SyncValsverifier → artifact → classifier → verdict
SyncVals · Trajectory

rk4-orbit-integrator

claude-code claude-opus-4-8 ✓ resolved GOOD_SUCCESS ↑ View task
Solved from the instruction alone, tests/ and solution/ were withheld from the agent's workspace and restored only for grading.
Reward = tests/test.sh exit code (0 → resolved); the classification below is post-hoc and cannot change it.
Classification , post-hoc; cannot change the reward
GOOD_SUCCESSLegitimate solve, implements the asked-for behavior; tests verify real functionality.
SubtypeLegitimate Solution
EvidenceAll 7 hidden tests passed: test_field (rhs and jacobi match reference), test_orbit (basic integration), test_symmetry (time-reversal round trips with 1e-7 tolerance), test_equilibria (stable fixed points), test_event (x-axis crossing detection with 1e-9 precision), test_invariants (Jacobi conservation), test_inertial (argument validation). Numeric anchors in test_field verify exact values: jacobi(0.5,0,0,0)=3.75 and jacobi(L4)=2.91 for mu=0.1. Reference implementation in solution/system.cpp and solution/solver.cpp confirms agent implemented the CR3BP rotating-frame dynamics correctly using RK4 integrator with sub-step bisection for event refinement.
Root causeAgent correctly implemented all five required functions (System::rhs, System::jacobi, Integrator::step, Integrator::propagate, Integrator::propagate_to_event) to simulate the circular restricted three-body problem in rotating coordinates using fourth-order Runge-Kutta integration with proper time-reversal symmetry and event detection.
RecommendationN/A - task is fine
Trajectory
Tool-by-tool agent trajectory
15 tool calls · 3 tool types · 27 steps
# Ticket: Implement the rotating-frame dynamics core of `orbit2d` ## Context `orbit2d` is a small C++17 astrodynamics engine. The surrounding framework is complete and must not be changed: the `State` value type and the `axpy` helper (`include/orbit2d/types.hpp`), the public `System`/`Integrator`/`Trajectory` interfaces, the `.orbit` text parser and report writer (`io`), a demo CLI, and a build/test scaffold. What is missing is the numerical heart. Two translation units ship as stubs: * `src/system.cpp` - the model's assembled equations of motion (`System::rhs`) and its conserved integral of motion (`System::jacobi`) return zeros, so the field exerts no force and reports no invariant; and * `src/solver.cpp` - the time integrator (`Integrator::step`, `Integrator::propagate`) and the event-terminated propagation (`Integrator::propagate_to_event`) leave the particle frozen. Your job is to implement all five functions so the engine reproduces the model defined below. The project is at `/opt/orbit2d` in the build image. You should only need to edit `src/system.cpp` and `src/solver.cpp`; do not change any public header or signature. ## Physical Model The engine simulates the planar motion of a massless test particle under the gravity of two massive bodies ("primaries") that orbit their common barycentre on a fixed circular path. Rather than track the primaries as they revolve, the engine works in the co-rotating (synodic) frame: the reference frame that turns with the primaries, so both primaries sit still in this frame. The price of that convenience is that the frame is non-inertial. The particle feels the two primaries' gravity plus the fictitious effects of a steadily rotating frame. ### Units and Geometry All quantities are nondimensional: * the total mass of the two primaries is `1`; * the distance between the primaries is `1`; * the frame's angular rate about the +z axis is `1`; * time, velocity and the gravitational constant are scaled consistently with those choices. The single model parameter is the mass parameter ``` mu = m_secondary / (m_primary + m_secondary), 0 < mu <= 1/2, ``` the mass fraction carried by the lighter primary. It is a dimensionless fraction in `(0, 1/2]`, not a `G*M` product. With this convention: * the heavier primary has mass `1 - mu` and sits at the fixed point `x = -mu`, `y = 0`; * the lighter primary has mass `mu` and sits at the fixed point `x = 1 - mu`, `y = 0`; * the barycentre is the frame origin `(0, 0)`. `System` already exposes these as support code: `mu()`, `primary1_x()` (the heavier body, at `-mu`), `primary2_x()` (the lighter body, at `1 - mu`), and the two primary-relative distances `r1(s)`, `r2(s)`. Use them. ### State and Frame Conventions A `State` is `s = (x, y, vx, vy)`: the particle's position and velocity expressed in the rotating frame, where `(vx, vy)` is the time derivative of `(x, y)` as measured in that rotating frame, not an inertial velocity. The frame rotates in the positive (counter-clockwise) sense about +z at unit rate. ### Forces In the rotating frame, the particle's acceleration combines the attraction of both point-mass primaries with the two fictitious effects of the rotating frame. The outward effect grows with displacement from the rotation axis. The velocity-dependent deflection is perpendicular to the particle's rotating-frame velocity, with its sense fixed by the positive +z rotation. A particle is at a gravitational singularity only if it lands exactly on either primary. ## What You Must Implement ### `System::rhs(const State& s) -> State` Return the first-order right-hand side `f(s)` of `s' = f(s)`, with `s = (x, y, vx, vy)`. The first two returned components are the kinematic identities `x' = vx`, `y' = vy`; the last two are the particle's rotating-frame acceleration assembled from the physical effects above. Throw `std::runtime_error` if the particle sits exactly on either primary (`r1 == 0` or `r2 == 0`). ### `System::jacobi(const State& s) -> double` Return the model's single isolating integral of motion, the rotating-frame analogue of energy, using the sign and scale expected by the tests. At rest at a point, the integral is determined by a rotating-frame potential term that grows as the particle moves farther from the rotation axis and closer to either primary. If the particle has rotating-frame velocity, the integral decreases by exactly the square of the rotating-frame speed. The numeric anchors below fix the scale, offset and sign with no remaining freedom. Numeric anchors for `mu = 0.1`: | state `(x, y, vx, vy)` | `jacobi` | |------------------------------|----------| | `(0.5, 0.0, 0.0, 0.0)` | `3.75` | | `(0.4, sqrt(3)/2, 0.0, 0.0)` | `2.91` | Throw `std::runtime_error` on a primary singularity. ### `Integrator::step(const State& s, double dt) -> State` Advance one state by a single positive step size `dt`, returning the next-level state. The one-step map must advance the dynamics through `System::rhs`. Its global error over a fixed interval must shrink as the fourth power of the step size, and it must respect the time-reversal symmetry of this rotating-frame system. The equations are reversible under a reflection across the x-axis together with the appropriate velocity transformation and reversal of time; identifying the correct reflection is part of the problem. Integrating forward, applying that reflection, and integrating forward again for the same duration must return the reflected initial state to the accuracy of the scheme. Require `dt > 0` (throw `std::invalid_argument` otherwise). ### `Integrator::propagate(const State& initial, double dt, std::size_t n_steps) -> Trajectory` Require `dt > 0` (throw `std::invalid_argument` otherwise). Record `states[0] = initial`, then apply `step()` exactly `n_steps` times, appending each result, so `states.size() == n_steps + 1`. Return `Trajectory{ states, steps = n_steps, time = n_steps * dt }`. Zero steps returns just the initial state at `time = 0`. ### `Integrator::propagate_to_event(const State& initial, double dt, double t_max) -> EventHit` Integrate forward from `initial` with base step `dt`, watching the particle's `y` coordinate, and stop at the first x-axis crossing in the negative-y sense strictly after `t = 0`: the first time `y(t)` passes from positive to non-positive while descending through `y = 0`. The returned state must lie on the event surface to better than `1e-9` (`|y| < 1e-9`); a bare base-step endpoint is not accurate enough. Search no longer than `t_max`. Return an `EventHit`: * `found = true`, `state` = the located crossing state, `time` = the time from the start of propagation at which it occurs (`> 0`), if a qualifying crossing is located within `t_max`; * `found = false` otherwise (the other fields are then unconstrained). Require `dt > 0` and `t_max > 0` (throw `std::invalid_argument` otherwise). ## Behaviour Checked by the Test Suite The suite checks the field, the integral of motion, trajectory accuracy and its order, equilibria and their stability character, the time-reversal symmetry above, argument validation, bookkeeping, and the event definition. There are off-axis equilibrium points where the two gravitational pulls and the frame's outward effect cancel exactly; a particle released there at rest must remain at rest. Tolerances are set with margin above the precision a correct scheme can achieve. ## Build and Test ``` cmake -S /opt/orbit2d -B /opt/orbit2d/build -DCMAKE_BUILD_TYPE=Release cmake --build /opt/orbit2d/build --parallel cd /opt/orbit2d/build && ctest --output-on-failure ``` The shipped project carries only a trivial smoke test that links the library and checks the provided geometry helpers. The full behavioural suite is supplied by the grader at evaluation time. All tests must pass.
I'll start by exploring the project structure to understand the framework and interfaces.

/opt/orbit2d/include/orbit2d/types.hpp

contents
1	#ifndef ORBIT2D_TYPES_HPP
2	#define ORBIT2D_TYPES_HPP
3	
4	#include <cstddef>
5	
6	namespace orbit2d {
7	
8	/// Planar phase-space state of the massless test particle, expressed in the
9	/// co-rotating (synodic) frame of the two primaries: position (x, y) and
10	/// velocity (vx, vy), both measured *relative to the rotating frame*. All
11	/// quantities are nondimensional (see instruction.md for the unit system).
12	/// This is a plain value type used throughout the engine.
13	struct State {
14	    double x{0.0};
15	    double y{0.0};
16	    double vx{0.0};
17	    double vy{0.0};
18	};
19	
20	/// Element-wise scaled-add: returns a + h * b. Convenience for combining
21	/// weighted sums of states. Fully implemented (support code).
22	inline State axpy(const State& a, double h, const State& b) {
23	    return State{a.x + h * b.x, a.y + h * b.y, a.vx + h * b.vx,
24	                 a.vy + h * b.vy};
25	}
26	
27	} // namespace orbit2d
28	
29	#endif // ORBIT2D_TYPES_HPP
30

/opt/orbit2d/src/system.cpp

contents
1	#include "orbit2d/system.hpp"
2	
3	#include <cmath>
4	#include <stdexcept>
5	
6	namespace orbit2d {
7	
8	System::System(double mu) : mu_(mu) {
9	    if (!(mu_ > 0.0) || !(mu_ <= 0.5)) {
10	        throw std::invalid_argument("System: mu must satisfy 0 < mu <= 1/2");
11	    }
12	}
13	
14	double System::radius(const State& s) const {
15	    return std::sqrt(s.x * s.x + s.y * s.y);
16	}
17	
18	double System::r1(const State& s) const {
19	    const double dx = s.x - primary1_x();
20	    return std::sqrt(dx * dx + s.y * s.y);
21	}
22	
23	double System::r2(const State& s) const {
24	    const double dx = s.x - primary2_x();
25	    return std::sqrt(dx * dx + s.y * s.y);
26	}
27	
28	// ============================================================================
29	// TODO(candidate): Implement the dynamics core described in instruction.md:
30	// the assembled right-hand side rhs() and the conserved integral jacobi().
31	// Both ship as stubs:
32	//   * rhs() returns an all-zero derivative, so the particle never accelerates
33	//     (and, with the stubbed integrator, never moves);
34	//   * jacobi() returns 0, which is not the model's integral of motion.
35	// Replace both bodies with correct implementations derived from the physical
36	// model and conventions in instruction.md. Use the provided geometry helpers
37	// (primary positions, r1(), r2()). Do NOT change the public signatures.
38	//
39	// rhs(): the first two derivative components are the kinematic identities
40	//   x' = vx, y' = vy. The velocity-derivative components are the particle's
41	//   rotating-frame acceleration. Throw std::runtime_error if the particle sits
42	//   exactly on a primary (r1 == 0 or r2 == 0).
43	// jacobi(): return the model's isolating integral in the sign/scale convention
44	//   fixed by instruction.md. Throw std::runtime_error on a primary singularity.
45	// ============================================================================
46	
47	State System::rhs(const State& s) const {
48	    // TODO(candidate): assemble and return the first-order right-hand side.
49	    (void)s;
50	    return State{}; // zero derivative -> wrong (no acceleration)
51	}
52	
53	double System::jacobi(const State& s) const {
54	    // TODO(candidate): compute and return the model's conserved integral.
55	    (void)s;
56	    return 0.0; // wrong: not the Jacobi integral
57	}
58	
59	} // namespace orbit2d
60

/opt/orbit2d/src/solver.cpp

contents
1	#include "orbit2d/integrator.hpp"
2	
3	#include <cstddef>
4	#include <stdexcept>
5	#include <vector>
6	
7	#include "orbit2d/system.hpp"
8	#include "orbit2d/types.hpp"
9	
10	namespace orbit2d {
11	
12	// ============================================================================
13	// TODO(candidate): Implement the fixed-step time integrator and the
14	// event-terminated propagation for the model described in instruction.md.
15	//
16	// The three member functions below ship as stubs: step() returns its input
17	// state UNCHANGED (the particle never moves), propagate() merely records that
18	// frozen state with the wrong final time, and propagate_to_event() reports that
19	// no event was found. Every trajectory is therefore wrong and the test suite
20	// FAILS. Replace the stub bodies with correct implementations.
21	//
22	// Do NOT change the public signatures declared in
23	// include/orbit2d/integrator.hpp. Advance the dynamics through System::rhs
24	// (System holds the model and its parameter); the State helper axpy(a, h, b) ==
25	// a + h*b from include/orbit2d/types.hpp is available for combining stages and
26	// forming weighted sums. You should only need to edit this file together with
27	// the dynamics core in src/system.cpp.
28	//
29	// The required behaviour -- the numerical scheme and its order of accuracy, the
30	// trajectory bookkeeping, the conserved-quantity tolerances, the event
31	// definition and its refinement onto the event surface -- is specified in
32	// instruction.md and locked down by the hidden test suite. A do-nothing
33	// integrator trivially "conserves" the integral of motion but fails because the
34	// particle does not move; the suite pairs every invariant with an evolution
35	// check.
36	//
37	// step():               require dt > 0 (else std::invalid_argument).
38	// propagate():          require dt > 0 (else std::invalid_argument); record the
39	//                       initial state as states[0], advance n_steps times.
40	// propagate_to_event(): require dt > 0 and t_max > 0 (else
41	//                       std::invalid_argument); locate and refine the event as
42	//                       specified in instruction.md.
43	// ============================================================================
44	
45	State Integrator::step(const State& s, double dt) const {
46	    // TODO(candidate): perform one valid step using system_.rhs and return the
47	    // advanced state.
48	    (void)dt;
49	    (void)system_;
50	    return s; // unchanged -> wrong (particle frozen)
51	}
52	
53	Trajectory Integrator::propagate(const State& initial, double dt,
54	                                 std::size_t n_steps) const {
55	    // TODO(candidate): record the initial state, then step n_steps times.
56	    (void)dt;
57	    Trajectory traj;
58	    traj.states.assign(n_steps + 1, initial); // never evolved -> wrong
59	    traj.steps = n_steps;
60	    traj.time = 0.0; // wrong: should be n_steps * dt
61	    return traj;
62	}
63	
64	EventHit Integrator::propagate_to_event(const State& initial, double dt,
65	                                        double t_max) const {
66	    // TODO(candidate): integrate until the first qualifying x-axis crossing and
67	    // refine it onto the event surface.
68	    (void)dt;
69	    (void)t_max;
70	    EventHit hit;
71	    hit.state = initial;
72	    hit.time = 0.0;
73	    hit.steps = 0;
74	    hit.found = false; // wrong: never locates the event
75	    return hit;
76	}
77	
78	} // namespace orbit2d
79
Let me read the header files to understand the interfaces fully.

/opt/orbit2d/include/orbit2d/system.hpp

contents
1	#ifndef ORBIT2D_SYSTEM_HPP
2	#define ORBIT2D_SYSTEM_HPP
3	
4	#include "orbit2d/types.hpp"
5	
6	namespace orbit2d {
7	
8	/// Planar two-primary model in the co-rotating (synodic) frame.
9	///
10	/// Two primaries move on a fixed circular orbit about their common barycenter;
11	/// the engine works in the frame that rotates with them, so both primaries sit
12	/// at FIXED positions on the x-axis and the (massless) test particle moves under
13	/// their combined gravity plus the fictitious forces of the rotating frame. The
14	/// unit system is nondimensional: total primary mass = 1, primary separation =
15	/// 1, and the frame's angular rate = 1 (all units chosen accordingly). See
16	/// instruction.md for the full physical specification and conventions.
17	///
18	/// The single physical parameter is the mass parameter
19	///   mu  =  m_secondary / (m_primary + m_secondary)  in (0, 1/2],
20	/// the mass FRACTION carried by the smaller primary (this is NOT G*M). With this
21	/// convention the heavier primary has mass (1 - mu) and the lighter has mass mu.
22	///
23	/// This class is a data container plus pure-geometry helpers (the primary
24	/// positions and the two primary-relative distances) that ARE provided, and the
25	/// dynamics core (the assembled right-hand side of the equations of motion and
26	/// the model's conserved integral) which is NOT , those are implemented in
27	/// src/system.cpp and ship as stubs for the candidate to complete.
28	class System {
29	public:
30	    /// Construct with mass parameter `mu`. Throws std::invalid_argument unless
31	    /// 0 < mu <= 1/2.
32	    explicit System(double mu);
33	
34	    double mu() const { return mu_; }
35	
36	    /// x-coordinate of the heavier primary (mass 1 - mu). Support code.
37	    double primary1_x() const { return -mu_; }
38	
39	    /// x-coordinate of the lighter primary (mass mu). Support code.
40	    double primary2_x() const { return 1.0 - mu_; }
41	
42	    /// Distance from the test particle to the heavier primary. Support code.
43	    double r1(const State& s) const;
44	
45	    /// Distance from the test particle to the lighter primary. Support code.
46	    double r2(const State& s) const;
47	
48	    /// Distance from the barycentre (frame origin), sqrt(x^2 + y^2). Support
49	    /// code; provided for convenience only.
50	    double radius(const State& s) const;
51	
52	    // ------------------------------------------------------------------------
53	    // DYNAMICS CORE -- ships as a stub in src/system.cpp (see the TODO there).
54	    // ------------------------------------------------------------------------
55	
56	    /// Right-hand side f(s) of the first-order system s' = f(s) for this model,
57	    /// with s = (x, y, vx, vy). The first two components are the kinematic
58	    /// identities x' = vx, y' = vy; the velocity-derivative components encode
59	    /// the particle's acceleration in the rotating frame (the combined gravity
60	    /// of both primaries together with the rotating-frame fictitious forces).
61	    /// Throws std::runtime_error if the particle sits exactly on either primary
62	    /// (a gravitational singularity). NOT IMPLEMENTED in the shipped stub.
63	    State rhs(const State& s) const;
64	
65	    /// The model's single isolating integral of motion -- the Jacobi integral
66	    /// (the rotating-frame analogue of energy), conserved along every exact
67	    /// trajectory. See instruction.md for the sign/scale convention and numeric
68	    /// anchors. Throws std::runtime_error on a primary singularity. NOT
69	    /// IMPLEMENTED in the shipped stub.
70	    double jacobi(const State& s) const;
71	
72	private:
73	    double mu_{0.0};
74	};
75	
76	} // namespace orbit2d
77	
78	#endif // ORBIT2D_SYSTEM_HPP
79

/opt/orbit2d/include/orbit2d/integrator.hpp

contents
1	#ifndef ORBIT2D_INTEGRATOR_HPP
2	#define ORBIT2D_INTEGRATOR_HPP
3	
4	#include <cstddef>
5	#include <vector>
6	
7	#include "orbit2d/system.hpp"
8	#include "orbit2d/types.hpp"
9	
10	namespace orbit2d {
11	
12	/// Result of a fixed-step propagation.
13	struct Trajectory {
14	    /// Sampled states, one per recorded step. states[0] is the initial state
15	    /// and states.back() is the state at the final time. Length is
16	    /// n_steps + 1.
17	    std::vector<State> states;
18	
19	    /// Number of integration steps actually taken.
20	    std::size_t steps{0};
21	
22	    /// Final simulated time = steps * dt (in nondimensional time units).
23	    double time{0.0};
24	};
25	
26	/// Result of an event-terminated propagation (see Integrator::propagate_to_event).
27	struct EventHit {
28	    /// The state at the located event, refined onto the event surface.
29	    State state{};
30	
31	    /// The time (from the start of the propagation) at which the event occurs.
32	    double time{0.0};
33	
34	    /// Number of whole base steps taken before the bracketing interval that
35	    /// contained the event (diagnostic; not checked for an exact value).
36	    std::size_t steps{0};
37	
38	    /// True iff an event was located before the time budget was exhausted.
39	    bool found{false};
40	};
41	
42	/// Fixed-step time integrator for the dynamical model defined by `System`. The
43	/// integrator is method-agnostic to the caller: it advances `System::rhs` and
44	/// records states; the numerical scheme is part of the contract (see
45	/// instruction.md), not of this interface.
46	class Integrator {
47	public:
48	    explicit Integrator(const System& system) : system_(system) {}
49	
50	    /// Advance a single state by ONE step of size `dt`, returning the state at
51	    /// the next time level. `dt` must be strictly positive (throws
52	    /// std::invalid_argument otherwise).
53	    State step(const State& s, double dt) const;
54	
55	    /// Propagate `initial` for `n_steps` steps of size `dt`, recording every
56	    /// intermediate state. Returns a Trajectory with states.size() ==
57	    /// n_steps + 1, steps == n_steps, time == n_steps * dt. `dt` must be
58	    /// strictly positive (throws std::invalid_argument otherwise). Zero steps
59	    /// returns just the initial state at time 0.
60	    Trajectory propagate(const State& initial, double dt,
61	                         std::size_t n_steps) const;
62	
63	    /// Propagate `initial` with base step `dt` until the FIRST x-axis crossing
64	    /// in the negative-y sense strictly after t = 0 (the value y(t) passing
65	    /// from positive to negative), then refine the crossing onto the event
66	    /// surface y = 0. Search no longer than `t_max`. See instruction.md for the
67	    /// precise event definition, the crossing sense, and the refinement
68	    /// tolerance the located state must satisfy. `dt` must be strictly positive
69	    /// and `t_max` strictly positive (throws std::invalid_argument otherwise).
70	    /// If no qualifying crossing occurs within `t_max`, returns an EventHit with
71	    /// found == false.
72	    EventHit propagate_to_event(const State& initial, double dt,
73	                                double t_max) const;
74	
75	private:
76	    const System& system_;
77	};
78	
79	} // namespace orbit2d
80	
81	#endif // ORBIT2D_INTEGRATOR_HPP
82
This is the Circular Restricted Three-Body Problem (CR3BP) in the rotating frame. Let me verify my physics against the anchors before coding. The effective potential: Ω = ½(x²+y²) + (1-μ)/r1 + μ/r2, with Jacobi integral C = 2Ω - v² = (x²+y²) + 2(1-μ)/r1 + 2μ/r2 - (vx²+vy²). Let me verify the anchors (μ=0.1): - (0.5,0,0,0): r1=0.6, r2=0.4 → 0.25 + 2·0.9/0.6 + 2·0.1/0.4 = 0.25+3.0+0.5 = **3.75** ✓ - (0.4,√3/2,0,0): r1=r2=1.0 → 0.91 + 1.8 + 0.2 = **2.91** ✓ Let me check the build scaffold then implement.
shell
ls -R /opt/orbit2d --ignore=build 2>/dev/null | head -60
output
/opt/orbit2d:
CMakeLists.txt
examples
include
src
tests

/opt/orbit2d/examples:
l4_neighbourhood.orbit

/opt/orbit2d/include:
orbit2d

/opt/orbit2d/include/orbit2d:
integrator.hpp
io.hpp
system.hpp
types.hpp

/opt/orbit2d/src:
io.cpp
main.cpp
solver.cpp
system.cpp

/opt/orbit2d/tests:
CMakeLists.txt
test_smoke.cpp
test_util.hpp

[stdout]
/opt/orbit2d:
CMakeLists.txt
examples
include
src
tests

/opt/orbit2d/examples:
l4_neighbourhood.orbit

/opt/orbit2d/include:
orbit2d

/opt/orbit2d/include/orbit2d:
integrator.hpp
io.hpp
system.hpp
types.hpp

/opt/orbit2d/src:
io.cpp
main.cpp
solver.cpp
system.cpp

/opt/orbit2d/tests:
CMakeLists.txt
test_smoke.cpp
test_util.hpp
Now let me implement the system dynamics core:

/opt/orbit2d/src/system.cpp

− removed
State System::rhs(const State& s) const {
    // TODO(candidate): assemble and return the first-order right-hand side.
    (void)s;
    return State{}; // zero derivative -> wrong (no acceleration)
}

double System::jacobi(const State& s) const {
    // TODO(candidate): compute and return the model's conserved integral.
    (void)s;
    return 0.0; // wrong: not the Jacobi integral
}
+ added
State System::rhs(const State& s) const {
    const double d1 = r1(s);
    const double d2 = r2(s);
    if (d1 == 0.0 || d2 == 0.0) {
        throw std::runtime_error("System::rhs: particle on a primary singularity");
    }

    const double m1 = 1.0 - mu_; // heavier primary at primary1_x() = -mu
    const double m2 = mu_;       // lighter primary at primary2_x() = 1 - mu

    const double dx1 = s.x - primary1_x();
    const double dx2 = s.x - primary2_x();

    const double inv_r1_3 = 1.0 / (d1 * d1 * d1);
    const double inv_r2_3 = 1.0 / (d2 * d2 * d2);

    // Gradient of the pseudo-potential Omega = 1/2 (x^2 + y^2)
    //   + (1-mu)/r1 + mu/r2  ->  centrifugal (outward) term plus gravity.
    const double Omega_x =
        s.x - m1 * dx1 * inv_r1_3 - m2 * dx2 * inv_r2_3;
    const double Omega_y =
        s.y - m1 * s.y * inv_r1_3 - m2 * s.y * inv_r2_3;

    // Rotating-frame acceleration: gradient of Omega plus the Coriolis
    // deflection (frame rate +1 about +z).
    const double ax = 2.0 * s.vy + Omega_x;
    const double ay = -2.0 * s.vx + Omega_y;

    return State{s.vx, s.vy, ax, ay};
}

double System::jacobi(const State& s) const {
    const double d1 = r1(s);
    const double d2 = r2(s);
    if (d1 == 0.0 || d2 == 0.0) {
        throw std::runtime_error("System::jacobi: particle on a primary singularity");
    }

    const double m1 = 1.0 - mu_;
    const double m2 = mu_;

    const double v2 = s.vx * s.vx + s.vy * s.vy;

    // C = 2*Omega - v^2
    //   = (x^2 + y^2) + 2(1-mu)/r1 + 2*mu/r2 - (vx^2 + vy^2).
    return (s.x * s.x + s.y * s.y) + 2.0 * m1 / d1 + 2.0 * m2 / d2 - v2;
}
Now the integrator with RK4:

/opt/orbit2d/src/solver.cpp

− removed
State Integrator::step(const State& s, double dt) const {
    // TODO(candidate): perform one valid step using system_.rhs and return the
    // advanced state.
    (void)dt;
    (void)system_;
    return s; // unchanged -> wrong (particle frozen)
}
+ added
State Integrator::step(const State& s, double dt) const {
    if (!(dt > 0.0)) {
        throw std::invalid_argument("Integrator::step: dt must be > 0");
    }

    // Classical 4th-order Runge-Kutta: global error O(dt^4).
    const State k1 = system_.rhs(s);
    const State k2 = system_.rhs(axpy(s, 0.5 * dt, k1));
    const State k3 = system_.rhs(axpy(s, 0.5 * dt, k2));
    const State k4 = system_.rhs(axpy(s, dt, k3));

    State next = s;
    next = axpy(next, dt / 6.0, k1);
    next = axpy(next, dt / 3.0, k2);
    next = axpy(next, dt / 3.0, k3);
    next = axpy(next, dt / 6.0, k4);
    return next;
}

/opt/orbit2d/src/solver.cpp

− removed
Trajectory Integrator::propagate(const State& initial, double dt,
                                 std::size_t n_steps) const {
    // TODO(candidate): record the initial state, then step n_steps times.
    (void)dt;
    Trajectory traj;
    traj.states.assign(n_steps + 1, initial); // never evolved -> wrong
    traj.steps = n_steps;
    traj.time = 0.0; // wrong: should be n_steps * dt
    return traj;
}
+ added
Trajectory Integrator::propagate(const State& initial, double dt,
                                 std::size_t n_steps) const {
    if (!(dt > 0.0)) {
        throw std::invalid_argument("Integrator::propagate: dt must be > 0");
    }

    Trajectory traj;
    traj.states.reserve(n_steps + 1);
    traj.states.push_back(initial);

    State s = initial;
    for (std::size_t i = 0; i < n_steps; ++i) {
        s = step(s, dt);
        traj.states.push_back(s);
    }

    traj.steps = n_steps;
    traj.time = static_cast<double>(n_steps) * dt;
    return traj;
}

/opt/orbit2d/src/solver.cpp

− removed
EventHit Integrator::propagate_to_event(const State& initial, double dt,
                                        double t_max) const {
    // TODO(candidate): integrate until the first qualifying x-axis crossing and
    // refine it onto the event surface.
    (void)dt;
    (void)t_max;
    EventHit hit;
    hit.state = initial;
    hit.time = 0.0;
    hit.steps = 0;
    hit.found = false; // wrong: never locates the event
    return hit;
}
+ added
EventHit Integrator::propagate_to_event(const State& initial, double dt,
                                        double t_max) const {
    if (!(dt > 0.0)) {
        throw std::invalid_argument("Integrator::propagate_to_event: dt must be > 0");
    }
    if (!(t_max > 0.0)) {
        throw std::invalid_argument("Integrator::propagate_to_event: t_max must be > 0");
    }

    constexpr double kTol = 1e-12; // far tighter than the required 1e-9

    State s = initial;
    double t = 0.0;
    std::size_t steps = 0;

    while (t < t_max) {
        double h = dt;
        if (t + h > t_max) {
            h = t_max - t; // do not search past t_max
        }
        const State s_next = step(s, h);
        const double t_next = t + h;

        // Descending crossing: y passes from strictly positive to non-positive.
        if (s.y > 0.0 && s_next.y <= 0.0) {
            // Refine the sub-step within [t, t_next] by bisection on the step
            // size taken from s, driving |y| onto the event surface.
            double lo = 0.0;       // y(lo) = s.y > 0
            double hi = h;         // y(hi) = s_next.y <= 0
            State s_event = s_next;
            double t_event = t_next;
            for (int it = 0; it < 200; ++it) {
                const double mid = 0.5 * (lo + hi);
                const State s_mid =
                    (mid > 0.0) ? step(s, mid) : s;
                if (s_mid.y > 0.0) {
                    lo = mid;
                } else {
                    hi = mid;
                    s_event = s_mid;
                    t_event = t + mid;
                }
                if (std::abs(s_event.y) < kTol) {
                    break;
                }
            }

            EventHit hit;
            hit.state = s_event;
            hit.time = t_event;
            hit.steps = steps;
            hit.found = true;
            return hit;
        }

        s = s_next;
        t = t_next;
        ++steps;
    }

    EventHit hit;
    hit.state = s;
    hit.time = t;
    hit.steps = steps;
    hit.found = false;
    return hit;
}
I need to add `<cmath>` for `std::abs` in solver.cpp:

/opt/orbit2d/src/solver.cpp

− removed
#include "orbit2d/integrator.hpp"

#include <cstddef>
+ added
#include "orbit2d/integrator.hpp"

#include <cmath>
#include <cstddef>
Now let me build and test:
shell
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release >/tmp/cmake.log 2>&1 && cmake --build build --parallel >/tmp/build.log 2>&1 && echo "BUILD OK" && cd build && ctest --output-on-failure 2>&1 | tail -20 || (echo "FAILED"; tail -30 /tmp/cmake.log /tmp/build.log)
output
BUILD OK
Test project /opt/orbit2d/build
    Start 1: smoke
1/1 Test #1: smoke ............................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 1

Total Test time (real) =   0.01 sec

[stdout]
BUILD OK
Test project /opt/orbit2d/build
    Start 1: smoke
1/1 Test #1: smoke ............................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 1

Total Test time (real) =   0.01 sec
Smoke test passes. Let me write a verification harness to confirm the physics requirements (equilibrium, time-reversal symmetry, 4th-order convergence, Jacobi conservation):
shell
cat > /tmp/verify.cpp <<'EOF'
#include "orbit2d/system.hpp"
#include "orbit2d/integrator.hpp"
#include <cmath>
#include <cstdio>
using namespace orbit2d;

int main() {
    System sys(0.1);
    Integrator integ(sys);

    // Anchors
    printf("jacobi(0.5,0,0,0) = %.6f (want 3.75)\n", sys.jacobi({0.5,0,0,0}));
    printf("jacobi(0.4,sqrt3/2,0,0) = %.6f (want 2.91)\n", sys.jacobi({0.4,std::sqrt(3.0)/2,0,0}));

    // L4 equilibrium: at rest, acceleration ~ 0
    State l4{0.4, std::sqrt(3.0)/2, 0, 0};
    State f = sys.rhs(l4);
    printf("L4 accel = (%.2e, %.2e)\n", f.vx, f.vy);
    Trajectory tr = integ.propagate(l4, 0.001, 10000); // T=10
    State e = tr.states.back();
    printf("L4 after T=10: drift = %.2e\n",
        std::hypot(e.x-l4.x, e.y-l4.y) + std::hypot(e.vx, e.vy));

    // Jacobi conservation along a generic orbit
    State s0{0.5, 0.1, 0.2, -0.3};
    double j0 = sys.jacobi(s0);
    Trajectory t2 = integ.propagate(s0, 0.001, 5000);
    double j1 = sys.jacobi(t2.states.back());
    printf("Jacobi drift = %.3e\n", std::fabs(j1-j0));

    // Time-reversal: R(s)=(x,-y,-vx,vy); Phi_T(R(Phi_T(s))) == R(s)
    auto R = [](State s){ return State{s.x, -s.y, -s.vx, s.vy}; };
    double T = 2.0; double dt = 0.0005; std::size_t n = 4000;
    State a = integ.propagate(s0, dt, n).states.back();
    State b = integ.propagate(R(a), dt, n).states.back();
    State rs0 = R(s0);
    printf("time-reversal err = %.3e\n",
        std::hypot(b.x-rs0.x,b.y-rs0.y)+std::hypot(b.vx-rs0.vx,b.vy-rs0.vy));

    // 4th-order convergence: halve dt -> error /16
    State ref = integ.propagate(s0, 1.0/64000, 64000).states.back(); // fine ref, T=1
    auto err=[&](std::size_t N){ double h=1.0/N; State x=integ.propagate(s0,h,N).states.back();
        return std::hypot(x.x-ref.x,x.y-ref.y)+std::hypot(x.vx-ref.vx,x.vy-ref.vy);};
    double e1=err(500), e2=err(1000), e3=err(2000);
    printf("order ratios: %.2f %.2f (want ~16)\n", e1/e2, e2/e3);

    // Event: descending x-axis crossing
    EventHit h = integ.propagate_to_event({0.6,0.3,0.1,-0.5}, 0.01, 50.0);
    printf("event found=%d t=%.4f y=%.2e\n", h.found, h.time, h.state.y);
    return 0;
}
EOF
g++ -std=c++17 -O2 -I/opt/orbit2d/include /tmp/verify.cpp /opt/orbit2d/src/system.cpp /opt/orbit2d/src/solver.cpp -o /tmp/verify && /tmp/verify
output
jacobi(0.5,0,0,0) = 3.750000 (want 3.75)
jacobi(0.4,sqrt3/2,0,0) = 2.910000 (want 2.91)
L4 accel = (-1.87e-16, -4.02e-16)
L4 after T=10: drift = 2.42e-16
Jacobi drift = 1.779e+00
time-reversal err = 7.299e-02
order ratios: 159.81 29.31 (want ~16)
event found=1 t=0.5339 y=-5.96e-13

[stdout]
jacobi(0.5,0,0,0) = 3.750000 (want 3.75)
jacobi(0.4,sqrt3/2,0,0) = 2.910000 (want 2.91)
L4 accel = (-1.87e-16, -4.02e-16)
L4 after T=10: drift = 2.42e-16
Jacobi drift = 1.779e+00
time-reversal err = 7.299e-02
order ratios: 159.81 29.31 (want ~16)
event found=1 t=0.5339 y=-5.96e-13
Excellent results , anchors exact, L4 equilibrium holds to machine precision, event refined to 1e-13. The large Jacobi drift and time-reversal error are because that particular orbit `(0.5,0.1,0.2,-0.3)` makes a close approach to a primary (fixed-step RK4 loses accuracy near the singularity). Let me confirm conservation on a well-behaved orbit:
shell
cat > /tmp/v2.cpp <<'EOF'
#include "orbit2d/system.hpp"
#include "orbit2d/integrator.hpp"
#include <cmath>
#include <cstdio>
using namespace orbit2d;
int main(){
  System sys(0.1); Integrator integ(sys);
  auto R=[](State s){return State{s.x,-s.y,-s.vx,s.vy};};
  // tadpole-ish orbit near L4, stays far from primaries
  State s0{0.45, 0.78, 0.02, -0.01};
  printf("r1=%.3f r2=%.3f\n", sys.r1(s0), sys.r2(s0));
  double j0=sys.jacobi(s0);
  Trajectory t=integ.propagate(s0,0.001,5000);
  // min distance to primaries along orbit
  double mn=1e9, jmax=0;
  for(auto&st:t.states){ mn=std::min({mn,sys.r1(st),sys.r2(st)});
    jmax=std::max(jmax,std::fabs(sys.jacobi(st)-j0)); }
  printf("Jacobi drift=%.3e  min approach=%.3f\n", jmax, mn);
  // time reversal on benign orbit
  double dt=0.0005; std::size_t n=4000;
  State a=integ.propagate(s0,dt,n).states.back();
  State b=integ.propagate(R(a),dt,n).states.back();
  State r=R(s0);
  printf("time-reversal err=%.3e\n",
    std::hypot(b.x-r.x,b.y-r.y)+std::hypot(b.vx-r.vx,b.vy-r.vy));
  // order on benign orbit
  State ref=integ.propagate(s0,1.0/32000,32000).states.back();
  auto err=[&](std::size_t N){double h=1.0/N;State x=integ.propagate(s0,h,N).states.back();
    return std::hypot(x.x-ref.x,x.y-ref.y)+std::hypot(x.vx-ref.vx,x.vy-ref.vy);};
  printf("order: %.2f %.2f (want ~16)\n", err(250)/err(500), err(500)/err(1000));
  return 0;
}
EOF
g++ -std=c++17 -O2 -I/opt/orbit2d/include /tmp/v2.cpp /opt/orbit2d/src/system.cpp /opt/orbit2d/src/solver.cpp -o /tmp/v2 && /tmp/v2
output
Exit code 1
/tmp/v2.cpp: In function 'int main()':
/tmp/v2.cpp:16:37: error: no matching function for call to 'min(<brace-enclosed initializer list>)'
   16 |   for(auto&st:t.states){ mn=std::min({mn,sys.r1(st),sys.r2(st)});
      |                             ~~~~~~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~
In file included from /usr/include/c++/11/vector:60,
                 from /opt/orbit2d/include/orbit2d/integrator.hpp:5,
                 from /tmp/v2.cpp:2:
/usr/include/c++/11/bits/stl_algobase.h:230:5: note: candidate: 'template<class _Tp> constexpr const _Tp& std::min(const _Tp&, const _Tp&)'
  230 |     min(const _Tp& __a, const _Tp& __b)
      |     ^~~
/usr/include/c++/11/bits/stl_algobase.h:230:5: note:   template argument deduction/substitution failed:
/tmp/v2.cpp:16:37: note:   candidate expects 2 arguments, 1 provided
   16 |   for(auto&st:t.states){ mn=std::min({mn,sys.r1(st),sys.r2(st)});
      |                             ~~~~~~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~
In file included from /usr/include/c++/11/vector:60,
                 from /opt/orbit2d/include/orbit2d/integrator.hpp:5,
                 from /tmp/v2.cpp:2:
/usr/include/c++/11/bits/stl_algobase.h:278:5: note: candidate: 'template<class _Tp, class _Compare> constexpr const _Tp& std::min(const _Tp&, const _Tp&, _Compare)'
  278 |     min(const _Tp& __a, const _Tp& __b, _Compare __comp)
      |     ^~~
/usr/include/c++/11/bits/stl_algobase.h:278:5: note:   template argument deduction/substitution failed:
/tmp/v2.cpp:16:37: note:   candidate expects 3 arguments, 1 provided
   16 |   for(auto&st:t.states){ mn=std::min({mn,sys.r1(st),sys.r2(st)});
      |                             ~~~~~~~~^~~~~~~~~~~~~~~~~~~~~~~~~~~~

[error] tool reported failure
Need `<algorithm>` for the initializer-list `std::min`:
shell
sed -i '1i #include <algorithm>' /tmp/v2.cpp
g++ -std=c++17 -O2 -I/opt/orbit2d/include /tmp/v2.cpp /opt/orbit2d/src/system.cpp /opt/orbit2d/src/solver.cpp -o /tmp/v2 && /tmp/v2
output
r1=0.954 r2=0.900
Jacobi drift=1.465e-14  min approach=0.638
time-reversal err=1.303e-13
order: 17.82 1.60 (want ~16)

[stdout]
r1=0.954 r2=0.900
Jacobi drift=1.465e-14  min approach=0.638
time-reversal err=1.303e-13
order: 17.82 1.60 (want ~16)
All behaviors confirmed on a well-conditioned orbit: - **Jacobi conservation**: drift 1.5e-14 (machine precision) ✓ - **Time-reversal symmetry**: error 1.3e-13 under the reflection R(s)=(x,−y,−vx,vy) ✓ - **4th-order accuracy**: ratio 17.8 ≈ 16 (the second ratio 1.6 is just the reference hitting its own round-off floor) ✓ The earlier large drifts were purely from that test orbit's close approach to a primary , expected for fixed-step RK4 and not a correctness issue. ## Summary Implemented the rotating-frame (CR3BP) dynamics core. Only `src/system.cpp` and `src/solver.cpp` were edited; no headers or signatures changed. **`src/system.cpp`** - `rhs`: rotating-frame equations of motion. Acceleration = gradient of the pseudo-potential Ω = ½(x²+y²) + (1−μ)/r1 + μ/r2 (centrifugal + gravity) plus the Coriolis deflection (`+2·vy`, `−2·vx`, fixing the +z rotation sense). Throws `std::runtime_error` on a primary singularity. - `jacobi`: the Jacobi integral `C = (x²+y²) + 2(1−μ)/r1 + 2μ/r2 − (vx²+vy²)`, matching both numeric anchors exactly (3.75 and 2.91). **`src/solver.cpp`** - `step`: classical RK4 (global error O(dt⁴)), validates `dt > 0`. - `propagate`: records `states[0]=initial`, applies `step` `n_steps` times → `size == n_steps+1`, `time = n_steps·dt`. - `propagate_to_event`: marches in base steps watching `y`, brackets the first descending crossing (`y_prev > 0 && y_next ≤ 0`) strictly after t=0, then bisects on the sub-step size from the bracket's start to drive `|y|` below 1e-12 (well within the required 1e-9), without searching past `t_max`. Validates `dt > 0` and `t_max > 0`. Build is clean and the smoke test passes. Verified independently: anchors exact, L4 equilibrium stays at rest to machine precision, Jacobi conserved to ~1e-14, time-reversal symmetry to ~1e-13, 4th-order convergence, and event refinement onto `y=0`.
Final diff , what the agent changed

No file changes, the agent left the workspace untouched.

Verifier output , the proof behind the reward
tests/test.sh
== Configuring (cmake) ==
-- The CXX compiler identification is GNU 11.4.0
-- Detecting CXX compiler ABI info
-- Detecting CXX compiler ABI info - done
-- Check for working CXX compiler: /usr/bin/c++ - skipped
-- Detecting CXX compile features
-- Detecting CXX compile features - done
-- Configuring done
-- Generating done
-- Build files have been written to: /tmp/tmp.N4k8mdKL8k/orbit2d/build_grade
== Building ==
[  5%] Building CXX object CMakeFiles/orbit2d.dir/src/system.cpp.o
[ 10%] Building CXX object CMakeFiles/orbit2d.dir/src/solver.cpp.o
[ 15%] Building CXX object CMakeFiles/orbit2d.dir/src/io.cpp.o
[ 20%] Linking CXX static library liborbit2d.a
[ 20%] Built target orbit2d
[ 25%] Building CXX object CMakeFiles/orbit2d_cli.dir/src/main.cpp.o
[ 30%] Building CXX object tests/CMakeFiles/test_orbit.dir/test_orbit.cpp.o
[ 35%] Building CXX object tests/CMakeFiles/test_field.dir/test_field.cpp.o
[ 40%] Building CXX object tests/CMakeFiles/test_event.dir/test_event.cpp.o
[ 45%] Building CXX object tests/CMakeFiles/test_inertial.dir/test_inertial.cpp.o
[ 50%] Building CXX object tests/CMakeFiles/test_equilibria.dir/test_equilibria.cpp.o
[ 55%] Building CXX object tests/CMakeFiles/test_symmetry.dir/test_symmetry.cpp.o
[ 60%] Building CXX object tests/CMakeFiles/test_invariants.dir/test_invariants.cpp.o
[ 65%] Linking CXX executable orbit2d_cli
[ 65%] Built target orbit2d_cli
[ 70%] Linking CXX executable test_field
[ 75%] Linking CXX executable test_inertial
[ 80%] Linking CXX executable test_symmetry
[ 85%] Linking CXX executable test_event
[ 85%] Built target test_inertial
[ 85%] Built target test_field
[ 90%] Linking CXX executable test_orbit
[ 95%] Linking CXX executable test_invariants
[ 95%] Built target test_event
[ 95%] Built target test_symmetry
[100%] Linking CXX executable test_equilibria
[100%] Built target test_orbit
[100%] Built target test_invariants
[100%] Built target test_equilibria
== Running hidden test suite (ctest) ==
Test project /tmp/tmp.N4k8mdKL8k/orbit2d/build_grade
    Start 1: test_field
1/7 Test #1: test_field .......................   Passed    0.00 sec
    Start 2: test_orbit
2/7 Test #2: test_orbit .......................   Passed    0.01 sec
    Start 3: test_symmetry
3/7 Test #3: test_symmetry ....................   Passed    0.03 sec
    Start 4: test_equilibria
4/7 Test #4: test_equilibria ..................   Passed    0.00 sec
    Start 5: test_event
5/7 Test #5: test_event .......................   Passed    0.00 sec
    Start 6: test_invariants
6/7 Test #6: test_invariants ..................   Passed    0.01 sec
    Start 7: test_inertial
7/7 Test #7: test_inertial ....................   Passed    0.00 sec

100% tests passed, 0 tests failed out of 7

Label Time Summary:
hidden    =   0.05 sec*proc (7 tests)

Total Test time (real) =   0.06 sec
PASS: all hidden tests passed

Reproduce this trial: git checkout 2f94510 && PYTHONPATH=src python3 scripts/build_site.py , then open trial/trial_0ae5d3fd96b84db5. Re-running the agent live requires EVAL_PLATFORM_ENABLE_OAUTH_SMOKE=1 and is non-deterministic.

Trial trial_0ae5d3fd96b84db5 · verifier authoritative; classifier explanatory.