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.

 

[PJC01]

Hi Nick,

That's a great question. The symmetries of the Lagrangian do indeed correspond to symmetries of the solution to the governing equations. This is because the Lagrangian itself is a function of the coordinates and velocities of the system, and any symmetries of the Lagrangian will affect the equations of motion derived from it.

Specifically, if the Lagrangian has a certain symmetry, say rotational symmetry, then the equations of motion will also have that symmetry. This means that any solutions to the equations will also exhibit this symmetry. In other words, if you rotate the system, the equations and solutions will still hold true.

This is a powerful tool in studying physical systems, as it allows us to simplify the equations and find solutions that are more easily understood. It also helps us to identify conserved quantities, as symmetries of the Lagrangian correspond to conserved quantities in the system.

For more information and examples, I would recommend looking into Noether's theorem, which provides a mathematical framework for understanding the connection between symmetries and conserved quantities in Lagrangian systems.

I hope this helps answer your question. Let me know if you need any further clarification or references.

Best,