It is frequently said that geodesics are "straight lines in curved space".
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.
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 Riemannian 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.
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 that maps an interval I of the real number line to the manifold M, one writes the energy where
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.
are the Christoffel symbols, and repeated indices imply the use of the summation convention.
The Hamiltonian is constructed from the metric on the manifold, and is thus a quadratic form consisting entirely of the kinetic term.
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 flow determined by these equations is called the cogeodesic flow; a simple substitution of one into the other obtains the Euler–Lagrange equations, which give the geodesic flow on the tangent bundle TM.