- Geodesics as Hamiltonian flows
In

mathematics , thegeodesic equation s are second-order non-lineardifferential equation s, and are commonly presented in the form ofEuler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equations, in the form ofHamilton's equations . This later formulation is developed in this article.**Overview**It is frequently said that

geodesics are "straight lines in curved space". By using the Hamilton-Jacobi approach to thegeodesic equation , this statement can be given a very intuitive meaning: geodesics describe the motions of particles that are not experiencing any forces. In flat space, it is well known that a particle moving in a straight line will continue to move in a straight line if it experiences no external forces; this isNewton's first law . The Hamiltonan describing such motion is well known to be $H=mv^2/2=p^2/2m$ with "p" being themomentum . It is theconservation of momentum that leads to the straight motion of a particle. On a curved surface, exactly the same ideas are at play, except that, in order to measure distances correctly, one must use the metric. To measure momenta correctly, one must use the inverse of the metric. The motion of a free particle on a curved surface still has exactly the same form as above, i.e. consisting entirely of akinetic term . The resulting motion is still, in a sense, a "straight line", which is why it is sometimes said that geodesics are "straight lines in curved space". This idea is developed in greater detail below.**Geodesics as an application of the principle of least action**Given a (pseudo-)

Riemannian manifold "M", ageodesic may be defined as the curve that results from the application of theprinciple of least action . A differential equation describing their shape may be derived, usingvariational principle s, by minimizing (or finding the extremum) of theenergy of a curve. Given asmooth curve :$gamma:I\; o\; M$that maps an interval "I" of the

real number line to the manifold "M", one writes the energy:$E(gamma)=frac\{1\}\{2\}int\_I\; g(dotgamma(t),dotgamma(t)),dt,$

where $dotgamma(t)$ is the

tangent vector to the curve $gamma$ at point $t\; in\; I$.Here, $g(cdot,cdot)$ is themetric tensor on the manifold "M". Using the energy given above as the action, one may choose to solve either theEuler–Lagrange equations , or the Hamilton-Jacobi equations. Both methods give thegeodesic equation as the solution; however, the Hamilton–Jacobi equations provide greater insight into the structure of the manifold, as shown below. In terms of thelocal coordinates on "M", the (Euler–Lagrange) geodesic equation is:$frac\{d^2x^a\}\{dt^2\}\; +\; Gamma^\{a\}\; \{\}\_\{bc\}frac\{dx^b\}\{dt\}frac\{dx^c\}\{dt\}\; =\; 0$

Here, the "x"

^{"a"}("t") are the coordinates of the curve γ("t") and $Gamma^\{a\}\; \{\}\_\{bc\}$ are theChristoffel symbol s. Repeated indices imply the use of thesummation convention .**Hamiltonian approach to the geodesic equations**Geodesics can be understood to be the

Hamiltonian flow s of a specialHamiltonian vector field defined on thecotangent space of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus aquadratic form consisting entirely of thekinetic term .The geodesic equations are second-order differential equations; they can be re-expressed as first-order ordinary differential equations taking the form of the Hamiltonian–Jacobi equations by introducing additional independent variables, as shown below. Start by finding a chart that trivializes the

cotangent bundle "T"^{∗}"M" ("i.e." a "local trivialization ")::$T^*M|\_\{U\}simeq\; U\; imes\; mathbb\{R\}^n$

where "U" is an open

subset of the manifold "M", and the tangent space is of rank "n". Label the coordinates of the chart as ("x"^{1}, "x"^{2}, …, "x"^{"n"}, "p"_{1}, "p"_{2}, …, "p"_{"n"}). Then introduce the Hamiltonian as:$H(x,p)=frac\{1\}\{2\}g^\{ab\}(x)p\_a\; p\_b.$

Here, "g"

^{"ab"}("x") is the inverse of themetric tensor : "g"^{"ab"}("x")"g"_{"bc"}("x") = $delta^a\_c$. This inverse almost always exists for a broad class of metric manifolds. The behavior of the metric tensor under coordinate transformations implies that "H" is invariant under a change of variable. The geodesic equations can then be written as:$dot\{x\}^a\; =\; frac\{partial\; H\}\{partial\; p\_a\}\; =\; g^\{ab\}(x)\; p\_b$

and

:$dot\{p\}\_a\; =\; -\; frac\; \{partial\; H\}\{partial\; x^a\}\; =\; -frac\{1\}\{2\}\; frac\; \{partial\; g^\{bc\}(x)\}\{partial\; x^a\}\; p\_b\; p\_c.$

The second order geodesic equations are easily obtained by substitution of one into the other. The flow determined by these equations is called the

**cogeodesic flow**. The first of the two equations gives the flow on the tangent bundle "TM", the**geodesic flow**. Thus, the geodesic lines are the integral curves of the geodesic flow onto the manifold "M". This is aHamiltonian flow , and that the Hamiltonian is constant along the geodesics::$frac\{dH\}\{dt\}\; =\; frac\; \{partial\; H\}\{partial\; x^a\}\; dot\{x\}^a\; +frac\{partial\; H\}\{partial\; p\_a\}\; dot\{p\}\_a\; =\; -\; dot\{p\}\_a\; dot\{x\}^a\; +\; dot\{x\}^a\; dot\{p\}\_a\; =\; 0.$

Thus, the geodesic flow splits the cotangent bundle into

level set s of constant energy:$M\_E\; =\; \{\; (x,p)\; in\; T^*M\; :\; H(x,p)=E\; \}$

for each energy "E" ≥ 0, so that

:$T^*M=igcup\_\{E\; ge\; 0\}\; M\_E$.

The

Hopf-Rinow theorem guarantees the completeness of the manifold. The positivity of the energy follows from the positivity of the metric tensor; this analysis is modified on pseudo-Riemannian manifolds.**References*** Ralph Abraham and Jarrold E. Marsden, "Foundations of Mechanics", (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X "See section 2.7".

* Jurgen Jost, "Riemannian Geometry and Geometric Analysis", (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 "See section 1.4".

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