# Introduction to Mathematical Physics/Relativity/Exercises

Exercice:

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

Exercice:

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

Exercice:

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

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

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

${\displaystyle \Gamma _{hk}^{i}={\frac {1}{2}}g^{ij}(\partial _{h}g_{kj}+\partial _{k}g_{hj}-\partial _{j}g_{hk})}$

Exercice:

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

${\displaystyle ds^{2}=g_{ij}dq_{i}dq_{j}}$

Show that the kinetic energy of the system is:

${\displaystyle E_{c}={\frac {1}{2}}g_{ij}{\dot {q}}_{i}{\dot {q}}_{j}}$

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

${\displaystyle {\frac {D{\dot {q}}_{i}}{Dt}}=-{\frac {\partial V}{\partial q_{i}}}}$