The God has created a man in order that he creates that the God fails to do

Sunday, 19 October 2014

Foundations of Modern Physics 7: Non-relativistic time-dependent mechanics

The technique of symplectic manifolds is well known to provide the adequate Hamiltonian formulation of autonomous mechanics. Its realistic example is a mechanical system whose configuration space is a manifold M and whose phase space is the cotangent bundle T*M of M provided with the canonical symplectic form on T*M. Any autonomous Hamiltonian system locally is of this type.

However, this geometric formulation of autonomous mechanics is not extended to mechanics under time-dependent transformations because the symplectic form Om fails to be invariant under these transformations. As a palliative variant, one has developed time-dependent mechanics on a configuration space Q=RxM where R is the time axis. Its phase space RxT*M is provided with the pull-back presymplectic form. However, this presymplectic form also is broken by time-dependent transformations.

We address non-relativistic mechanics in a case of arbitrary time-dependent transformations. Its configuration space is a fibre bundle Q->R. Its velocity space is the first order jet manifold of sections of Q->R. A phase space is the vertical cotangent bundle V*Q of Q->R.

This formulation of non-relativistic mechanics is similar to that of classical field theory on fibre bundles over a base of dimension >1 (#). A difference between mechanics and field theory however lies in the fact that connections on bundles over R are flat, and they fail to be dynamic variables, but describe reference frames.


Advanced mechanics. Mathematical introductionarXiv: 0911.0411

Thursday, 2 October 2014

Who is who among universities in 2014-15 by THE World University Rankings

New world ranking of universities "Times Higher Education World University Rankings 2014-2015" (#) has been published. It contains 400 universities.

The top ten positions are occupied by 7 universities of USA and the 3 ones of United Kingdom.

In the top twenty: 15 - USA, 3 - United Kingdom, 1 – Switzerland and Canada.

In the first 50 Universities: 28 – USA; 7 - United Kingdom; 3 – Canada; 2 – Switzerland, Australia and China; 1 - Hong-Kong, Japan, Sweden, Germany and Korea.