Glossary
This glossary defines key terms used in HITEN and dynamical systems analysis.
A
- Attractor
A set of points in phase space that trajectories approach as time goes to infinity.
B
- Bifurcation
A qualitative change in the behavior of a dynamical system as a parameter is varied.
- Bifurcation Point
A parameter value at which a bifurcation occurs.
- Birkhoff Normal Form
A canonical form for Hamiltonian systems near equilibrium points.
C
- Center Manifold
A manifold of equilibrium points that contains all the bounded solutions near the equilibrium.
- Circular Restricted Three-Body Problem (CR3BP)
A simplified model of three-body dynamics where one body has negligible mass and the other two bodies move in circular orbits.
- Collinear Points
The L1, L2, and L3 Lagrange points, which lie on the line connecting the two primary bodies.
D
- Dynamical System
A system whose state evolves over time according to a set of differential equations.
- Dynamical System Theory
The mathematical study of dynamical systems and their properties.
E
- Equilibrium Point
A point in phase space where the time derivative of the state is zero.
F
- Family of Orbits
A continuous set of orbits parameterized by one or more parameters.
- First Integral
A function that remains constant along trajectories of a dynamical system.
- Fourier Series
A representation of a periodic function as a sum of sinusoidal functions.
G
- Gravitational Parameter
The product of the gravitational constant and the mass of a celestial body.
H
- Hamiltonian
A function that describes the total energy of a dynamical system in terms of position and momentum variables.
- Hamiltonian System
A dynamical system whose equations of motion can be derived from a Hamiltonian function.
- Halo Orbit
A three-dimensional periodic orbit around a Lagrange point.
I
- Invariant Manifold
A manifold that is invariant under the flow of a dynamical system.
- Island of Stability
A region in phase space where trajectories remain bounded for long times.
J
- Jacobi Constant
A conserved quantity in the CR3BP that plays a role similar to energy.
- Jacobi Integral
Another name for the Jacobi constant.
L
- Lagrange Points
Five equilibrium points in the CR3BP where the gravitational and centrifugal forces balance.
- Libration
Oscillatory motion around an equilibrium point.
- Libration Points
Another name for Lagrange points.
M
- Mass Parameter
The ratio of the smaller primary mass to the total mass in the CR3BP.
- Monodromy Matrix
A matrix that describes how small perturbations evolve over one period of a periodic orbit.
N
- Normal Form
A simplified form of a dynamical system near an equilibrium point or periodic orbit.
- Normal Mode
A mode of oscillation in a linearized system.
O
- Orbit
The path followed by a particle in phase space.
- Orbital Period
The time required for one complete orbit.
P
- Periodic Orbit
An orbit that returns to its initial state after a finite time.
- Phase Space
The space of all possible states of a dynamical system.
- Poincare Map
A map that describes how trajectories intersect a surface of section.
- Poincare Section
A surface in phase space used to study the dynamics of a system.
- Poisson Bracket
A binary operation on functions in phase space that is fundamental to Hamiltonian mechanics.
R
- Restricted Three-Body Problem
A three-body problem where one body has negligible mass.
- Rotating Frame
A coordinate system that rotates with the primary bodies.
S
- Stable Manifold
The set of points that approach an equilibrium point or periodic orbit as time goes to infinity.
- Surface of Section
Another name for Poincare section.
- Symplectic
A property of transformations that preserve the symplectic structure of phase space.
T
- Triangular Points
The L4 and L5 Lagrange points, which form equilateral triangles with the two primary bodies.
- Trajectory
The path followed by a particle in phase space over time.
U
- Unstable Manifold
The set of points that approach an equilibrium point or periodic orbit as time goes to negative infinity.
V
- Variational Equations
Linearized equations that describe how small perturbations evolve in a dynamical system.
W
- Weak Stability Boundary
A region in phase space where trajectories are weakly stable.
Z
- Zero-Velocity Surface
A surface in configuration space where the velocity is zero for a given value of the Jacobi constant.