# Introduction to Mathematical Physics/Relativity/Introduction

In this chapter we focus on the ideas of symmetry and
transformations. More precisely, we study the consequences on the
physical laws of transformation invariance. The reading of the
appendices Tensors and Groups is thus recommended
for those who are not familiar with tensorial calculus and group theory.
In classical mechanics, a material point of mass is referenced by its
position and its momentum at each time . Time does not depend on the
reference frame used to evaluate position and momentum. The
Newton's law of motion is invariant under Galileean
transformations. In special and general
relativity, time depends on the considered reference frame. This yields
to modify
classical notions of position and momentum. Historicaly, special
relativity was proposed to describe the invariance of the light speed.
The group of transformations that leaves the new form of the dynamics
equations is the Lorentz group. Quantum, kinetic, and continuous
description of matter will be presented later in the
book.^{[1]}.

- ↑ In quantum mechanics, physical space considered is a functional space, and the state of a system is represented by a "wave" function of space and time. Quantity can be interpreted as the probability to have at time a particle in volume . Wave function notion can be generalized to systems more complex than those constituted by one particle. A kinetic description of system constituted by an large number of particles consists in representing the state of considered system by a function called "repartition" function, that represents the probability density to encounter a particle at position with momentum . Continuous description of matter refers to several functions of position and time to describe the state of the physical system considered.