Roger's Equations

In this two part entry, I will attempt to demonstrate the power of Lagrange's Equations by showing all the information they can provide, specifically for the Kepler central potential, by just knowing the energies involved. The Langrangian is defined to be:

L=T - U

where T is the kinetic energy and U is the potential energy of the system in question.

Using the Lagrangian, we can find the equations of motion for a system with the equation:

where L is the Lagrangian of the system, q_i is a generalized coordinate (could be x, y, z, r, θ, φ, etc.), and q_i^' is the first time derivative of the generalized coodinate.

An example to get a feel for how it works:

Imagine a mass mass m moving at velocity v in a gravitational potential U_(x) would have the Lagrangian:

L= K.E. - P.E. = ¹/₂mv² - mgx

where g is 9.8 m/s and x is the distance the mass is above the ground. Lagrange's equation for this system is:

where

which gives us;

This equation is equivalent to F=-mg, the familiar equation for this sort of problem.

Orbits of a Central Potential:

A great example of the power of the Lagrangian is in calculating orbits. Consider a central potential U_(r) and a mass m. Since an orbiting mass moves along a plane, we can define the kinetic energy to be:

K.E.= ¹/₂ mv_x² +¹/₂ mv_y² (In xyz coordinates)

The central potential is spherically symmetric, so it's probably a good idea to express our kinetic energy in spherical coordinates as well. That gives us a Lagrangian of:

(I used x=rcosθsinφ and y=rsinθsinφ to convert the K.E. to spherical coordinates)

Lagrange's equations for this system and coordinates are:

Using Lagrange's equations from above we get:

and

Already we have a significant result. Notice the the time derivative of the second result (mr2(dθ/dt)) is equal to zero. When something doesn't change over time, it's constant. This equation is saying the Angular Momentum of the orbiting object remains constant. Conservation of Angular Momentum comes right out in the equation. We'll call the quantity (mr2(dθ/dt)) = L (L is the usually symbol for angular momentum), so that now our second equation is dL/dt=0.

Next we'll use our constant L to help solve our other lagrange equation:

which becomes;

All I"ve done above is sub in L. Please note that the potential is now V_(r) instead of U_(r) (just changing variables, it means the same thing)

The equation above is in terms of time t and distance r. It is convenient to express the equation in terms of angle θ and distance r instead. To do this, we can change variables from time(t) to angle(θ) with the relation for L:

which gives us a new equation independent of time in terms of r and θ;

Using

and multiplying both sides by

we get

The central potential we're intererested in is the Kepler Potential;

Where k=Gm Which we can set up to solve for θ by substituting the Kepler Potential and differentiating;

which gives us a second order differential equation with the solution;

Where e is the eccentricity and θ₀ is your starting angle (you've got to start measuring angle somewhere in the orbit) and θ is your finishing angle.

What does this equation tell us?

If e=0, that is eccentricity is 0, the orbit is circular and the cosine term goes away. This means the radius is constant, just as you would expect for a circular orbit.

If 0<e<1, the orbit is elliptical with maximum r (aphelion) when cos(θ-θ₀)=-1 and has a minimum r (perihelion) when cos(θ-θ₀)=1 so aphelion when (θ-θ₀)=180° or 270° (nπ) and (θ-θ₀)=0º or 360º (2nπ). So the distance from the center potential varies but the orbit is closed (repeats itself), just as you would expect for an ellipse.

So our answer in terms of r and θ seems consistent with what we would expect. In part two, We'll carve these equations up to get Keplers Laws and other interesting stuff. Hopefully part two will show why all this work in part one was worthwhile.

I needed a lot of help on this derivation. I'd like to thank the following websites from which I took pieces of the derivation above.

http://www.worldforge.org/project/newsletters/July2002/LagrangianP3
http://farside.ph.utexas.edu/teaching/336k/lectures/node80.html
http://en.wikipedia.org/wiki/Bertrand's_theorem