Laplace-Legendre expansion of the planar planetary Hamiltonian.

We present a detailed analytical framework for the Laplace-Legendre expansion of the planar planetary Hamiltonian, a fundamental tool in celestial mechanics for studying the long-term evolution and stability of planetary systems. This work refines the perturbative approach, offering improved accuracy for calculations of gravitational interactions and resonant dynamics, which is essential for precise orbital predictions and understanding the architecture of exoplanetary systems.