Example of geodesics as the critical point of the energy functional!












1














I know that the critical points of the energy functional



$$E(gamma) = frac{1}{2}int_a^b |gamma'(t)|^2, dt$$



are geodesics. As finding geodesics sometimes involves us with some complicated system of equations, this trick is useful to find the critical points of $E$.



Can someone give some example to see how it works in the Riemannian case?






differential-geometry riemannian-geometry







share|cite|improve this question






















  • Can you be more explicit about what you want to know?
    – Arctic Char
    yesterday










  • @ArcticChar I want to see some cases in which using energy the geodesics are calculated. Actually, I am interested in the cases that the system equations of the geodesics is so complicated and one has to use some techniques to find the geodesics.
    – Majid
    yesterday
















1














I know that the critical points of the energy functional



$$E(gamma) = frac{1}{2}int_a^b |gamma'(t)|^2, dt$$



are geodesics. As finding geodesics sometimes involves us with some complicated system of equations, this trick is useful to find the critical points of $E$.



Can someone give some example to see how it works in the Riemannian case?






differential-geometry riemannian-geometry







share|cite|improve this question






















  • Can you be more explicit about what you want to know?
    – Arctic Char
    yesterday










  • @ArcticChar I want to see some cases in which using energy the geodesics are calculated. Actually, I am interested in the cases that the system equations of the geodesics is so complicated and one has to use some techniques to find the geodesics.
    – Majid
    yesterday














1












1








1


2





I know that the critical points of the energy functional



$$E(gamma) = frac{1}{2}int_a^b |gamma'(t)|^2, dt$$



are geodesics. As finding geodesics sometimes involves us with some complicated system of equations, this trick is useful to find the critical points of $E$.



Can someone give some example to see how it works in the Riemannian case?






differential-geometry riemannian-geometry







share|cite|improve this question













I know that the critical points of the energy functional



$$E(gamma) = frac{1}{2}int_a^b |gamma'(t)|^2, dt$$



are geodesics. As finding geodesics sometimes involves us with some complicated system of equations, this trick is useful to find the critical points of $E$.



Can someone give some example to see how it works in the Riemannian case?






differential-geometry riemannian-geometry




differential-geometry riemannian-geometry






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked yesterday









Majid

1,8691925




1,8691925












  • Can you be more explicit about what you want to know?
    – Arctic Char
    yesterday










  • @ArcticChar I want to see some cases in which using energy the geodesics are calculated. Actually, I am interested in the cases that the system equations of the geodesics is so complicated and one has to use some techniques to find the geodesics.
    – Majid
    yesterday


















  • Can you be more explicit about what you want to know?
    – Arctic Char
    yesterday










  • @ArcticChar I want to see some cases in which using energy the geodesics are calculated. Actually, I am interested in the cases that the system equations of the geodesics is so complicated and one has to use some techniques to find the geodesics.
    – Majid
    yesterday
















Can you be more explicit about what you want to know?
– Arctic Char
yesterday




Can you be more explicit about what you want to know?
– Arctic Char
yesterday












@ArcticChar I want to see some cases in which using energy the geodesics are calculated. Actually, I am interested in the cases that the system equations of the geodesics is so complicated and one has to use some techniques to find the geodesics.
– Majid
yesterday




@ArcticChar I want to see some cases in which using energy the geodesics are calculated. Actually, I am interested in the cases that the system equations of the geodesics is so complicated and one has to use some techniques to find the geodesics.
– Majid
yesterday










1 Answer
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A typical example of this formula is the case whre you have Gauss coordinates, namely cordinates $x_1,x_2,..., x_n$ with $g= dx_1^2 + sum _{i,jgeq 2}f_{i,j} dx_i.dx_j =dx_1^2+ g_1$. The tensor $g_1$ is positive, and stricly positive in any direction but the vertical direction.
This prove that the vertical lines $x_2(t)=c_2,..,x_n(t)=c_n$ are geodesic indeed, for every curve $gamma(t)=(x_1(t),...,x_n(t))$ between $(c_1,c_2,..c_n)$ and $(c'_1,c_2,..c_n)$ the energy is greater that $int _ {t_1} ^ {t_2} {x'}_1^2(t) dt$, so the vertical line minimize the arc length..






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    A typical example of this formula is the case whre you have Gauss coordinates, namely cordinates $x_1,x_2,..., x_n$ with $g= dx_1^2 + sum _{i,jgeq 2}f_{i,j} dx_i.dx_j =dx_1^2+ g_1$. The tensor $g_1$ is positive, and stricly positive in any direction but the vertical direction.
    This prove that the vertical lines $x_2(t)=c_2,..,x_n(t)=c_n$ are geodesic indeed, for every curve $gamma(t)=(x_1(t),...,x_n(t))$ between $(c_1,c_2,..c_n)$ and $(c'_1,c_2,..c_n)$ the energy is greater that $int _ {t_1} ^ {t_2} {x'}_1^2(t) dt$, so the vertical line minimize the arc length..






    share|cite|improve this answer


























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      A typical example of this formula is the case whre you have Gauss coordinates, namely cordinates $x_1,x_2,..., x_n$ with $g= dx_1^2 + sum _{i,jgeq 2}f_{i,j} dx_i.dx_j =dx_1^2+ g_1$. The tensor $g_1$ is positive, and stricly positive in any direction but the vertical direction.
      This prove that the vertical lines $x_2(t)=c_2,..,x_n(t)=c_n$ are geodesic indeed, for every curve $gamma(t)=(x_1(t),...,x_n(t))$ between $(c_1,c_2,..c_n)$ and $(c'_1,c_2,..c_n)$ the energy is greater that $int _ {t_1} ^ {t_2} {x'}_1^2(t) dt$, so the vertical line minimize the arc length..






      share|cite|improve this answer
























        0












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        0






        A typical example of this formula is the case whre you have Gauss coordinates, namely cordinates $x_1,x_2,..., x_n$ with $g= dx_1^2 + sum _{i,jgeq 2}f_{i,j} dx_i.dx_j =dx_1^2+ g_1$. The tensor $g_1$ is positive, and stricly positive in any direction but the vertical direction.
        This prove that the vertical lines $x_2(t)=c_2,..,x_n(t)=c_n$ are geodesic indeed, for every curve $gamma(t)=(x_1(t),...,x_n(t))$ between $(c_1,c_2,..c_n)$ and $(c'_1,c_2,..c_n)$ the energy is greater that $int _ {t_1} ^ {t_2} {x'}_1^2(t) dt$, so the vertical line minimize the arc length..






        share|cite|improve this answer












        A typical example of this formula is the case whre you have Gauss coordinates, namely cordinates $x_1,x_2,..., x_n$ with $g= dx_1^2 + sum _{i,jgeq 2}f_{i,j} dx_i.dx_j =dx_1^2+ g_1$. The tensor $g_1$ is positive, and stricly positive in any direction but the vertical direction.
        This prove that the vertical lines $x_2(t)=c_2,..,x_n(t)=c_n$ are geodesic indeed, for every curve $gamma(t)=(x_1(t),...,x_n(t))$ between $(c_1,c_2,..c_n)$ and $(c'_1,c_2,..c_n)$ the energy is greater that $int _ {t_1} ^ {t_2} {x'}_1^2(t) dt$, so the vertical line minimize the arc length..







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 8 hours ago









        Thomas

        3,864510




        3,864510






























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