Physics+ Lagrangian for Gravitation: UNIZOR.COM - Physics+ 4 All - Lagrangian

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Euler-Lagrange Equations
for the Gravitational Field

The following is an illustration of using Lagrangian Mechanics to analyze the movement of a planet in a central gravitational field of the Sun.
It will also show how the Kepler's laws of planetary movements are derived from Euler-Lagrange equations.

Let the Sun be modeled as a point mass M and a planet be modeled as a point mass m.
In the lecture about a central field we proved that the orbit of a planet lies in a plane. That allows us to choose polar coordinates r(t) and θ(t) with the Sun at the origin as generalized coordinates.

To apply Euler-Lagrange equation, we have to express kinetic energy T and potential energy U in terms of generalized coordinates r and θ.

Kinetic energy depends on the square of the magnitude of velocity v. In polar coordinates this is expressed as a sum of squares of radial speed vr and perpendicular to it tangential speed vθ:
vr = r'
vθ = r·θ'
v² = vr²+vθ² = (r')²+(r·θ')²
T = ½mv² = ½m[(r')²+(r·θ')²]

Potential energy of a planet in the gravitational field (you can refer to lectures in
Physics 4 Teens → Energy → Energy of Gravitational Field) is
U = −G·M·m/r
where G is the universal Gravitational Constant

Lagrangian of a planet is
L = T − U =
= ½m[(r')²+(r·θ')²] + G·M·m/r

The Euler-Lagrange equations for generalized coordinate are
∂L/∂r = d/dt ∂L/∂r'
∂L/∂θ = d/dt ∂L/∂θ'