Introduction to mathematical physics by Cossey L

By Cossey L

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We will not present here details about stability and precision calculation of the numerical integration schemes, however the reader should always keep in mind this crucial problem. Generally speaking, knowledge of mathematical properties of solutions should always be used to verify numerical results. For instance, if the solution of problem probmathevovec is known to be bounded, and that the solution obtained numerically diverges, then numerical solution will be obviously considered as bad. This problem of stability is the first that the numerician meet.

The element considered depend on the scale chosen to describe the matter. Each field of physics has its favourite N body problem: • • • • • nuclear physics deals with sets of nucleons. atomic physics deals with interactions between nucleus and nucleons. molecular chemistry, solid state physics, physics of liquids, gas, plasmas deals with interactions between atoms. classical mechanics and electrodynamics, N body problems can arise (planet movement for instance). en astrophysics one studies interactions between stars, galaxies, and clusters of galaxies.

From, the metrics, a "position vector" have to be defined. It is called fourvector position, and two formalisms are possible to define it:{four--vector}. • Either coordinates of four-vector position are taken equal to R = (x,y,z,ct) and space is equipped by pseudo scalar product defined by matrix: Then: (R | R) = RtDR where Rt represents the transposed of four-vector position R. • Then: Or coordinates of four-vector position are taken equal to R = (x,y,z,ict) and space is equipped by pseudo scalar product defined by matrix: (R | R) = RtR where Rt represents the transposed four-vector position R.

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