Hamiltonian

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

The Hamiltonian $H(\mathbf{q},\mathbf{p},t)$ is the scalar function on phase space that, together with the canonical equations $\dot{\mathbf{q}}=\partial H/\partial\mathbf{p}$, $\dot{\mathbf{p}}=-\partial H/\partial\mathbf{q}$, governs the dynamics of any Hamiltonian system. In conservative mechanics $H$ equals the total energy; in optimal control $H=L+\boldsymbol{\lambda}^{\!\top}\mathbf{f}$ anchors Pontryagin's principle; in CR3BP $H=-C/2$ where $C$ is the Jacobi constant. The same object thus unifies analytical mechanics, perturbation theory, and trajectory optimization (Meyer & Offin 2017; Szebehely 1967).

From Lagrangian to Hamiltonian

Given a Lagrangian $L(\mathbf{q},\dot{\mathbf{q}},t)$, define the conjugate momenta $\mathbf{p}=\partial L/\partial\dot{\mathbf{q}}$ and perform the Legendre transform

This yields the canonical equations. When $H$ is independent of $t$, $dH/dt=\partial H/\partial t=0$, i.e. $H$ is conserved. A natural Hamiltonian has the form $H=T(\mathbf{p})+V(\mathbf{q})$ (kinetic plus potential energy), the prototype of mechanical systems.

Hamiltonian matrix and symplectic structure

The coefficient matrix $A(t)$ of the linearization $\dot{\mathbf{z}}=A\mathbf{z}$ of a Hamiltonian system on phase space $\mathbf{z}=(\mathbf{q},\mathbf{p})$ satisfies $A^{\!\top}J+JA=0$ where $J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}$. Such $A$ are Hamiltonian matrices; their spectra are symmetric under $\lambda\mapsto -\lambda$ (Proposition 2.3.1, Meyer & Offin 2017). The matrix exponential of a Hamiltonian matrix is a symplectic matrix $M$ satisfying $M^{\!\top}JM=J$, whose eigenvalues come in reciprocal pairs $\lambda,1/\lambda$. The state-transition matrix of any Hamiltonian flow is symplectic; this is the algebraic origin of the saddle×center×center structure at collinear libration points (see Symplectic Structure & Hamiltonian Normal Form).

CR3BP Hamiltonian and the Jacobi constant

In the synodic frame of the circular restricted three-body problem, the autonomous Hamiltonian is

with $r_1,r_2$ the distances to the primaries (Szebehely 1967; Meyer & Offin 2017). The Coriolis and centrifugal terms appear as $yp_x-xp_y$. The Jacobi constant is $C=-2H$, the sole integral of CR3BP, gating the Hill region through the zero-velocity surfaces (see Jacobi Integral).

Cox et al. (2021) split $H$ into the natural Hamiltonian

(conserved in unperturbed CR3BP) and the low-thrust Hamiltonian $H_{lt}=H_{nat}-\mathbf{r}^{\!\top}\mathbf{a}_{lt}$, which becomes a new integral when the low-thrust acceleration $\mathbf{a}_{lt}$ is fixed in the rotating frame. The relative Hamiltonian describes the motion of a chaser relative to a target in the synodic frame under both primaries' gravity.

Nearly-integrable Hamiltonian systems

A system is integrable if it has $n$ independent integrals in involution (mutual Poisson brackets zero). A nearly-integrable Hamiltonian has the form $H=H_0(I)+\varepsilon H_1(I,\varphi)$, where $(I,\varphi)$ are action-angle variables of the integrable part and $\varepsilon\ll 1$. KAM and Nekhoroshev theorems describe its long-term dynamics (Celletti 2010; see KAM Theory). Hamiltonian integrable approximation averages $H_1$ over fast angles to obtain an integrable $\bar H$ that captures mean-motion resonances, used to seed continuation of resonant orbit families in cislunar space (Ding et al. 2025).

Hamiltonian reduction

When $H$ admits symmetries, Noether's theorem yields conserved quantities (linear momentum, angular momentum, Jacobi integral). Hamiltonian reduction quotients the phase space by the symmetry group to lower the effective degrees of freedom: Meyer–Marsden–Weinstein reduction is the standard framework (Meyer & Ofin 2017, Ch.7). The planar CR3BP with fixed Jacobi constant reduces to a 2-D Hamiltonian flow on the Hill region.

Optimal-control Hamiltonian

For an optimal-control problem, the Hamiltonian is built from the dynamics and running cost:

and the optimal control pointwise minimizes $H$ over $\mathbf{u}$. This Hamiltonian shares the canonical-equation structure of the mechanical one, with the costate $\boldsymbol{\lambda}$ playing the role of momentum (see Pontryagin's Maximum Principle, Costate Variables).

Application notes

• Conservation of $H$ in autonomous problems is a cheap and stringent accuracy check for numerical integration.
• CR3BP-LT with rotating-frame-fixed thrust inherits a conserved $H_{lt}$, enabling energy-based trajectory design analogous to the Jacobi-constant gating of CR3BP.
• The symplectic spectrum of the monodromy matrix, read off from $H$'s second variation, determines stable/unstable/center manifolds of libration-point orbits.

Related concepts

• CR3BP
• Jacobi Integral
• Monodromy Matrix
• Symplectic Structure & Hamiltonian Normal Form
• KAM Theory
• Pontryagin's Maximum Principle
• Costate Variables and Adjoint Equations
• Canonical Variables

References

• Szebehely, V. (1967). Theory of Orbits: The Restricted Problem of Three Bodies.
• Meyer, K. R., & Offin, D. C. (2017). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 3rd ed., Ch. 1–2, 7.
• Celletti, A. (2010). Stability and Chaos in Celestial Mechanics.
• Cox, A. B., et al. (2021). CR3BP with low-thrust.
• Ding, Y., et al. (2025). Cislunar SSA via Earth-Moon resonant orbits.

Definition

哈密顿函数 $H(\mathbf{q},\mathbf{p},t)$ 是相空间上的标量函数，与正则方程 $\dot{\mathbf{q}}=\partial H/\partial\mathbf{p}$, $\dot{\mathbf{p}}=-\partial H/\partial\mathbf{q}$ 一起决定任何哈密顿系统的演化。在保守力学中 $H$ 等于系统总能量；在最优控制中 $H=L+\boldsymbol{\lambda}^{\!\top}\mathbf{f}$ 是庞特里亚金原理的核心；在 CR3BP 中 $H=-C/2$，$C$ 为雅可比常数。同一对象统一了分析力学、摄动理论与轨迹优化（Meyer & Offin 2017；Szebehely 1967）。