Geodesics and a general pregeodesic equation
Let $(M,g)$ be a Riemannian manifold, and let $nabla$ denote the Levi-Civita connection. Then we say a smooth curve $gamma:Jto M, tmapstogamma(t)$ is a geodesic if
$$D_tgamma'=0.$$
We say a smooth curve $hat{gamma}:Ito M, smapstohat{gamma}(s)$ is a pregeodesic if there exists a diffeomorphism $phi:Jto I$ such that $gamma:=hat{gamma}circphi$ is a geodesic for some open interval $Jsubseteqmathbb{R}$.
Let's now turn to the local representation of the above, that is, suppose $(U,x^j)$ are coordinates on $M$ with Christoffel symbols $Gamma_{ij}^k$. Then then geodesic equation reads
$$ddot{gamma}^k+dot{gamma}^idot{gamma}^jGamma_{ij}^k=0.$$
Then a fairly straightforward application of the chain rule yields the result: A curve $hat{gamma}:Ito U$ is a pregeodesic if and only if
$$frac{d^2hat{gamma}^k}{ds^2}+frac{dhat{gamma}^i}{ds}frac{dhat{gamma}^j}{ds}Gamma_{ij}^k=f(s)frac{dhat{gamma}^k}{ds},qquad (*)$$
for some continuous $f:Itomathbb{R}$. Indeed (for the relevant direction), suppose $gamma=hat{gamma}circphi$ is a geodesic for some diffeomorphism $phi:Jto I, s=phi(t)$. Then $hat{gamma}$ satisfies the above system $(*)$ with
$$f(phi(t))=-frac{frac{d^2phi}{dt^2}}{left(frac{dphi}{dt}right)^2}.$$
This leads to my question:
I've come across in the literature that $(*)$ is equivalent to the equation
$$frac{d^2hat{gamma}^k}{ds^2}+frac{dhat{gamma}^i}{ds}frac{dhat{gamma}^j}{ds}Gamma_{ij}^k=F(gamma')frac{dhat{gamma}^k}{ds},qquad (**)$$
for some continuous $F:TUtomathbb{R}$ which is homogeneous of degree $1$ in the tangent variable.
I don't understand what this function $F$ is. Clearly, any curve $hat{gamma}$ satisfying $(**)$ is a pregeodesic. However, the result is saying that there exists some $F:TUtomathbb{R}$ with the above properties so that all pregeodesics satisfy $(**)$ with that specific $F$.
Now, the function $f(s)$ depends on the diffeomorphism $phi$, which in turn depends on the geodesic $gamma$ with starting point $(x,xi)in TU$. There is some homogeneity of geodesics when dealing with the initial condition, so this certainly seems reasonable, but I can't piece all of this together in coherent form.
I think this may actually be related to general sprays, and this $F$ is a reformulation of the Liouville vector field associated to the geodesic spray, but this is a bit outside my field (at the moment).
Any help or references would be appreciated.
differential-geometry riemannian-geometry geodesic finsler-geometry
asked Dec 6 at 10:56
Matt
51228
add a comment |
Let $(M,g)$ be a Riemannian manifold, and let $nabla$ denote the Levi-Civita connection. Then we say a smooth curve $gamma:Jto M, tmapstogamma(t)$ is a geodesic if
$$D_tgamma'=0.$$
We say a smooth curve $hat{gamma}:Ito M, smapstohat{gamma}(s)$ is a pregeodesic if there exists a diffeomorphism $phi:Jto I$ such that $gamma:=hat{gamma}circphi$ is a geodesic for some open interval $Jsubseteqmathbb{R}$.
Let's now turn to the local representation of the above, that is, suppose $(U,x^j)$ are coordinates on $M$ with Christoffel symbols $Gamma_{ij}^k$. Then then geodesic equation reads
$$ddot{gamma}^k+dot{gamma}^idot{gamma}^jGamma_{ij}^k=0.$$
Then a fairly straightforward application of the chain rule yields the result: A curve $hat{gamma}:Ito U$ is a pregeodesic if and only if
$$frac{d^2hat{gamma}^k}{ds^2}+frac{dhat{gamma}^i}{ds}frac{dhat{gamma}^j}{ds}Gamma_{ij}^k=f(s)frac{dhat{gamma}^k}{ds},qquad (*)$$
for some continuous $f:Itomathbb{R}$. Indeed (for the relevant direction), suppose $gamma=hat{gamma}circphi$ is a geodesic for some diffeomorphism $phi:Jto I, s=phi(t)$. Then $hat{gamma}$ satisfies the above system $(*)$ with
$$f(phi(t))=-frac{frac{d^2phi}{dt^2}}{left(frac{dphi}{dt}right)^2}.$$
This leads to my question:
I've come across in the literature that $(*)$ is equivalent to the equation
$$frac{d^2hat{gamma}^k}{ds^2}+frac{dhat{gamma}^i}{ds}frac{dhat{gamma}^j}{ds}Gamma_{ij}^k=F(gamma')frac{dhat{gamma}^k}{ds},qquad (**)$$
for some continuous $F:TUtomathbb{R}$ which is homogeneous of degree $1$ in the tangent variable.
I don't understand what this function $F$ is. Clearly, any curve $hat{gamma}$ satisfying $(**)$ is a pregeodesic. However, the result is saying that there exists some $F:TUtomathbb{R}$ with the above properties so that all pregeodesics satisfy $(**)$ with that specific $F$.
Now, the function $f(s)$ depends on the diffeomorphism $phi$, which in turn depends on the geodesic $gamma$ with starting point $(x,xi)in TU$. There is some homogeneity of geodesics when dealing with the initial condition, so this certainly seems reasonable, but I can't piece all of this together in coherent form.
I think this may actually be related to general sprays, and this $F$ is a reformulation of the Liouville vector field associated to the geodesic spray, but this is a bit outside my field (at the moment).
Any help or references would be appreciated.
differential-geometry riemannian-geometry geodesic finsler-geometry
asked Dec 6 at 10:56
Matt
51228
add a comment |
Let $(M,g)$ be a Riemannian manifold, and let $nabla$ denote the Levi-Civita connection. Then we say a smooth curve $gamma:Jto M, tmapstogamma(t)$ is a geodesic if
$$D_tgamma'=0.$$
We say a smooth curve $hat{gamma}:Ito M, smapstohat{gamma}(s)$ is a pregeodesic if there exists a diffeomorphism $phi:Jto I$ such that $gamma:=hat{gamma}circphi$ is a geodesic for some open interval $Jsubseteqmathbb{R}$.
Let's now turn to the local representation of the above, that is, suppose $(U,x^j)$ are coordinates on $M$ with Christoffel symbols $Gamma_{ij}^k$. Then then geodesic equation reads
$$ddot{gamma}^k+dot{gamma}^idot{gamma}^jGamma_{ij}^k=0.$$
Then a fairly straightforward application of the chain rule yields the result: A curve $hat{gamma}:Ito U$ is a pregeodesic if and only if
$$frac{d^2hat{gamma}^k}{ds^2}+frac{dhat{gamma}^i}{ds}frac{dhat{gamma}^j}{ds}Gamma_{ij}^k=f(s)frac{dhat{gamma}^k}{ds},qquad (*)$$
for some continuous $f:Itomathbb{R}$. Indeed (for the relevant direction), suppose $gamma=hat{gamma}circphi$ is a geodesic for some diffeomorphism $phi:Jto I, s=phi(t)$. Then $hat{gamma}$ satisfies the above system $(*)$ with
$$f(phi(t))=-frac{frac{d^2phi}{dt^2}}{left(frac{dphi}{dt}right)^2}.$$
This leads to my question:
I've come across in the literature that $(*)$ is equivalent to the equation
$$frac{d^2hat{gamma}^k}{ds^2}+frac{dhat{gamma}^i}{ds}frac{dhat{gamma}^j}{ds}Gamma_{ij}^k=F(gamma')frac{dhat{gamma}^k}{ds},qquad (**)$$
for some continuous $F:TUtomathbb{R}$ which is homogeneous of degree $1$ in the tangent variable.
I don't understand what this function $F$ is. Clearly, any curve $hat{gamma}$ satisfying $(**)$ is a pregeodesic. However, the result is saying that there exists some $F:TUtomathbb{R}$ with the above properties so that all pregeodesics satisfy $(**)$ with that specific $F$.
Now, the function $f(s)$ depends on the diffeomorphism $phi$, which in turn depends on the geodesic $gamma$ with starting point $(x,xi)in TU$. There is some homogeneity of geodesics when dealing with the initial condition, so this certainly seems reasonable, but I can't piece all of this together in coherent form.
I think this may actually be related to general sprays, and this $F$ is a reformulation of the Liouville vector field associated to the geodesic spray, but this is a bit outside my field (at the moment).
Any help or references would be appreciated.
differential-geometry riemannian-geometry geodesic finsler-geometry
asked Dec 6 at 10:56
Matt
51228
Let $(M,g)$ be a Riemannian manifold, and let $nabla$ denote the Levi-Civita connection. Then we say a smooth curve $gamma:Jto M, tmapstogamma(t)$ is a geodesic if
$$D_tgamma'=0.$$
We say a smooth curve $hat{gamma}:Ito M, smapstohat{gamma}(s)$ is a pregeodesic if there exists a diffeomorphism $phi:Jto I$ such that $gamma:=hat{gamma}circphi$ is a geodesic for some open interval $Jsubseteqmathbb{R}$.
Let's now turn to the local representation of the above, that is, suppose $(U,x^j)$ are coordinates on $M$ with Christoffel symbols $Gamma_{ij}^k$. Then then geodesic equation reads
$$ddot{gamma}^k+dot{gamma}^idot{gamma}^jGamma_{ij}^k=0.$$
Then a fairly straightforward application of the chain rule yields the result: A curve $hat{gamma}:Ito U$ is a pregeodesic if and only if
$$frac{d^2hat{gamma}^k}{ds^2}+frac{dhat{gamma}^i}{ds}frac{dhat{gamma}^j}{ds}Gamma_{ij}^k=f(s)frac{dhat{gamma}^k}{ds},qquad (*)$$
for some continuous $f:Itomathbb{R}$. Indeed (for the relevant direction), suppose $gamma=hat{gamma}circphi$ is a geodesic for some diffeomorphism $phi:Jto I, s=phi(t)$. Then $hat{gamma}$ satisfies the above system $(*)$ with
$$f(phi(t))=-frac{frac{d^2phi}{dt^2}}{left(frac{dphi}{dt}right)^2}.$$
This leads to my question:
I've come across in the literature that $(*)$ is equivalent to the equation
$$frac{d^2hat{gamma}^k}{ds^2}+frac{dhat{gamma}^i}{ds}frac{dhat{gamma}^j}{ds}Gamma_{ij}^k=F(gamma')frac{dhat{gamma}^k}{ds},qquad (**)$$
for some continuous $F:TUtomathbb{R}$ which is homogeneous of degree $1$ in the tangent variable.
I don't understand what this function $F$ is. Clearly, any curve $hat{gamma}$ satisfying $(**)$ is a pregeodesic. However, the result is saying that there exists some $F:TUtomathbb{R}$ with the above properties so that all pregeodesics satisfy $(**)$ with that specific $F$.
Now, the function $f(s)$ depends on the diffeomorphism $phi$, which in turn depends on the geodesic $gamma$ with starting point $(x,xi)in TU$. There is some homogeneity of geodesics when dealing with the initial condition, so this certainly seems reasonable, but I can't piece all of this together in coherent form.
I think this may actually be related to general sprays, and this $F$ is a reformulation of the Liouville vector field associated to the geodesic spray, but this is a bit outside my field (at the moment).
Any help or references would be appreciated.
differential-geometry riemannian-geometry geodesic finsler-geometry
differential-geometry riemannian-geometry geodesic finsler-geometry
asked Dec 6 at 10:56
Matt
51228
asked Dec 6 at 10:56
Matt
51228
asked Dec 6 at 10:56
Matt
51228
asked Dec 6 at 10:56
Matt
51228
asked Dec 6 at 10:56
Matt
51228
51228
add a comment |
add a comment |
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