Variational Formulation of Distance
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Variational formulation of distance[edit
The Euclidean distance between two points in space ( minimum value of an integral:
source | editbeta]
and ) may be written in a variational form where the distance is the
Here
is the trajectory (path) between the two points. The value of the integral (D) represents the length of this trajectory. The distance is the where is the optimal trajectory. In the familiar Euclidean case (the above integral)
minimal value of this integral and is obtained when
this optimal trajectory is simply a straight line. It is well known that the shortest path between two points is a straight line. Straight lines can formally be obtained by solving the Euler-Lagrange equations for the above functional. In non-Euclideanmanifolds (curved spaces) where the nature of the space is
represented by a metric
the integrand has be to modified to
, where Einstein summation convention has been used.
The Euclidean distance between two points in space ( minimum value of an integral:
source | editbeta]
and ) may be written in a variational form where the distance is the
Here
is the trajectory (path) between the two points. The value of the integral (D) represents the length of this trajectory. The distance is the where is the optimal trajectory. In the familiar Euclidean case (the above integral)
minimal value of this integral and is obtained when
this optimal trajectory is simply a straight line. It is well known that the shortest path between two points is a straight line. Straight lines can formally be obtained by solving the Euler-Lagrange equations for the above functional. In non-Euclideanmanifolds (curved spaces) where the nature of the space is
represented by a metric
the integrand has be to modified to
, where Einstein summation convention has been used.
