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Projection Methods for the Kepler Problem

A comprehensive study and implementation of structure-preserving numerical integration methods applied to the classical Kepler Problem (modeling two-body planetary motion under an inverse-square gravitational force).

This project explores why traditional numerical solvers (like Forward/Backward Euler) suffer from long-term geometric degradation—causing orbits to unrealistically spiral inward or outward—and implements Manifold Projection Methods to enforce the strict preservation of physical invariants like Total Energy and Angular Momentum.

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Project Overview

When simulating orbital mechanics, standard numerical integrators introduce cumulative truncation errors that destroy the system's geometric structure over time. This project implements and analyzes a variety of numerical solutions, contrasting traditional methods against structure-preserving alternatives.

Key Objectives

• Mathematical Validation: Proving the boundaries of existence and uniqueness of the Kepler problem via local vs. global variants of the Picard-Lindelöf Theorem.
• Error Analysis: Analyzing the local and global truncation errors ($O(h)$ vs $$O(h^4)$$) of explicit, implicit, and higher-order integrators.
• Invariant Mapping: Tracking the unphysical gain/loss of system invariants over extended iterations.
• Manifold Projections: Utilizing the Lagrange Multipliers Method to correct numerical drift by shifting drifted states back onto the exact Energy ($E$) and Angular Momentum ($L$) manifolds.

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Numerical Solvers Implemented

The repository contains implementations of several distinct ordinary differential equation (ODE) solvers to benchmark their performance:

Solver Category	Method	Order (Global)	Structural Stability
Standard Explicit	Forward Euler	$O(h)$	Poor (Spirals Outward, Gains Energy)
Standard Implicit	Backward Euler	$O(h)$	Poor (Spirals Inward, Loses Energy)
Higher-Order	Runge-Kutta 4 (RK4)	$O(h^4)$	High Precision (Drifts very slowly)
Geometric/Symplectic	Symplectic Euler, Störmer-Verlet, Implicit Midpoint	Variable	Excellent (Preserves phase-space volume)
Structure-Enforced	Projected Integrators (Energy / Angular Momentum / Both)	Dependent	Perfect Invariant Conservation

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Projection Framework

To eliminate numerical drift without drastically diminishing step sizes ($h$), we solve a constrained optimization problem at each time step using Lagrange multipliers:

$$\min \left( \frac{1}{2} |u_{n+1} - \tilde{u}_{n+1}|^2 \right) \quad \text{subject to} \quad g(u_{n+1}) = 0$$

Where $\tilde{u}^{n+1}$ is the uncorrected state produced by a standard step, $u^{n+1}$ is the corrected target state, and $g(u)$ represents our physical constraints:

1. Projection on Energy Manifold: Enforces $$E(u_{n+1}) - E(u_n) = 0$$
2. Projection on Angular Momentum (AM) Manifold: Enforces $$L(u_{n+1}) - L(u_n) = 0$$
3. Simultaneous Projection (Energy & AM): Solves both nonlinear geometric constraints simultaneously using root-finding algorithms (e.g., fsolve).

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Key Findings & Visualizations

• The Classical Drift: Standard Forward Euler trajectories artificially spiral away from the central mass due to exponential energy growth, while Backward Euler collapses inward onto the singularity.
• The Projection Remedy: Applying state projections forces the system to preserve its underlying physics. Even when using a highly unstable baseline solver like Forward Euler with a large step size ($h = 0.005$), enforcing energy and angular momentum constraints keeps the planet locked into stable, perfectly closed elliptical orbits over long temporal durations.

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References & Literature

This implementation is heavily informed by core mathematical principles and reference materials in geometric integration, including:

• Geometric Numerical Integration: Structure-Preserving Algorithms for ODEs (Hairer, Lubich, Wanner).
• Grundlagen der Numerischen Mathematick (Hanke-Bourgeois).
• Classical applications of Noether's Theorem regarding continuous symmetries and conservation laws.