Symmetries of Lagrangian and governing equations

[nickthequick]

Hi,

I have a quick question: Let's say I have a Lagrangian \mathcal{L}. From Hamilton's principle I find a governing equation for my system, call it N\phi=0 where N is some operator and \phi represents the dependent variable of the system. If \mathcal{L} has a particular symmetry, how does that (or does it at all) correspond to symmetries of the solution \phi? ie does this symmetry map solutions to solutions?

Basically the essence of the question is this: Do the symmetries of the Lagrangian give us additional information about solutions to the governing equations?

Any help/references is appreciated.

Thanks,

Nick

 

[Vargo]

Hi Nick,

Here are a couple answers.

1. Symmetries in the Lagrangian do map a solution to another solution. For example, if your system is translation invariant then translations of solutions are still solutions.

2. See Noether's Theorem on wikipedia or in any book on calculus of variations. Symmetries in the Lagrangian imply conserved quantities. If your Lagrangian is symmetric with respect to translation, then linear momentum is conserved. Symmetry in time implies Energy is conserved.

Conserved quantities allow us to reduce the order of the equations of motion and if there are enough conserved quantities we can find explicit solutions. There is an explicit procedure for how to do this, but it is delicate and depends on the symmetries commuting with each other to a certain extent.

Example: Two point masses in space attracted by a Force that is proportional to the inverse square of the distance between the points. This is a 12 dimensional system of ODE because each particle has 3 position coordinates and 3 velocity/momentum coordinates. Let's count symmetries/conserved quantities.

1. translation (linear momentum conserved)
2. rotationally symmetric (angular momentum is conserved)
3. Symmetric w/respect to inertial changes of coordinates (center of mass minus t*initial velocity is conserved)
4. If the Force is inverse square, then there are "hidden symmetries" that imply conservation of the Runge-Lenz vector.

That is 12 conserved quantities. So the solution (your 12 unknown functions) satisfies 12 equations that can be used to explicitly solve the equations.

Here is how you can exploit those things. Using an inertial change of coordinates plus a translation and rotation, you can assume the center of mass is at the origin (for all t) and that the angular momentum is a constant vector pointing in the z direction. That implies the motion is planar in the x,y plane. Using 9 symmetries we reduced the problem from 12 to 3 unknown functions. r1(t), r2(t), and theta(t). Note both masses must move with the same angular velocity since the center of mass remains at the origin. Then you use conservation of Runge Lenz to finish the problem.

 

[nickthequick]

Vargo,

Thanks for the reply! It was very helpful.

Do you have a reference for "There is an explicit procedure for how to do this, but it is delicate and depends on the symmetries commuting with each other to a certain extent."?

I would love to dive into this stuff a bit more.


Nick

 

[Vargo]

Almost everything I know about Lagrangians and Hamiltonians I learned from "Mathematical methods of classical mechanics", by V.I. Arnold. Another book which is good is "Calculus of Variations" by Gelfand and Fomin. That is an easier book, but it is a bit less thorough and I dont' think it discusses symmetries as clearly.

They are both by mathematicians. I don't know of any books on the subject that are geared more for physicists, but someone around here must.