Abstract
Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.
Highlights
Several classic one-dimensional problems of mechanics solved with variational calculus have analogues in spatially homogeneous and isotropic cosmology
One can build in the laboratory many physical systems that are analogous to curved spacetimes describing black holes or universes, using which one can study curved space quantum effects such as Hawking radiation, particle creation, or superradiance
These systems include Bose-Einstein condensates and other condensed matter systems [1,2,3,4,5,6,7,8,9,10,11,12,13,14], fluid-dynamical systems [15,16,17,18,19,20,21,22,23,24,25,26,27], and optical systems [28,29,30,31,32] and they have originated the field of research known as analogue gravity (e.g., References [33,34,35,36,37]), part of which focuses on analogues of FLRW cosmology in Bose-Einstein condensates [2,3,4,5,6,7,10,34,38,39,40,41]
Summary
Several classic one-dimensional problems of mechanics solved with variational calculus have analogues in spatially homogeneous and isotropic (or Friedmann-Lemaître-Robertson-Walker, hereafter “FLRW“) cosmology. It may seem that a complete analogy between the Einstein-Friedmann equations and the physical systems that we will consider is not complete because in the latter the dynamics is described by a single differential equation (analogous to the Friedmann equation), while the universe is described by two equations of the set (4)–(6) This is not the case because the information contained in the second equation (say, the covariant conservation Equation (6)) has already been inserted into the first one (the Friedmann equation (4) by using the scaling (7) of the perfect fluid energy density and of the curvature term with the scale factor a on the right hand side of Equation (4), or a similar functional dependence ρ( a) that characterizes the perfect fluid (for example a non-linear equation of state). In the following we review the most celebrated textbook problems of variational calculus in one dimension, building cosmological analogies where possible
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