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14: General Perturbation Theory

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    • 14.1: Introduction to General Perturbation Theory
      This page covers the motion of a particle in orbit around a mass, defined by a gravitational potential of −\(GM/r\) leading to keplerian ellipses. It also discusses how perturbations, such as −(GM/r + R), lead to deviations from perfect ellipses, influencing orbital elements like semi-major axis and eccentricity.
    • 14.2: Contact Transformations and General Perturbation Theory
      This page explores Hamiltonian dynamics and perturbation theory in orbital mechanics, linking an initial Hamiltonian \(H_0\) to a perturbed Hamiltonian \(H\) via a contact transformation. It examines the evolution of orbital elements \(α_i\) and \(β_i\) in response to perturbations, deriving formulas for their time variations using derivatives and Poisson brackets. The section underscores the interdependence of these elements within the dynamic system affected by perturbations.
    • 14.3: The Poisson Brackets for the Orbital Elements
      This page covers the computation of Poisson brackets for orbital elements \(A_i\), presenting a detailed example with \(Ω\) and \(i\). It demonstrates the calculation of \(\{ Ω , i \}\) and concludes with the expression \(\frac{1}{\sqrt{GMm^2 a (1-e^2) \sin i}}\). Additional brackets for pairs \( \{ a , T \} \), \( \{ e , T \} \), \( \{ i , ω \} \), and \( \{ e , ω \} \) are derived, highlighting the inherent symmetrical relationships between the elements.
    • 14.4: Lagrange's Planetary Equations
      Lagrange’s Planetary Equations enable us to calculate the rates of change of the orbital elements if we know the form of the perturbing function.
    • 14.5: Motion Around an Oblate Symmetric Top
      This page discusses the gravitational potential of an oblate spheroid and its impact on orbiting bodies, highlighting that long-term average rates of various orbital elements remain stable while detailing specific changes in nodes and apsides influenced by orbital inclination.

    Thumbnail: Animation showing the relationship between the five Lagrangian points (red) of a planet (blue) orbiting a star (yellow), and the gravitational potential in the plane containing the orbit (grey surface with purple contours of equal potential). (CC BY-SA 4.0; cmglee).

    This page titled 14: General Perturbation Theory was last modified on Wed, 30 Dec 2020 20:59:06 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.