An Example of Lagrangian Mechanics

A planet is in motion around a central star. The star’s mass is so great in comparison to the planet that it is assumed not to move as the planet orbits around the star. In Cartesian coordinates, using Newtonian mechanics, the system would be described by the y and x components of the force vectors between the star and the planet. Alternatively, we can describe the system using the Lagrangian method, as follows.

First, generalized coordinates are defined. Instead of using Cartesian coordinates, we choose the distance between the planet and the star r, and the angle between the positive y axis and r, which here we call theta.


The next step is to determine the Lagrangian of the system. The kinetic energy of the system will contain both the radial and tangential velocities of the planet. The radial velocity is simply the time rate of change of r, denoted r* here. The tangential velocity at any given instant is r times the time rate of change of theta (written as theta*).

The kinetic energy is thus given by:
K=.5m(r*^2+r^2theta*^2)
The potential energy is the same as between any two gravitating objects:
U=-GMmr^-2
Plugging these into the Lagrangian:
L=.5m(r*^2+r^2theta*^2)+GMm/r

Then writing down Lagrange’s equations:

d/dt(partial L/partial r*)=partial L/partial r
and partially differentiating: r**=rtheta*^2-GM/r
The first term in 1.b is, by Newton’s 2nd Law, the net force acting radially. It is equated with the sum of -GMm/r^2, which is clearly the force of gravity, and a second term that is the fictitious inertial force corresponding to centripetal acceleration.

Equation 2.b, when integrated yields:
mr^2theta*
C being a constant of integration. This is the Conservation of Angular Momentum.
Introduction - Core Terminology and Concepts - Principle of Least Action - Euler-Lagrange Equation

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