Lagrangian Geodesics and Euler–Lagrange Equations

A guided research project in Geometry & Dynamics.

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Question

How does a rule of least energy determine the straightest path on a curved space?

The Euler–Lagrange equations convert a global path principle into local differential equations.
The Euler–Lagrange equations convert a global path principle into local differential equations.

Goals

  • Derive geodesic equations from a metric or Lagrangian.
  • Solve and compare geodesics on selected surfaces.

Method

Treat a geodesic as a path that makes an energy or length functional stationary.

  • Write the metric and its kinetic-energy Lagrangian in local coordinates.
  • Apply the Euler–Lagrange equations to obtain the geodesic equations.
  • Solve the equations analytically or numerically and draw the resulting paths.