Hamiltonian mechanics In Hamiltonian mechanics, a physical system is specified by a function, called
Hamiltonian and denoted \mathcal{H}, of
canonical coordinates in
phase space. The canonical coordinates consist of the
generalized coordinates x_k in
configuration space and their
conjugate momenta p_k, for k = 1, ... N, for the bodies in the system (N = 3 for the von Zeipel-Kozai–Lidov effect). The number of (x_k, p_k) pairs required to describe a given system is the number of its
degrees of freedom. The coordinate pairs are usually chosen in such a way as to simplify the calculations involved in solving a particular problem. One set of canonical coordinates can be changed to another by a
canonical transformation. The
equations of motion for the system are obtained from the Hamiltonian through ''Hamilton's canonical equations'', which relate time derivatives of the coordinates to partial derivatives of the Hamiltonian with respect to the conjugate momenta.
Three-body problem The dynamics of a system composed of three bodies system acting under their mutual gravitational attraction is
chaotic: its behavior over long periods of time is enormously sensitive to any slight changes in the initial conditions. This exposes computations to rapid deterioration from uncertainties in those conditions, in determining them, and then preserving them from rounding away in
computer arithmetic. The practical consequence is that, the
three-body problem cannot be solved analytically for an indefinite amount of time, except in special cases. Instead,
numerical methods are used for forecast-times limited by the available precision. The Lidov–Kozai mechanism is a feature of
hierarchical triple systems, that is systems in which one of the bodies, called the "perturber", is located far from the other two, which are said to comprise the
inner binary. The perturber and the centre of mass of the inner binary comprise the
outer binary. Such systems are often studied by using the methods of
perturbation theory to write the Hamiltonian of a hierarchical three-body system as a sum of two terms responsible for the isolated evolution of the inner and the outer binary, and a third term
coupling the two orbits, : \mathcal{H} = \mathcal{H}_{\rm in} + \mathcal{H}_{\rm out} + \mathcal{H}_{\rm pert}. The coupling term is then expanded in the orders of parameter \alpha, defined as the ratio of the
semi-major axes of the inner and the outer binary and hence small in a hierarchical system. Since the perturbative series
converges rapidly, the qualitative behaviour of a hierarchical three-body system is determined by the initial terms in the expansion, referred to as the
quadrupole (\propto\alpha^2),
octupole (\propto\alpha^3) and
hexadecapole (\propto\alpha^4) order terms, : \mathcal{H}_{\rm pert} = \mathcal{H}_{\rm quad} + \mathcal{H}_{\rm oct} + \mathcal{H}_{\rm hex} + O(\alpha^5). For many systems, a satisfactory description is found already at the lowest, quadrupole order in the perturbative expansion. The octupole term becomes dominant in certain regimes and is responsible for a long-term variation in the amplitude of the Lidov–Kozai oscillations.
Secular approximation The Lidov–Kozai mechanism is a
secular effect, that is, it occurs on timescales much longer compared to the orbital periods of the inner and the outer binary. In order to simplify the problem and make it more tractable computationally, the hierarchical three-body Hamiltonian can be
secularised, that is, averaged over the rapidly varying mean anomalies of the two orbits. Through this process, the problem is reduced to that of two interacting massive wire loops. ==Overview of the mechanism==