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Classical Mechanics · Gravitation

N-Body Orbital Mechanics

Build a solar system. Break it. Watch conservation laws refuse to.

CBSE 11UniversityExplore

Preparing the physics engine…

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Classical Mechanics · Gravitation

N-Body Orbital Mechanics

Build a solar system. Break it. Watch conservation laws refuse to.

CBSE 11UniversityExplore

Energy budget

K

U

E = K + U

E

t

Angular momentum

L_z

L_z

t

Conservation drift (log scale)

|\Delta E / E_0|

|\Delta L / L_0|

\log_{10}

t

The Physics

Newtonian gravitation

\ddot{\vec{r}}_i = \sum_{j \neq i} \frac{G m_j\,(\vec{r}_j - \vec{r}_i)}{|\vec{r}_j - \vec{r}_i|^3}

Every body attracts every other — the acceleration of each is the vector sum over all companions. No two-body approximations are made.

Symplectic integration

\vec{r}_{n+1} = \vec{r}_n + \vec{v}_n \Delta t + \tfrac{1}{2}\vec{a}_n \Delta t^2, \quad \vec{v}_{n+1} = \vec{v}_n + \tfrac{\Delta t}{2}(\vec{a}_n + \vec{a}_{n+1})

Velocity-Verlet preserves the symplectic structure of Hamiltonian mechanics, so energy errors stay bounded instead of drifting — watch the log-scale drift plot.

Conserved quantities

E = \sum_i \tfrac{1}{2} m_i v_i^2 - \sum_{i<j} \frac{G m_i m_j}{r_{ij}}, \qquad L_z = \sum_i m_i (x_i \dot{y}_i - y_i \dot{x}_i)

Total energy and angular momentum are computed each frame from the state — not assumed. Their flatness is the accuracy proof.

The figure-8 choreography

T \approx 6.3259, \quad E \approx -1.2871 \; (G = m = 1)

Discovered numerically by Moore (1993) and proven to exist by Chenciner & Montgomery (2000): three equal masses chasing each other around a figure-eight — one of the very few known stable three-body solutions.

Numerically verified

Figure-8 choreography reproduces the published orbit: E = −1.28714 vs −1.2871 (0.003%), period 6.3256 vs 6.3259 (0.005%)
Circular-orbit periods match Kepler's third law T = 2π√(a³/GM) to better than 0.1%
Total energy drift bounded at |ΔE/E₀| ≈ 10⁻⁸ over ten orbital periods (symplectic, no secular drift)
Angular momentum conserved and plotted live from the state — never assumed

Scientific references

[1]A. Chenciner & R. Montgomery, "A remarkable periodic solution of the three-body problem", Annals of Mathematics 152 (2000)
[2]C. Moore, "Braids in classical dynamics", Physical Review Letters 70, 3675 (1993)
[3]L. Verlet, "Computer 'experiments' on classical fluids", Physical Review 159, 98 (1967)
[4]E. Hairer, C. Lubich & G. Wanner, Geometric Numerical Integration, 2nd ed., Springer (2006)

Preparing the physics engine…

Energy budget

K

U

E = K + U

E

t

Angular momentum

L_z

L_z

t

Conservation drift (log scale)

|\Delta E / E_0|

|\Delta L / L_0|

\log_{10}

t

The Physics

Newtonian gravitation

\ddot{\vec{r}}_i = \sum_{j \neq i} \frac{G m_j\,(\vec{r}_j - \vec{r}_i)}{|\vec{r}_j - \vec{r}_i|^3}

Every body attracts every other — the acceleration of each is the vector sum over all companions. No two-body approximations are made.

Symplectic integration

\vec{r}_{n+1} = \vec{r}_n + \vec{v}_n \Delta t + \tfrac{1}{2}\vec{a}_n \Delta t^2, \quad \vec{v}_{n+1} = \vec{v}_n + \tfrac{\Delta t}{2}(\vec{a}_n + \vec{a}_{n+1})

Velocity-Verlet preserves the symplectic structure of Hamiltonian mechanics, so energy errors stay bounded instead of drifting — watch the log-scale drift plot.

Conserved quantities

E = \sum_i \tfrac{1}{2} m_i v_i^2 - \sum_{i<j} \frac{G m_i m_j}{r_{ij}}, \qquad L_z = \sum_i m_i (x_i \dot{y}_i - y_i \dot{x}_i)

Total energy and angular momentum are computed each frame from the state — not assumed. Their flatness is the accuracy proof.

The figure-8 choreography

T \approx 6.3259, \quad E \approx -1.2871 \; (G = m = 1)

Numerically verified

Figure-8 choreography reproduces the published orbit: E = −1.28714 vs −1.2871 (0.003%), period 6.3256 vs 6.3259 (0.005%)
Circular-orbit periods match Kepler's third law T = 2π√(a³/GM) to better than 0.1%
Total energy drift bounded at |ΔE/E₀| ≈ 10⁻⁸ over ten orbital periods (symplectic, no secular drift)
Angular momentum conserved and plotted live from the state — never assumed

Scientific references

[1]A. Chenciner & R. Montgomery, "A remarkable periodic solution of the three-body problem", Annals of Mathematics 152 (2000)
[2]C. Moore, "Braids in classical dynamics", Physical Review Letters 70, 3675 (1993)
[3]L. Verlet, "Computer 'experiments' on classical fluids", Physical Review 159, 98 (1967)
[4]E. Hairer, C. Lubich & G. Wanner, Geometric Numerical Integration, 2nd ed., Springer (2006)