Variational method for geodesics - I'm stuck!

  1. 1. The problem statement, all variables and given/known data
    Hi,
    I am reading Ray d'Inverno's book, 'Introducing Einstein's Relativity' and there is a particular derivation of the geodesic equation that I get stumped on (chapter 7). It is a variational method and the final equation is

    df/dx_alpha-d/du{df/dx_alpha_dot}=0

    where f is the Lagrangian, x_alpha, beta etc are the coordinates
    and _dot denotes differenation with respect to an affine parameter.

    Now you can carry this computation and derive the equivalent equation but using christoffel symbols. But in this derivation, one line is

    f=g[beta,gamma]*dx_beta/du*dx_gamma/du

    This equation is a sum over all indices (alpha, beta etc).

    df/dx_alpha=dg[beta,gamma]/dx_xlpha*dx_beta/du*dx_gamma/du
    My question is shouldn't this last term be differentiated with the product rule, so that

    df/dx_alpha=dg[beta,gamma]/dx_xlpha*dx_beta/du*dx_gamma/du +
    g[beta,gamma]*d/dx_alpha{dx_beta/du*dx_gamma/du}

    f is a summation over all indices (ALPHA INCLUDED), so do we not need extra terms to determine these if we are differentiating with respect to alpha??
    Why can we jump to just

    df/dx_alpha=dg[beta,gamma]/dx_xlpha*dx_beta/du*dx_gamma/du

    ???

    Please help. I feel I'm missing a point here.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. gabbagabbahey

    gabbagabbahey 5,009
    Homework Helper
    Gold Member

    Hi Corals, welcome to PF.

    Your post is really hard to read. This forum supports LaTeX, so you could for example write your first equation like

    Code (Text):

     [&tex] \frac{\partial L}{\partial x^{\alpha}}- \frac{d}{du} \left( \frac{\partial L}{\partial \dot{x}^{\alpha}} \right)=0[&/tex]
     
    without the & signs to obtain:

    [tex] \frac{\partial L}{\partial x^{\alpha}} - \frac{d}{du} \left( \frac{\partial L}{\partial \dot{x}^{\alpha}} \right)=0
    [/tex]

    Anyways...I will try to decipher your post and see if I can help...
     
  4. Cheers for that. I'll change the layout tonight. Any light on my question in the meantime would be appreciated. I'm a banker by day, reminiscing my background of physics, contemplating a move back...
     
  5. gabbagabbahey

    gabbagabbahey 5,009
    Homework Helper
    Gold Member

    Is your question about this term:

    [tex]\frac{\partial}{\partial x^{\alpha}} (g_{\beta \gamma}\dot{x}^{\beta}\dot{x}^{\gamma})[/tex]

    If so, the product rule gives:

    [tex]\frac{\partial}{\partial x^{\alpha}} (g_{\beta \gamma}\dot{x}^{\beta}\dot{x}^{\gamma})=\frac{\partial g_{\beta \gamma}}{\partial x^{\alpha}} (\dot{x}^{\beta}\dot{x}^{\gamma})+\frac{\partial \dot{x}^{\beta}}{\partial x^{\alpha}} (g_{\beta \gamma}\dot{x}^{\gamma})+\frac{\partial \dot{x}^{\gamma}}{\partial x^{\alpha}} (g_{\beta \gamma}\dot{x}^{\beta})[/tex]

    But(!) [itex] \frac{\partial \dot{x}^{\beta}}{\partial x^{\alpha}}[/itex] and [itex]\frac{\partial \dot{x}^{\gamma}}{\partial x^{\alpha}}[/itex] are both zero so it simplifies to

    [tex]\frac{\partial g_{\beta \gamma}}{\partial x^{\alpha}} (\dot{x}^{\beta}\dot{x}^{\gamma})[/tex]
     
  6. Thanks for that, that helps.
    But just to explore this a bit.

    The first term for the Lagrangian is the below.

    [tex] \frac{\partial L}{\partial x^{\alpha}}- \frac{d}{du} \left( \frac{\partial L}{\partial \dot{x}^{\alpha}} \right)=0[/tex]

    [tex] (g_{\beta \gamma}\dot{x}^{\beta}\dot{x}^{\gamma})[/tex]

    I can imagine writing this out explicity as a big sum in which case there may well be instances of x^alpha in the sum, in which case they would need to be differentiated to.

    I'm explicity trying to solve a problem in Ray D'inverno's book, one in which you use the euler lagrange equation you showed me to calculate the christoffel symbols for spherical coordinates.

    There the Lagrangian is [tex]L=e^{\nu}\dot{t}^{2}[/tex]

    And now differentiating with respect to t

    gives a term [tex]\frac{\partial \dot{t}}{\partial \t}[/tex]

    i.e the derivative of dt/du with respect to t which may or may not be zero.

    (Ps this may well be an embarassing question on my part)
    ps i'm trying to latex the equations, but they're previewing as text - how can I activate them>
     
    Last edited by a moderator: Oct 14, 2008
  7. gabbagabbahey

    gabbagabbahey 5,009
    Homework Helper
    Gold Member

    Get rid of the & signs in your [&tex] and [&/tex] and make sure that you use a forward slash in the closing [&/tex].
     
  8. Thanks for that, that helps.
    But just to explore this a bit.

    The first term for the Lagrangian is the below.

    [tex]g_{\beta\gamma}\dot{x}^{\beta}\dot{x}^{\gamma}[/tex]

    I can imagine writing this out explicity as a big sum in which case there may well be instances of x^alpha in the sum, in which case they would need to be differentiated too.

    I'm explicity trying to solve a problem in Ray D'inverno's book, one in which you use the euler lagrange equation you showed me to calculate the christoffel symbols for spherical coordinates.

    There the Lagrangian is [tex]2L=e^{\nu}\dot{t}^{2}[/tex]

    And now differentiating with respect to t
    gives a term [tex]e^{\nu}\dot{t}\frac{\partial \dot{t} }{\partial t}[/tex]


    and the derivative of dt/du with respect to t which may or may not be zero, surely.

    I suspect I am being naive somewhere and making a basic error. Can you help?
     
  9. gabbagabbahey

    gabbagabbahey 5,009
    Homework Helper
    Gold Member

    The derivatives are not ordinary derivatives, but partial derivatives. This is very important, because if a function has no explicit dependence on x, then its partial derivative with respect to x is zero. For ordinary derivatives, this is not always the case; if f had an implicit dependence on x, then the ordinary derivative would not necessarily be zero.

    The quantities [itex]\dot{x}^{\beta}[/itex] are the time derivatives of the coordinates [itex]x^{\beta}[/itex] and they have no explicit dependence on the coordinates and so the partial derivatives I posted above are zero.

    The same thing applies to [itex]\frac{\partial \dot{t} }{\partial t}[/itex], If [itex]\dot{t}[/itex] has no explicit dependence on [itex]t[/itex] then [itex]\frac{\partial \dot{t} }{\partial t}=0[/itex]....Out of curiosity, what exactly do [itex]t[/itex] and [itex]\dot{t}[/itex] represent here? (surely [itex]t[/itex] isn't time because then [itex]\dot{t}=\frac{dt}{dt}=1[/itex])
     
  10. Well t is in fact time and [tex]\dot{t}[/tex] is the derivative of t with respect to u (the curve along which the integration takes place).

    Thanks for your answer - it's all clear now...
     
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