In mathematics, the geodesic equations are second-order non-linear differential equations, 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 later formulation is developed in this article. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... An illustration of a differential equation. ... In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ...

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 H = mv² / 2 = p² / 2m 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. In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... Newtons laws of motion are three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In classical mechanics, momentum (pl. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... In mathematics a metric or distance function is a function which defines a distance between elements of a set. ... In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields (this does not include the mass term!) (and for nonlinear sigma models, they are not even bilinear), and usually contains two derivatives with respect to time (or space); in the case of fermions...

Geodesics as an application of the principle of least action

Given a (pseudo-)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 principles, by minimizing (or finding the extremum) of the energy of a curve. Given a smooth curve In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ... A variational principle is a principle in physics which is expressed in terms of the calculus of variations. ... Smooth could mean many things, including: Smooth function, a function that is infinitely differentiable, used in calculus and topology. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ...

that maps an interval I of the real number line to the manifold M, one writes the energy In mathematics, the real line is simply the set of real numbers. ...

where is the tangent vector to the curve γ at point . Here, is the metric tensor on the manifold M. In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

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 In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. ... Local coordinates are measurement indices into a local coordinate system or a local coordinate space. ...

Here, the x^a(t) are the coordinates of the curve γ(t) and Γ^a_bc are the Christoffel symbols. Repeated indecies imply the use of the summation convention. In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

Hamiltonian approach to the geodesic equations

Geodesics can be understood to be the Hamiltonian flows of a special Hamiltonian vector field 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. In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. ... In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. ... In differential geometry, one can attach to every point p of a differentiable manifold a vector space called the cotangent space at p. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In physics, a kinetic term is the part of the Lagrangian that is bilinear in the fields (this does not include the mass term!) (and for nonlinear sigma models, they are not even bilinear), and usually contains two derivatives with respect to time (or space); in the case of fermions...

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): In topology, an atlas describes how a complicated space is glued together from simpler pieces. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In algebraic topology, a fibration is a continuous mapping Y → X satisfying the homotopy lifting property. ...

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, …, p_n). Then introduce the Hamiltonian as A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. ...

Here, g^ab(x) is the inverse of the metric tensor: g^ab(x)g_bc(x) = . 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 In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... In mathematics, an invariant is something that does not change under a set of transformations. ...

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 integral curves of the geodesic flow onto the manifold M. This is a Hamiltonian flow, and that the Hamiltonian is constant along the geodesics: In mathematics, flow refers to the group action of a one-parameter group on a set. ... In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds. ...

Thus, the geodesic flow splits the cotangent bundle into level sets of constant energy In mathematics, a level set of a real-valued function f of n variables is a set of the form { (x1,...,xn) | f(x1,...,xn) = c } where c is a constant. ...

for each energy E ≥ 0, so that

.

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. If two objects are at a distance one mile from each other, it should be possible to construct a road of length one mile between them. ...

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.