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

UNIZOR.COM - Creative Mind through Art of Mathematics Read full text of notes for this lecture on UNIZOR.COM - Physics+ 4 All - Lagrangian - Gravity Example Notes to a video lecture on UNIZOR.COM 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'