# Introduction to Mathematical Physics/Relativity/Exercises

**Exercice:**

Show that a rocket of mass that ejects at speed (with respect to itself) a part of its mass by time unit moves in the sense opposed to . Give the movement law for a rocket with speed zero at time , located in a earth gravitational field considered as constant.

**Exercice:**

*Doppler effect.* Consider a light source moving at constant speed with respect to reference frame . Using wave four-vector give the relation between frequencies measured by an experimentator moving with and another experimentater attached to . What about sound waves?

**Exercice:**

For a cylindrical coordinates system, metrics of the space is:

Calculate the Christoffel symbols defined by:

**Exercice:**

Consider a unit mass in a three dimensional reference frame whose metrics is:

Show that the kinetic energy of the system is:

Show that the fundamental equation of dynamics is written here (forces are assumed to derive from a potential ) :