Example of geodesics as the critical point of the energy functional!
$begingroup$
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
asked Jan 3 at 12:59
MajidMajid
1,8731926
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add a comment |
$begingroup$
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
asked Jan 3 at 12:59
MajidMajid
1,8731926
$endgroup$
$begingroup$
Can you be more explicit about what you want to know?
$endgroup$
– Arctic Char
Jan 3 at 15:13
$begingroup$
@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.
$endgroup$
– Majid
Jan 3 at 15:45
add a comment |
$begingroup$
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
asked Jan 3 at 12:59
MajidMajid
1,8731926
$endgroup$
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
asked Jan 3 at 12:59
MajidMajid
1,8731926
asked Jan 3 at 12:59
MajidMajid
1,8731926
asked Jan 3 at 12:59
MajidMajid
1,8731926
asked Jan 3 at 12:59
MajidMajid
1,8731926
asked Jan 3 at 12:59
MajidMajid
1,8731926
1,8731926
$begingroup$
Can you be more explicit about what you want to know?
$endgroup$
– Arctic Char
Jan 3 at 15:13
$begingroup$
@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.
$endgroup$
– Majid
Jan 3 at 15:45
add a comment |
$begingroup$
Can you be more explicit about what you want to know?
$endgroup$
– Arctic Char
Jan 3 at 15:13
$begingroup$
@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.
$endgroup$
– Majid
Jan 3 at 15:45
$begingroup$
Can you be more explicit about what you want to know?
$endgroup$
– Arctic Char
Jan 3 at 15:13
$begingroup$
Can you be more explicit about what you want to know?
$endgroup$
– Arctic Char
Jan 3 at 15:13
$begingroup$
@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.
$endgroup$
– Majid
Jan 3 at 15:45
$begingroup$
@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.
$endgroup$
– Majid
Jan 3 at 15:45
add a comment |
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..
answered Jan 4 at 17:55
ThomasThomas
4,102510
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add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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..
answered Jan 4 at 17:55
ThomasThomas
4,102510
$endgroup$
add a comment |
$begingroup$
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..
answered Jan 4 at 17:55
ThomasThomas
4,102510
$endgroup$
add a comment |
$begingroup$
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..
answered Jan 4 at 17:55
ThomasThomas
4,102510
$endgroup$
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..
answered Jan 4 at 17:55
ThomasThomas
4,102510
answered Jan 4 at 17:55
ThomasThomas
4,102510
answered Jan 4 at 17:55
ThomasThomas
4,102510
answered Jan 4 at 17:55
ThomasThomas
4,102510
4,102510
add a comment |
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$begingroup$
Can you be more explicit about what you want to know?
$endgroup$
– Arctic Char
Jan 3 at 15:13
$begingroup$
@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.
$endgroup$
– Majid
Jan 3 at 15:45