# Introduction to Mathematical Physics/Relativity/Exercises

Exercice:

Show that a rocket of mass $m$ that ejects at speed $u$ (with respect to itself) a part of its mass $dm$ by time unit $dt$ moves in the sense opposed to $u$. Give the movement law for a rocket with speed zero at time $t=0$, located in a earth gravitational field considered as constant.

Exercice:

Doppler effect. Consider a light source $S$ moving at constant speed with respect to reference frame $R$. Using wave four-vector $(k,\omega/c)$ give the relation between frequencies measured by an experimentator moving with $S$ and another experimentater attached to $R$. What about sound waves?

Exercice:

For a cylindrical coordinates system, metrics $g_{ij}$ of the space is:

$ds^2=dr^2+r^2d\theta^2+dz^2$

Calculate the Christoffel symbols $\Gamma^{i}_{hk}$ defined by:

$\Gamma^{i}_{hk}=\frac{1}{2}g^{ij} (\partial_hg_{kj}+ \partial_kg_{hj}- \partial_jg_{hk})$

Exercice:

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

$ds^2=g_{ij}dq_idq_j$

Show that the kinetic energy of the system is:

$E_c=\frac{1}{2}g_{ij}\dot q_i\dot q_j$

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

$\frac{D\dot q_i}{Dt}=-\frac{\partial V}{\partial q_i}$