Specialized Lab · Dedicated ViewCelestial Mechanics & Astrodynamics

Lagrange Points & Restricted Three-Body Problem (CR3BP)

RK4 numerical integrator in corotating frame with Hill equipotential surfaces and Halo orbits

Interactive orbital simulator of the Circular Restricted Three-Body Problem (CR3BP). Solves the five equilibrium Lagrange points (L1-L5), renders Hill effective potential contours, computes the Jacobi constant in real time, and tests space missions like JWST (L2), SOHO (L1), and Trojans (L4).

Fundamental Law
$$\ddot{x} - 2\dot{y} = \frac{\partial U_{\text{eff}}}{\partial x}, \quad \ddot{y} + 2\dot{x} = \frac{\partial U_{\text{eff}}}{\partial y}$$
Celestial Mechanics & Dynamical Systems

Lagrange Points Orbital Simulator (CR3BP & Jacobi Potential)

$\nabla U(x, y) = 0 \iff L_1, \dots, L_5$

{lang === 'es' ? 'En 1772, Joseph-Louis Lagrange descubrió que en el Problema Restringido de Tres Cuerpos Circulares (CR3BP) existen cinco puntos de equilibrio en el sistema de referencia corrotante donde la atracción gravitatoria combinada de dos masas principales equivale exactamente a la fuerza centrífuga necesaria para acompañar su rotación orbital. Los puntos colineales ($L_1, L_2, L_3$) son puntos de ensilladura inestables que albergan observatorios como el James Webb o SOHO mediante maniobras continuas de mantenimiento de estación (station-keeping). Los puntos triangulares ($L_4, L_5$) son máximos del potencial estabilizados dinámicamente por la aceleración de Coriolis siempre que la razón de masas cumpla el Criterio de Routh ($\\mu < 0.03852$), albergando los enjambres de asteroides troyanos.' : 'In 1772, Joseph-Louis Lagrange discovered that in the Circular Restricted Three-Body Problem (CR3BP), exactly five equilibrium points exist in the corotating frame where the combined gravitational pull of two primary masses balances the centrifugal force of their orbital rotation. The collinear points ($L_1, L_2, L_3$) are unstable saddle points hosting space observatories such as the James Webb or SOHO through active station-keeping maneuvers. The triangular points ($L_4, L_5$) are effective potential maxima dynamically stabilized by the Coriolis acceleration provided the mass ratio satisfies the Gascheau-Routh Criterion ($\\mu < 0.03852$), sheltering the Trojan asteroid swarms.'}

Missions & Orbits:
💡 Click & drag in space to launch a probe with a velocity vector
Position (x, y):0.00, 0.00
Speed |v|:0.000
Jacobi Constant CJ:3.000
Routh Stability:Stable (μ < 0.0385)
0.0300
0.001 (Satélite)μ_crit = 0.03850.500 (Binaria)
Field Layers & Vectors
Integration Speed (RK4)
Export & Telemetry
Dynamical Formulation

The Corotating Frame & Jacobi Potential

{lang === 'es' ? 'Al transformar las ecuaciones de Newton a un marco de referencia centrado en el baricentro que gira a velocidad angular unitaria $\\mathbf{\\Omega} = (0, 0, 1)$, surge la aceleración ficticia de Coriolis $2(\\mathbf{\\Omega} \\times \\mathbf{v})$ y la fuerza centrífuga $-\\mathbf{\\Omega} \\times (\\mathbf{\\Omega} \\times \\mathbf{r})$. El movimiento de una partícula de masa infinitesimal $m \\ll M_2 < M_1$ queda gobernado por el potencial efectivo escalar $U(x, y)$:' : 'Transforming Newton’s equations to a barycentric reference frame rotating with unit angular velocity $\\mathbf{\\Omega} = (0, 0, 1)$ introduces the fictitious Coriolis acceleration $2(\\mathbf{\\Omega} \\times \\mathbf{v})$ and centrifugal force $-\\mathbf{\\Omega} \\times (\\mathbf{\\Omega} \\times \\mathbf{r})$. The motion of an infinitesimal test mass $m \\ll M_2 < M_1$ is governed by the scalar effective potential $U(x, y)$:'}

$$U(x, y) = \frac{1}{2}(x^2 + y^2) + \frac{1-\mu}{r_1} + \frac{\mu}{r_2}$$ $$\ddot{x} - 2\dot{y} = \frac{\partial U}{\partial x}, \quad \ddot{y} + 2\dot{x} = \frac{\partial U}{\partial y}$$
Energy Conservation

The Jacobi Integral & Hill Regions

{lang === 'es' ? 'Aunque la energía mecánica ordinaria no se conserva en el marco rotante, la contracción del tensor de aceleración produce una constante de movimiento exacta: la Integral de Jacobi $C_J$. La velocidad relativa cumple $v^2 = 2U(x, y) - C_J$. La condición de velocidad nula ($v = 0$) define las Curvas de Velocidad Cero (Superficies de Hill), delimitando regiones espacialmente prohibidas que ningún vehículo orbital puede penetrar con dicha energía.' : 'Although ordinary mechanical energy is not conserved in the rotating frame, contracting the acceleration vector yields an exact constant of motion: the Jacobi Integral $C_J$. The relative speed satisfies $v^2 = 2U(x, y) - C_J$. Setting $v = 0$ defines the Zero-Velocity Curves (Hill Surfaces), carving out forbidden topological domains that no trajectory can cross with that energy level.'}

$$C_J = 2U(x, y) - (\dot{x}^2 + \dot{y}^2) = \text{constante}$$ $$\mathcal{ZVC} = \{ (x, y) \in \mathbb{R}^2 \mid 2U(x, y) = C_J \}$$
Asymptotic Stability

Gascheau-Routh Stability for L₄ and L₅

{lang === 'es' ? 'La linealización en torno a los puntos triangulares $L_4$ y $L_5$ (vértices de triángulos equiláteros con los cuerpos masivos) produce una ecuación característica cuártica. Para que el movimiento sea puramente cuasiperiódico (sin modos de divergencia exponencial), el discriminante debe ser estrictamente no negativo, requiriendo que la masa secundaria no supere el 3.85% de la masa total:' : 'Linearizing around the equilateral triangular points $L_4$ and $L_5$ yields a fourth-degree characteristic polynomial. For bounded quasi-periodic libration without exponential runaway, the discriminant must remain non-negative, imposing the requirement that the secondary mass cannot exceed 3.85% of total system mass:'}

$$\lambda^4 + \lambda^2 + \frac{27}{4}\mu(1-\mu) = 0$$ $$\mu < \mu_{\text{crit}} = \frac{1}{2}\left(1 - \sqrt{\frac{23}{27}}\right) \approx 0.0385208$$
Modern Astronautics

Mission Guide to the Lagrange Points

  • L₁ (Unstable): Uninterrupted view of the Sun. Home to SOHO, DSCOVR, and solar wind monitor ACE.
  • L₂ (Unstable): Deep space vista with Earth shielding solar glare. Hosts James Webb Space Telescope (JWST), Gaia, and future Nancy Grace Roman.
  • L₃ (Unstable): Hidden behind the primary body. Popular in theoretical orbital resonance and classic science fiction.
  • L₄ & L₅ (Stable): Equilateral triangular points at 60°. Sheltering thousands of Jupiter Trojan asteroids and visited by NASA’s Lucy spacecraft.