive=True)

    # Plane wave: E in x-direction, propagating in z-direction
    # E_x = E0 * cos(k*z - omega*t)
    E0 = sp.Symbol("E_0", real=True)
    E_x = E0 * sp.cos(k_val * z - omega_k * t)
    E_field = E_x * N.i + 0 * N.j + 0 * N.k

    # Corresponding B field from Faraday's law: B_y = E0/c * cos(k*z - omega*t)
    B_y = E0 / c * sp.cos(k_val * z - omega_k * t)
    B_field = 0 * N.i + B_y * N.j + 0 * N.k

    # Verify div E = 0 (vacuum)
    div_E = divergence(E_field)
    print("div(E) =", sp.simplify(div_E))

    # Verify div B = 0
    div_B = divergence(B_field)
    print("div(B) =", sp.simplify(div_B))

    # Verify Faraday's law: curl E = -1/c * dB/dt
    curl_E = curl(E_field)
    dB_dt = (E0 / c) * omega_k * sp.sin(k_val * z - omega_k * t) * N.j
    faraday_lhs = curl_E
    faraday_rhs = -1/c * dB_dt
    print("\nFaraday's law verification:")
    print("  curl(E) =", faraday_lhs)
    print("  -1/c * dB/dt =", sp.simplify(faraday_rhs))
    # Verify they're equal when k = omega/c
    substituted = sp.simplify(
        sp.nsimplify(faraday_lhs.dot(N.j) - faraday_rhs.dot(N.j)).subs(k_val, omega_k / c)
    )
    print(f"  Difference (k=omega/c): {substituted}")

    # Dispersion relation: from Ampere's law, k^2 = omega^2/c^2
    print("\nDispersion relation for electromagnetic waves in vacuum:")
    print("  k = omega/c  =>  v_phase = omega/k = c")

    return E_field, B_field


if __name__ == "__main__":
    maxwell_equations_demo()

---

9. Complete Example A — Double Pendulum: Symbolic → Numerical

import sympy as sp
import numpy as np
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt


def double_pendulum_symbolic_to_numerical():
    """
    (a) Derive double pendulum EOM symbolically
    (b) Lambdify to fast numerical functions
    (c) Simulate and plot
    """
    t = sp.Symbol("t")
    g_sym, l_sym, m_sym = sp.symbols("g l m", positive=True)

    q1 = sp.Function("q1")(t)
    q2 = sp.Function("q2")(t)
    dq1 = q1.diff(t)
    dq2 = q2.diff(t)

    # Equal masses and lengths for simplicity: m1=m2=m, l1=l2=l
    # Positions
    x1 = l_sym * sp.sin(q1)
    y1 = -l_sym * sp.cos(q1)
    x2 = x1 + l_sym * sp.sin(q2)
    y2 = y1 - l_sym * sp.cos(q2)

    # Kinetic energy
    T = sp.Rational(1, 2) * m_sym * (
        (x1.diff(t))**2 + (y1.diff(t))**2 +
        (x2.diff(t))**2 + (y2.diff(t))**2
    )
    T = sp.trigsimp(sp.expand(T))

    # Potential energy
    V = m_sym * g_sym * (y1 + y2)

    L = sp.expand(T - V)

    # Euler-Lagrange for q1
    dL_dq1 = L.diff(q1)
    dL_ddq1 = L.diff(dq1)
    EL1 = sp.expand(dL_ddq1.diff(t) - dL_dq1)

    # Euler-Lagrange for q2
    dL_dq2 = L.diff(q2)
    dL_ddq2 = L.diff(dq2)
    EL2 = sp.expand(dL_ddq2.diff(t) - dL_dq2)

    # Extract second derivatives
    ddq1, ddq2 = sp.symbols("ddq1 ddq2")
    EL1_sub = EL1.subs({q1.diff(t, 2): ddq1, q2.diff(t, 2): ddq2})
    EL2_sub = EL2.subs({q1.diff(t, 2): ddq1, q2.diff(t, 2): ddq2})

    sol = sp.solve([EL1_sub, EL2_sub], [ddq1, ddq2])
    ddq1_expr = sol[ddq1]
    ddq2_expr = sol[ddq2]

    # Substitute state variables: q1→Q1, dq1→W1, q2→Q2, dq2→W2
    Q1, W1, Q2, W2 = sp.symbols("Q1 W1 Q2 W2", real=True)
    subs_dict = {q1: Q1, dq1: W1, q2: Q2, dq2: W2}
    ddq1_num = ddq1_expr.subs(subs_dict)
    ddq2_num = ddq2_expr.subs(subs_dict)

    # Numerical parameters
    g_val, l_val, m_val = 9.81, 1.0, 1.0
    param_subs = {g_sym: g_val, l_sym: l_val, m_sym: m_val}
    ddq1_np = sp.lambdify([Q1, W1, Q2, W2], ddq1_num.subs(param_subs), modules="numpy")
    ddq2_np = sp.lambdify([Q1, W1, Q2, W2], ddq2_num.subs(param_subs), modules="numpy")

    def ode_system(t_val, state):
        q1v, w1v, q2v, w2v = state
        return [
            w1v,
            ddq1_np(q1v, w1v, q2v, w2v),
            w2v,
            ddq2_np(q1v, w1v, q2v, w2v),
        ]

    # Simulate
    y0 = [np.pi / 2, 0.0, np.pi / 4, 0.0]
    t_span = (0, 20)
    t_eval = np.linspace(*t_span, 5000)

    sol_num = solve_ivp(ode_system, t_span, y0, method="DOP853",
                        t_eval=t_eval, rtol=1e-9, atol=1e-11)

    # Convert to Cartesian for animation-ready output
    q1_t = sol_num.y[0]
    q2_t = sol_num.y[2]
    x1_t = l_val * np.sin(q1_t)
    y1_t = -l_val * np.cos(q1_t)
    x2_t = x1_t + l_val * np.sin(q2_t)
    y2_t = y1_t - l_val * np.cos(q2_t)

    fig, axes = plt.subplots(1, 2, figsize=(13, 5))
    axes[0].plot(sol_num.t, q1_t, label="q₁(t)", lw=1)
    axes[0].plot(sol_num.t, q2_t, label="q₂(t)", lw=1, ls="--")
    axes[0].set_xlabel("Time (s)")
    axes[0].set_ylabel("Angle (rad)")
    axes[0].legend()
    axes[0].set_title("Double Pendulum Angles")

    axes[1].plot(x2_t, y2_t, lw=0.4, alpha=0.7, color="steelblue")
    axes[1].set_aspect("equal")
    axes[1].set_xlabel("x₂ (m)")
    axes[1].set_ylabel("y₂ (m)")
    axes[1].set_title("Trajectory of Bob 2")

    plt.tight_layout()
    plt.show()

    return sol_num


if __name__ == "__main__":
    double_pendulum_symbolic_to_numerical()

---

10. Complete Example B — Symbolic ODE Solve then Plot

import sympy as sp
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp


def symbolic_then_numerical_ode():
    """
    (b) Solve a 2D system of ODEs symbolically, then compare with numerical solution.

    System (predator-prey linearization about fixed point):
      dx/dt = a*x - b*x*y  → linearized: dx/dt = a*x - b*y
      dy/dt = -c*y + d*x*y → linearized: dy/dt = d*x - c*y
    """
    t = sp.Symbol("t", positive=True)
    x = sp.Function("x")
    y = sp.Function("y")
    a, b, c, d = sp.symbols("a b c d", positive=True)

    # Linearized Lotka-Volterra
    eq1 = sp.Eq(x(t).diff(t), a * x(t) - b * y(t))
    eq2 = sp.Eq(y(t).diff(t), d * x(t) - c * y(t))

    print("Linearized Lotka-Volterra system:")
    print("  ", eq1)
    print("  ", eq2)

    # Solve symbolically
    system = [eq1, eq2]
    sol = sp.dsolve(system, [x(t), y(t)])
    print("\nSymbolic solution:")
    for s in sol:
        print("  ", s)

    # Numerical values
    params = {a: 0.5, b: 0.1, c: 0.5, d: 0.1}
    x_sol_sym = sol[0].rhs.subs(params)
    y_sol_sym = sol[1].rhs.subs(params)

    # Apply ICs numerically: x(0)=10, y(0)=5
    C1, C2, C3, C4 = sp.symbols("C1 C2 C3 C4")
    x0_val, y0_val = 10.0, 5.0
    ic_eqs = [
        sp.Eq(x_sol_sym.subs(t, 0), x0_val),
        sp.Eq(y_sol_sym.subs(t, 0), y0_val),
    ]
    # Try to extract constants (may depend on sympy version)
    try:
        const_sol = sp.solve(ic_eqs, [C1, C2, C3, C4])
        x_particular = x_sol_sym.subs(const_sol)
        y_particular = y_sol_sym.subs(const_sol)
        x_np = sp.lambdify(t, x_particular, modules="numpy")
        y_np = sp.lambdify(t, y_particular, modules="numpy")
        t_arr = np.linspace(0, 20, 1000)
        x_sym_vals = x_np(t_arr)
        y_sym_vals = y_np(t_arr)
        has_symbolic = True
    except Exception as e:
        print(f"Symbolic IC application failed: {e}")
        has_symbolic = False

    # Numerical solution for comparison
    def lotka_volterra_linear(t_val, state):
        xv, yv = state
        a_n, b_n, c_n, d_n = 0.5, 0.1, 0.5, 0.1
        return [a_n * xv - b_n * yv, d_n * xv - c_n * yv]

    t_arr = np.linspace(0, 20, 1000)
    sol_num = solve_ivp(lotka_volterra_linear, (0, 20), [x0_val, y0_val],
                        t_eval=t_arr, method="RK45", rtol=1e-10)

    fig, axes = plt.subplots(1, 2, figsize=(13, 5))

    if has_symbolic:
        axes[0].plot(t_arr, x_sym_vals.real, "r--", lw=1.5, label="x(t) symbolic")
        axes[0].plot(t_arr, y_sym_vals.real, "b--", lw=1.5, label="y(t) symbolic")

    axes[0].plot(sol_num.t, sol_num.y[0], "r-", lw=1, alpha=0.7, label="x(t) numerical")
    axes[0].plot(sol_num.t, sol_num.y[1], "b-", lw=1, alpha=0.7, label="y(t) numerical")
    axes[0].set_xlabel("Time")
    axes[0].set_ylabel("Population")
    axes[0].legend()
    axes[0].set_title("Linearized Lotka-Volterra")

    axes[1].plot(sol_num.y[0], sol_num.y[1], color="steelblue", lw=1.2)
    axes[1].scatter([x0_val], [y0_val], color="red", zorder=5, label="Initial state")
    axes[1].set_xlabel("x (prey)")
    axes[1].set_ylabel("y (predator)")
    axes[1].legend()
    axes[1].set_title("Phase Portrait")

    plt.tight_layout()
    plt.show()


if __name__ == "__main__":
    symbolic_then_numerical_ode()

---

Quick Reference

Task	SymPy API
Define symbol with assumptions	sp.Symbol("x", real=True, positive=True)
Expand expression	sp.expand(expr)
Factor polynomial	sp.factor(expr)
Simplify (general)	sp.simplify(expr)
Trig simplification	sp.trigsimp(expr)
Differentiate	sp.diff(expr, var, n)
Integrate (indefinite)	sp.integrate(expr, var)
Integrate (definite)	sp.integrate(expr, (var, a, b))
Power series	sp.series(expr, var, x0, n)
Solve algebraic equation	sp.solve(eq, var)
Solve ODE	sp.dsolve(ode, func, ics={})
Matrix operations	sp.Matrix([[...]]); M.det(), M.inv(), M.eigenvects()
Convert to numpy	sp.lambdify([vars], expr, modules="numpy")
Common subexpression elimination	sp.cse([expr1, expr2])
LaTeX output	sp.latex(expr)
Pretty print	sp.pprint(expr, use_unicode=True)

Performance Tips

• Use sp.lambdify with modules="numpy" for vectorized numerical evaluation.
• Apply sp.cse before lambdifying multi-output expressions to avoid redundant computation.
• Set assumptions on symbols (e.g., real=True, positive=True) — SymPy simplifies more aggressively.
• Use sp.nsimplify to convert floating-point results back to exact rational/algebraic form.
• For large matrix operations, consider sp.MatrixSymbol for abstract manipulation.