Gravity, a Geometrical Course by Pietro Giuseppe Frè

Author:Pietro Giuseppe Frè
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

(3.8.5)

Performing a variational calculation we get that the length is extremal if

(3.8.6)

We are free to use any parameter λ to parameterize the curves. Let us use the affine parameter λ=τ defined by the condition:

(3.8.7)

In this case equation (3.8.6) reduces to which is the standard variational equation for a Lagrangian where the affine parameter τ is the time and x μ are the Lagrangian coordinates q μ . It is a straightforward exercise to verify that the Euler-Lagrange equations of this system:

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