Htykuut

I'm trying to find the Christoffel Symbols for the Lorentz metric $${rm d}s^2 = cos(2pi x)({rm d}x^2-{rm d}y^2) - 2sin(2pi x),{rm d}x,{rm d}y$$by looking at the Euler-Lagrange equations for $$L(x,dot{x},y,dot{y}) = cos(2pi x)(dot{x}^2-dot{y}^2) - 2sin(2pi x),dot{x},dot{y}.$$I have already done my fair share of computations like this, but I must be making some algebraic mistake that I cannot find for the life of me. If we write $$begin{align}frac{partial L}{partial x} - frac{{rm d}}{{rm d}t}&left(frac{partial L}{partial dot{x}}right) = -2pisin(2pi x)(dot{x}^2-dot{y}^2)-4picos(2pi x)dot{x}dot{y} \ &qquad - frac{{rm d}}{{rm d}t}left(2dot{x}cos(2pi x) - 2dot{y}sin(2pi x)right),end{align} $$and we will have a term with $ddot{y}$. This is a problem, since as far as I understand the geodesic equation corresponding to the $x$-coordinate should have the form $$ddot{x} + Gamma(dot{x},dot{y})=0,$$maybe after dividing by something. What is going on?

1. 
   

   The infinitesimal variation of the Lagrangian
   $$ L(x,dot{x})~=~ g_{ij}(x)~ dot{x}^idot{x}^j tag{1} $$
   is
   $$ frac{1}{2}delta L~=~ -left{ color{blue}{ g_{kell}ddot{x}^{ell}} +color{red}{Gamma_{k,ij} dot{x}^idot{x}^j}right}delta x^k
   +frac{mathrm{d}}{mathrm{d}t}left{ color{green}{ g_{kell}dot{x}^{ell} delta x^k}right},tag{2} $$
   where we have introduced the lowered Levi-Civita Christoffel symbols
   $$Gamma_{k,ij}~:=~g_{kell}Gamma^{ell}_{ij}. tag{3} $$
   Note that eq. (2) contains three different types of terms (displayed in different colors), which are uniquely characterized by how the $t$-derivatives are distributed.

   
   
   

   In particular we see that the geodesic equations are multiplied with the metric, cf. OP's last question.

   
   


2. 
   

   Example. OP's Lagrangian reads
   $$L~=~c(dot{x}^2-dot{y}^2) - 2sdot{x}dot{y}, qquad c~:=~cos(2pi x), qquad s~:=~sin(2pi x) ,tag{4}$$
   corresponding to the metric
   $$ begin{pmatrix} g_{xx} & g_{xy} cr g_{yx} & g_{yy} end{pmatrix} ~=~begin{pmatrix} c & -s cr -s & -c end{pmatrix}. tag{5}$$
   We calculate the infinitesimal variation:
   $$begin{align}frac{1}{2} delta L
   ~=~&left{ color{blue}{sddot{y}-cddot{x}}
   + color{red}{pi s (dot{x}^2+dot{y}^2)} right} delta x
   + left{ color{blue}{sddot{x}+cddot{y}}
   + color{red}{2pi (cdot{x}^2-sdot{x}dot{y})} right} delta y cr
   & + frac{mathrm{d}}{mathrm{d}t}left{ color{green}{(cdot{x}-sdot{y})delta x -(sdot{x}+cdot{y})delta y}right},
   end{align}tag{6}$$
   which should be compared with the general formula (2).

   
   
   

   From the red terms in eq. (6) we can read off the non-zero lowered Christoffel symbols
   $$ Gamma_{x,xx}~=~-pi s ~=~Gamma_{x,yy}~=~-Gamma_{y,xy}, qquad Gamma_{y,xx}~=~-2pi c,tag{7} $$
   cf. OP's title.