Newtonian cosmology

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Problem 1: the first equation

Obtain the first Friedman equation basing only on Newtonian mechanics.

Problem 2: the second equation

Derive the analogue of the second Friedman equation in the Newtonian mechanics.

Problem 3: conservation law

Obtain the conservation equation for non-relativistic matter from the continuity equation for the ideal fluid.

Problem 4: no static Universe in nonrelativistic theory

Show that equations of Newtonian hydrodynamics and gravity prohibit the existence of a uniform, isotropic and static cosmological model, i.e. a Universe constant in time, uniformly filled by ideal fluid.

Problem 5: cosmic energy and the Layzer-Irvine equation

Find the generalization of the Newtonian energy conservation equation to an expanding cosmological background$^*$.

$^*$P.J.E. Peebles, Principles of Physical Cosmology, Princeton University Press, 1993.

Problem 6: the virial theorem

Using the Layzer-Irvene equation, discussed in the previous problem, recover the Newtonian virial theorem.