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Concept# Classical mechanics

Summary

Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical mechanics, if the present state is known, it is possible to predict how it will move in the future (determinism), and how it has moved in the past (reversibility).
The "classical" in "classical mechanics" does not refer classical antiquity, as it might in, say, classical architecture. On the contrary the development of classical mechanics involved substantial change in the methods and philosophy of physics. Instead, the qualifier distinguishes classical mechanics from physics developed after the revolutions of the early 20th century, which revealed limitations of classical mechanics.
The earliest formulation of classical mechanics is often referred to as Newtonian mechanics. It consists of the physical concepts based on foundational works of Sir Isaa

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A functional integration approach – whose main ingredient is the Hubbard-Stratonovich transformation – for the quantum nonrelativistic many-fermion problem is investigated. With this method, the ground state energy correponds to a systematic expansion in powers of a small parameter related to the number of fermions. It is a functional of a potential determined by a self-consistent equation. The semiclassical Hartree energy is obtained at lowest order of the expansion, the exchange energy at first order, and the correlation energy at second order. This approach is applied to large neutral atoms, for which the correlation energy is computed. This approach is also applied to many-electron quantum dots with harmonic confinement. The self-consistent equation is solved as a function of a small parameter depending on the confinement strength. The Hartree and exchange energies are computed in powers of this parameter, and the correlation energy is computed at lowest order. The energy oscillations, arising from the Hartree energy, are also evaluated; they are related to the periodic orbits of the classical dynamics of the self-consistent potential.

Miguel Alexandre Ribeiro Correia

Accretion disks surrounding compact objects, and other environmental factors, deviate satellites from geodetic motion. Unfortunately, setting up the equations of motion for such relativistic trajectories is not as simple as in Newtonian mechanics. The principle of general (or Lorentz) covariance and the mass-shell constraint make it difficult to parametrize physically adequate 4-forces. Here, we propose a solution to this old problem. We apply our framework to several conservative and dissipative forces. In particular, we propose covariant formulations for Hooke???s law and the constant force and compute the drag due to gravitational and hard-sphere collisions in dust, gas, and radiation media. We recover and covariantly extend known forces such as Epstein drag, Chandrasekhar???s dynamical friction, and Poynting-Robertson drag. Variable-mass effects are also considered, namely, Hoyle-Lyttleton accretion and the variable-mass rocket. We conclude with two applications: (1) The free-falling spring, where we find that Hooke???s law corrects the deviation equation by an effective anti???de Sitter tidal force and (2) black hole infall with drag. We numerically compute some trajectories on a Schwarzschild background supporting a dustlike accretion disk.

This is an overview of a program of stochastic deformation of the mathematical tools of classical mechanics, in the Lagrangian and Hamiltonian approaches. It can also be regarded as a stochastic version of Geometric Mechanics.The main idea is to construct well defined probability measures strongly inspired by Feynman Path integral method in Quantum Mechanics. In contrast with other approaches, this deformation preserves the invariance under time reversal of the underlying classical (conservative) dynamical systems.

2012