Hamiltonian

Author: Tianjiang Shuo
Contributing Institution: School of Astronautics, Harbin Institute of Technology, National Key Laboratory of Rapid Design and Intelligent Swarm of Small Spacecraft

Definition

The Hamiltonian is a core scalar function in analytical mechanics and optimal control theory, constructed from generalized coordinates and generalized momenta (or state variables and co-state variables). In orbital mechanics, the Hamiltonian serves both as a physical quantity describing energy conservation and as a mathematical tool for establishing necessary conditions in optimal control. Pontryagin's Maximum Principle uses the Hamiltonian as its central framework to derive optimal control laws.

Mathematical Description

Hamiltonian in Classical Mechanics

In the Hamiltonian mechanics framework, the Hamiltonian is defined as:

H(\mathbf{q}, \mathbf{p}, t) = \mathbf{p}^T \dot{\mathbf{q}} - L(\mathbf{q}, \dot{\mathbf{q}}, t)

where $\mathbf{q}$ is the generalized coordinate, $\mathbf{p} = \partial L / \partial \dot{\mathbf{q}}$ is the generalized momentum, and $L$ is the Lagrangian. The canonical equations are:

\dot{\mathbf{q}} = \frac{\partial H}{\partial \mathbf{p}}, \quad \dot{\mathbf{p}} = -\frac{\partial H}{\partial \mathbf{q}}

When $H$ does not explicitly depend on time $t$ , it is a conserved quantity corresponding to the total energy of the system.

Hamiltonian in CR3BP

In the rotating frame of the Circular Restricted Three-Body Problem (CR3BP), the Hamiltonian is:

H = \frac{1}{2}(p_x^2 + p_y^2 + p_z^2) + y p_x - x p_y - \frac{1-\mu}{r_1} - \frac{\mu}{r_2}

where $p_x, p_y, p_z$ are canonical momenta, and $r_1, r_2$ are the distances from the spacecraft to the two primary bodies. The Jacobi constant $C = -2H$ is the only conserved quantity in CR3BP.

Hamiltonian in Optimal Control

In optimal control problems, the Hamiltonian is constructed from the state equations, co-state variables, and the performance index:

H(\mathbf{x}, \boldsymbol{\lambda}, \mathbf{u}, t) = L(\mathbf{x}, \mathbf{u}, t) + \boldsymbol{\lambda}^T \mathbf{f}(\mathbf{x}, \mathbf{u}, t)

where $L$ is the instantaneous cost function, $\mathbf{f}$ is the right-hand side of the state equation, and $\boldsymbol{\lambda}$ is the co-state variable. The optimal control $\mathbf{u}^*$ extremizes $H$ :

\frac{\partial H}{\partial \mathbf{u}} = 0 \quad \text{(for continuous control)}

Applications in Cislunar Space

The Hamiltonian has broad applications in cislunar space missions:

Orbit design: In the CR3BP framework, the Hamiltonian is directly related to the Jacobi constant; zero-velocity surfaces are determined by level sets of $H$ , providing the foundation for orbit accessibility analysis
Fuel-optimal control: In Pontryagin's Maximum Principle, the extremization condition of the Hamiltonian derives the optimal control law for thrust direction and magnitude, serving as the starting point for indirect methods in trajectory optimization
Invariant manifold analysis: The symplectic structure of Hamiltonian systems guarantees phase space volume conservation, providing theoretical assurance for computing stable/unstable manifolds of periodic orbits such as DRO and NRHO