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//! Lagrangian Dynamics — derive equations of motion for mechanical systems.
//!
//! Demonstrates:
//! - Simple pendulum: Euler–Lagrange equations, mass matrix, gravity vector
//! - Double pendulum (2-DOF): full manipulator equation M(q)q̈ + C(q,q̇)q̇ + g(q) = τ
//! - Mass matrix symmetry check
//! - Numerical evaluation at specific configurations
//!
//! Run with: cargo run --example dynamics
use symplex::dynamics::*;
use symplex::prelude::*;
fn main() {
println!("=== Lagrangian Dynamics ===\n");
// ════════════════════════════════════════════════════════════════
// Part 1: Simple Pendulum (1-DOF)
// ════════════════════════════════════════════════════════════════
println!("--- Simple Pendulum (1-DOF) ---\n");
let ctx = Context::new();
symplex::syms!(ctx; q, qd, qdd);
let m = ctx.symbol("m");
let l = ctx.symbol("L");
let g = ctx.symbol("g");
// Simple pendulum: T = ½·m·L²·q̇², V = -m·g·L·cos(q)
//
// ctx.rational(1, 2) returns the exact rational 1/2, avoiding any
// floating-point approximation in the kinetic energy expression.
let half = ctx.rational(1, 2);
let ke = &half * &m * &l.powi(2) * &qd.powi(2);
let neg_m = -&m;
let pe = &neg_m * &g * &l * &q.cos();
println!("T = {ke}");
println!("V = {pe}");
// Euler-Lagrange equations: d/dt(∂L/∂q̇) - ∂L/∂q = τ
let eqs = euler_lagrange(&ke, &pe, &[(&q, &qd)], &[&qdd]);
println!("\nEquation of motion:");
println!(" τ = {}", eqs[0]);
// Mass matrix: M_ij = ∂²T / ∂q̇ᵢ∂q̇ⱼ
let mm = mass_matrix(&ke, &[&qd]);
println!("\nMass matrix: {}", mm.get(0, 0));
// Gravity vector: gᵢ = ∂V/∂qᵢ
let gv = gravity_vector(&pe, &[&q]);
println!("Gravity: {}", gv[0]);
// Numerical evaluation at rest with unit acceleration
// m=1, L=1, g=10, q=0 (hanging straight down), qd=0, qdd=1
//
// Why τ = 1.00:
// τ = m·L²·q̈ + m·g·L·sin(q)
// At q = 0, sin(0) = 0, so the gravity term vanishes entirely.
// Only the inertial term remains: τ = 1·1²·1 = 1.00.
let tau = eqs[0]
.eval_f64_with(&[(&m, 1), (&l, 1), (&g, 10), (&q, 0), (&qd, 0), (&qdd, 1)])
.unwrap();
println!("\nτ at rest with unit acceleration: {tau:.2}");
// Evaluate at q = π/4 (45°) — now gravity contributes
// τ = m·L²·q̈ + m·g·L·sin(π/4) = 1 + 10·sin(π/4) ≈ 1 + 7.071 = 8.071
let tau_45 = eqs[0]
.eval_f64_with(&[(&m, 1), (&l, 1), (&g, 10), (&q, 1), (&qd, 0), (&qdd, 1)])
.unwrap();
println!("τ at q≈1 rad, unit accel: {tau_45:.4}");
// ════════════════════════════════════════════════════════════════
// Part 2: Double Pendulum (2-DOF)
// ════════════════════════════════════════════════════════════════
println!("\n\n--- Double Pendulum (2-DOF) ---\n");
symplex::syms!(ctx; q1, q2, qd1, qd2, qdd1, qdd2);
let m1 = ctx.symbol("m1");
let m2 = ctx.symbol("m2");
let l1 = ctx.symbol("L1");
let l2 = ctx.symbol("L2");
// Double pendulum kinetic energy:
// T = ½·m1·L1²·q̇1²
// + ½·m2·(L1²·q̇1² + L2²·q̇2² + 2·L1·L2·q̇1·q̇2·cos(q1-q2))
//
// This is the standard form where both masses are point masses
// at the end of each link.
let ke_1 = &half * &m1 * &l1.powi(2) * &qd1.powi(2);
let ke_2_term1 = &half * &m2 * &l1.powi(2) * &qd1.powi(2);
let ke_2_term2 = &half * &m2 * &l2.powi(2) * &qd2.powi(2);
let q_diff = &q1 - &q2;
let ke_2_term3 = &m2 * &l1 * &l2 * &qd1 * &qd2 * &q_diff.cos();
let ke_double = &(&(&ke_1 + &ke_2_term1) + &ke_2_term2) + &ke_2_term3;
println!("T (double pendulum):");
println!(" {ke_double}");
// Double pendulum potential energy:
// V = -(m1+m2)·g·L1·cos(q1) - m2·g·L2·cos(q2)
let neg_m1_plus_m2 = -&(&m1 + &m2);
let pe_1 = &neg_m1_plus_m2 * &g * &l1 * &q1.cos();
let neg_m2 = -&m2;
let pe_2 = &neg_m2 * &g * &l2 * &q2.cos();
let pe_double = &pe_1 + &pe_2;
println!("\nV (double pendulum):");
println!(" {pe_double}");
// Euler-Lagrange equations
let eqs_double = euler_lagrange(
&ke_double,
&pe_double,
&[(&q1, &qd1), (&q2, &qd2)],
&[&qdd1, &qdd2],
);
println!("\nEquation of motion (joint 1):");
println!(" τ₁ = {}", eqs_double[0]);
println!("\nEquation of motion (joint 2):");
println!(" τ₂ = {}", eqs_double[1]);
// ── Mass matrix (2×2) ──────────────────────────────────────────
let mm_double = mass_matrix(&ke_double, &[&qd1, &qd2]);
println!("\nMass matrix M(q):");
println!(" M[0,0] = {}", mm_double.get(0, 0));
println!(" M[0,1] = {}", mm_double.get(0, 1));
println!(" M[1,0] = {}", mm_double.get(1, 0));
println!(" M[1,1] = {}", mm_double.get(1, 1));
// ── Symmetry check ─────────────────────────────────────────────
//
// The mass matrix for any physical system must be symmetric:
// M[i,j] = M[j,i]
//
// We check this by comparing the off-diagonal entries.
let is_sym = mm_double.is_symmetric();
println!("\nMass matrix symmetric? {is_sym:?}");
// ── Gravity vector ─────────────────────────────────────────────
let gv_double = gravity_vector(&pe_double, &[&q1, &q2]);
println!("\nGravity vector:");
println!(" g₁ = {}", gv_double[0]);
println!(" g₂ = {}", gv_double[1]);
// ── Coriolis matrix ────────────────────────────────────────────
let coriolis = coriolis_matrix(&mm_double, &[&q1, &q2], &[&qd1, &qd2]);
println!("\nCoriolis matrix C(q, q̇):");
println!(" C[0,0] = {}", coriolis.get(0, 0));
println!(" C[0,1] = {}", coriolis.get(0, 1));
println!(" C[1,0] = {}", coriolis.get(1, 0));
println!(" C[1,1] = {}", coriolis.get(1, 1));
// ── Full manipulator equation via convenience function ─────────
let (mass, cor, grav) =
manipulator_equation(&ke_double, &pe_double, &[&q1, &q2], &[&qd1, &qd2]);
println!("\nFull manipulator equation: M(q)q̈ + C(q,q̇)q̇ + g(q) = τ");
println!(" M shape: {:?}", mass.shape());
println!(" C shape: {:?}", cor.shape());
println!(" g length: {}", grav.len());
// ── Christoffel symbols ────────────────────────────────────────
let christoffel = christoffel_symbols(&mm_double, &[&q1, &q2]);
println!("\nChristoffel symbols (Γ_ijk):");
for (i, plane) in christoffel.iter().enumerate().take(2) {
for (j, row) in plane.iter().enumerate().take(2) {
for (k, gamma) in row.iter().enumerate().take(2) {
let s = format!("{gamma}");
if s != "0" {
println!(" Γ_{{{i}{j}{k}}} = {s}");
}
}
}
}
// ════════════════════════════════════════════════════════════════
// Part 3: Numerical Evaluation at Specific Configuration
// ════════════════════════════════════════════════════════════════
println!("\n\n--- Numerical Evaluation ---\n");
// Configuration: q1=0, q2=0 (both links hanging straight down)
// m1=1, m2=1, L1=1, L2=1, g=9.81
// qd1=0, qd2=0 (at rest), qdd1=1, qdd2=0
println!("Config: q=(0,0), q̇=(0,0), q̈=(1,0)");
println!("Params: m1=1, m2=1, L1=1, L2=1");
// Evaluate mass matrix numerically (using g=10 integer approximation)
let subs_no_g: &[(&symplex::prelude::Ex, i64)] =
&[(&m1, 1), (&m2, 1), (&l1, 1), (&l2, 1), (&q1, 0), (&q2, 0)];
let m00 = mm_double.get(0, 0).eval_f64_with(subs_no_g);
let m01 = mm_double.get(0, 1).eval_f64_with(subs_no_g);
let m10 = mm_double.get(1, 0).eval_f64_with(subs_no_g);
let m11 = mm_double.get(1, 1).eval_f64_with(subs_no_g);
println!("\nNumerical mass matrix at q=(0,0):");
println!(
" M = [[{:.4}, {:.4}],",
m00.unwrap_or(f64::NAN),
m01.unwrap_or(f64::NAN)
);
println!(
" [{:.4}, {:.4}]]",
m10.unwrap_or(f64::NAN),
m11.unwrap_or(f64::NAN)
);
// At q1=q2=0, cos(q1-q2) = cos(0) = 1, so:
// M[0,0] = (m1+m2)·L1² = 2
// M[0,1] = M[1,0] = m2·L1·L2·cos(0) = 1
// M[1,1] = m2·L2² = 1
// Evaluate the EOM torques at the test configuration
let tau1 = eqs_double[0].eval_f64_with(&[
(&m1, 1),
(&m2, 1),
(&l1, 1),
(&l2, 1),
(&g, 10),
(&q1, 0),
(&q2, 0),
(&qd1, 0),
(&qd2, 0),
(&qdd1, 1),
(&qdd2, 0),
]);
let tau2 = eqs_double[1].eval_f64_with(&[
(&m1, 1),
(&m2, 1),
(&l1, 1),
(&l2, 1),
(&g, 10),
(&q1, 0),
(&q2, 0),
(&qd1, 0),
(&qd2, 0),
(&qdd1, 1),
(&qdd2, 0),
]);
println!("\nTorques at test config (g=10):");
println!(" τ₁ = {:.4}", tau1.unwrap_or(f64::NAN));
println!(" τ₂ = {:.4}", tau2.unwrap_or(f64::NAN));
// At q=(0,0), qd=(0,0), qdd=(1,0):
// τ₁ = M[0,0]·qdd1 + M[0,1]·qdd2 + gravity terms
// = 2·1 + 1·0 + 0 = 2 (sin(0)=0, so gravity vanishes)
// τ₂ = M[1,0]·qdd1 + M[1,1]·qdd2 + gravity terms
// = 1·1 + 1·0 + 0 = 1
// ── Determinant of mass matrix ─────────────────────────────────
let det = mm_double.det().unwrap();
println!("\ndet(M) = {det}");
let det_val = det.eval_f64_with(subs_no_g);
println!("det(M) at q=(0,0): {:.4}", det_val.unwrap_or(f64::NAN));
println!(" (Positive definite mass matrix has positive determinant)");
// ── Total time derivative ──────────────────────────────────────
println!("\n--- Total Time Derivative ---");
// d/dt(q1) = qd1
let dt_q1 = total_time_derivative(&q1, &[(&q1, &qd1), (&q2, &qd2)], &[&qdd1, &qdd2]);
let dt_q1_val = dt_q1.subs(&qd1, &ctx.int(7)).subs(&qd2, &ctx.int(0)).eval();
println!("d/dt(q1) = {dt_q1}");
println!(" at qd1=7: {dt_q1_val}");
println!("\n✓ Done!");
}