The main theme in this chapter is Algorithmic Regularization.The procedures in this are the {Logarithmic Hamiltonian (LogH)} and the {Time Transformed Leapfrog (TTL)}, in both of which the use of {\bf leapfrog is compulsory for regularization}. In addition to this, auxiliary variables for velocity dependent perturbations are specified.

This chapter discusses the various problems of few-body dynamics, starting from the two-body and planetary systems and proceeds to stellar dynamics and artificial satellite motions.

This chapter discusses the basic concepts in many-body dynamics.From the Lagrangian, Hamiltonian, canonical transformations and time transformations to Hamilton-Jacobi equations. This content can be found in most classical dynamics textbooks

This chapter is essentially a partial copy of the manuscript for the paper Mikkola, Palmer, Hashida(2002). In fact the text is mostly from the manuscript for that paper. A method of high precision computation of the motion of a body in a potential witch is an expansion in terms ofspherical harmonics, is considered.

This chapter discusses the main points in the computation of the motion of two-body systems. The matter here is basic but gives many important expressions for computation of two-body motion.

In this chapter one finds an introduction to the classical variation of parameters methods. This means e.g. that perturbed two-body motion can be represented as two-body motion with slowly evolving orbital elements.

Among other things this section handles Extended Phase Space, Time Transformations, Kustaanheimo-Stiefel (KS) Transformation and Chain Regularization. These topics are important in connection with regularization of the equations of motion. A major reference for this chapter is the book \cite{Sverre2010book}, which discusses also regularization although it concentrates on the use of polynomial approximations.

In this chapter the linearized Riemann tensor correlator on a de Sitter background including one-loop corrections from conformal fields is derived. The Riemann tensor correlation function exhibits interesting features: it is gauge-invariant even when including contributions from loops of matter fields, but excluding graviton loops as it is implemented in the 1/N expansion, it is compatible with de Sitter invariance, and provides a complete characterization of the local geometry. The two-point correlator function of the Riemann tensor is computed by taking suitable derivatives of the metric correlator function found in the previous chapter, and the result is written in a manifestly de Sitter-invariant form. Moreover, given the decomposition of the Riemann tensor in terms of Weyl and Ricci tensors, we write the explicit results for the Weyl and Ricci tensors correlators as well as the Weyl–Ricci tensors correlator and study both their subhorizon and superhorizon behavior. These results are extended to general conformal field theories. We also derive the Riemann tensor correlator in Minkowski spacetime in a manifestly Lorentz-invariant form by carefully taking the flat-space limit of our result in de Sitter.

As a short introduction to this chapter we first briefly summarize the in-in or closed-time-path (CTP) functional formalism and evaluate the CTP effective action for a scalar field in Minkowski spacetime. We then consider N quantum matter fields interacting with the gravitational field assuming an effective field theory approach to quantum gravity and consider the quantization of metric perturbations around a semiclassical background in the CTP formalism. A suitable prescription is given to select an asymptotic initial vacuum state of the interacting theory; this prescription plays an important role in calculations in later chapters. We derive expressions for the two-point metric correlations, which are conveniently written in terms of the CTP effective action that results from integrating out the matter fields by rescaling the gravitational constant and performing a 1/N expansion. These correlations include loop corrections from matter fields but no graviton loops. This is achieved consistently in the 1/N expansion, and is illustrated in a simplified model of matter–gravity interaction.

Structure formation in the early universe is a key problem in modern cosmology. In this chapter we discuss stochastic gravity as an alternative framework for studying the generation of primordial inhomogeneities in inflationary models, which can easily incorporate effects that go beyond the linear perturbations of the inflaton field. We show that the correlation functions that follow from the Einstein–Langevin equation, which emerge in the framework of stochastic gravity, coincide with that obtained with the usual quantization procedures when both the metric perturbations and the inflaton fluctuations are linear. Stochastic gravity, however, can also deal very naturally with the fluctuations of the inflaton field beyond the linear approximation. Here, we illustrate the stochastic approach with one of the simplest chaotic inflationary models in which the background spacetime is a quasi de Sitter universe, and prove the equivalence of the stochastic and quantum correlations to the linear order.