Geodesics As Hamiltonian Flows - Geodesics As An Application of The Principle of Least Action

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

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.

Read more about this topic:  Geodesics As Hamiltonian Flows

Famous quotes containing the words application, principle and/or action:

    The application requisite to the duties of the office I hold [governor of Virginia] is so excessive, and the execution of them after all so imperfect, that I have determined to retire from it at the close of the present campaign.
    Thomas Jefferson (1743–1826)

    For me chemistry represented an indefinite cloud of future potentialities which enveloped my life to come in black volutes torn by fiery flashes, like those which had hidden Mount Sinai. Like Moses, from that cloud I expected my law, the principle of order in me, around me, and in the world.... I would watch the buds swell in spring, the mica glint in the granite, my own hands, and I would say to myself: “I will understand this, too, I will understand everything.”
    Primo Levi (1919–1987)

    Power is action; the electoral principle is discussion. No political action is possible when discussion is permanently established.
    Honoré De Balzac (1799–1850)