Geodesics as Hamiltonian flows
Encyclopedia
In mathematics
, the geodesic equations are second-order non-linear differential equation
s, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equations, in the form of Hamilton's equations. This latter formulation is developed in this article.
with p being the momentum
. It is the conservation 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 a kinetic 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.
-)Riemannian manifold
M, a geodesic
may be defined as the curve that results from the application of the principle of least action
. A differential equation describing their shape may be derived, using variational principle
s, by minimizing (or finding the extremum) of the energy
of a curve. Given a smooth
curve
that maps an interval I of the real number line to the manifold M, one writes the energy
where
is the tangent vector
to the curve
at point 
.
Here,
is the metric tensor
on the manifold M.
Using the energy given above as the action, one may choose to solve either the Euler–Lagrange equations, or the Hamilton-Jacobi equations. Both methods give the geodesic equation as the solution; however, the Hamilton–Jacobi equations provide greater insight into the structure of the manifold, as shown below. In terms of the local coordinates
on M, the (Euler–Lagrange) geodesic equation is
Here, the xa(t) are the coordinates of the curve γ(t) and
are the Christoffel symbols. Repeated indices imply the use of the summation convention.
defined on the cotangent space
of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form
consisting entirely of the kinetic 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):
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 (x1, x2, …, xn, p1, p2, …, pn). Then introduce the Hamiltonian
as
Here, gab(x) is the inverse of the metric tensor
: gab(x)gbc(x) =
. 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
and
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 projections of integral curves of the geodesic flow onto the manifold M. This is a Hamiltonian flow, and that the Hamiltonian is constant along the geodesics:
Thus, the geodesic flow splits the cotangent bundle into level set
s of constant energy
for each energy E ≥ 0, so that
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the geodesic equations are second-order non-linear differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be presented as a set of coupled first-order equations, in the form of Hamilton's equations. This latter 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 the geodesic 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 is Newton's first law. The Hamiltonan describing such motion is well known to be with p being the momentumMomentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...
. It is the conservation 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
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a 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 a kinetic term
Kinetic term
In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields , and usually contains two derivatives with respect to time ; in the case of fermions, the kinetic term usually has one derivative only...
. 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 (pseudoPseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
-)Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
M, a geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
may be defined as the curve that results from the application of the principle of least action
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
. A differential equation describing their shape may be derived, using variational principle
Variational principle
A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...
s, by minimizing (or finding the extremum) of the energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
of a curve. Given a smooth
Smooth
Smooth means having a texture that lacks friction. Not rough.Smooth may also refer to:-In mathematics:* Smooth function, a function that is infinitely differentiable; used in calculus and topology...
curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
that maps an interval I of the real number line to the manifold M, one writes the energy
where
is the tangent vectorTangent vector
A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
to the curve
at point .Here,
is the metric tensorMetric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
on the manifold M.
Using the energy given above as the action, one may choose to solve either the Euler–Lagrange equations, or the Hamilton-Jacobi equations. Both methods give the geodesic equation as the solution; however, the Hamilton–Jacobi equations provide greater insight into the structure of the manifold, as shown below. In terms of the local coordinates
Local coordinates
Local coordinates are measurement indices into a local coordinate system or a local coordinate space. A simple example is using house numbers to locate a house on a street; the street is a local coordinate system within a larger system composed of city townships, states, countries, etc.Local...
on M, the (Euler–Lagrange) geodesic equation is
Here, the xa(t) are the coordinates of the curve γ(t) and
are the Christoffel symbols. Repeated indices imply the use of the summation convention.Hamiltonian approach to the geodesic equations
Geodesics can be understood to be the Hamiltonian flows of a special Hamiltonian vector fieldHamiltonian vector field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations...
defined on the cotangent space
Cotangent space
In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
of the manifold. The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
consisting entirely of the kinetic term
Kinetic term
In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields , and usually contains two derivatives with respect to time ; in the case of fermions, the kinetic term usually has one derivative only...
.
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
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
T∗M (i.e. a local trivialization):
where U is an open subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of the manifold M, and the tangent space is of rank n. Label the coordinates of the chart as (x1, x2, …, xn, p1, p2, …, pn). Then introduce the Hamiltonian
Hamiltonian vector field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations...
as
Here, gab(x) is the inverse of the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
: gab(x)gbc(x) =
. The behavior of the metric tensor under coordinate transformations implies that H is invariantInvariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
under a change of variable. The geodesic equations can then be written as
and
The second order geodesic equations are easily obtained by substitution of one into the other. The flow
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over...
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 projections of integral curves of the geodesic flow onto the manifold M. This is a Hamiltonian flow, and that the Hamiltonian is constant along the geodesics:
Thus, the geodesic flow splits the cotangent bundle into level set
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
s of constant energy
for each energy E ≥ 0, so that
.