What is the Lie symmetry of this potential?

[CarlB]

The potential function:
$$V(x_r,y_r,\theta_r,x_g,y_g,\theta_g,x_b,y_b,\theta_b) = 3$$
$$- exp(-x_r^2-y_r^2)(cos(\theta_r) + \sqrt{3}sin(\theta_r))$$
$$- exp(-x_g^2-y_g^2)(cos(\theta_g) + \sqrt{3}sin(\theta_g))$$
$$- exp(-x_b^2-y_b^2)(cos(\theta_b) + \sqrt{3}sin(\theta_b))$$
where the nine variables are real numbers, and are subject to the subsidiary conditions:
$$x_r + x_g + x_b = 0,$$
$$y_r + y_g + y_b = 0,$$
$$\theta_r + \theta_g + \theta_b = 0,$$
and the region of interest is the neighborhood of $$(0,0,0,0,0,0,0,0,0)$$ has what continuous symmetry? By the way, I know that the minimum value of V is 0, and this is achieved at only two places, the origin and again at $$(0,0,2\pi/3,0,0,2\pi/3,0,0,-4\pi/3)$$.

By "neighborhood of", I do not mean to ask for the symmetry at the origin, which is kind of an unusual location, in terms of the vanishing of derivatives, but instead to suggest that taking a series expansion to 2nd order around the origin makes sense.

It is obvious that if you ignore the subsidiary conditions, one can rotate $$x_r$$ and $$y_r$$ into each other. You can do these to the g and b terms too, so you may satisfy the subsidiary condition by simultaneously counter-rotating r, g and b.

And it is clear that one can define an infinitesimal rotation of x into y into z, if one adjusts the amplitudes of the x, y, and z with the appropriate functions of the thetas.

My natural inclination is to make linear approximations around some arbitrary point (a,b,c,d,e,f), and write down the infinitesimal generators by solving the linear algebra problem. Then I can calculate the commutators. But at that point I'm not sure what I'll do next. Probably I'll go searching around for my copy of Georgi.

But I'm wondering if, when you have an explicit formula, there is an easy way of programming my computer to work out the symmetry.

Carl

 

[Nate810]

Bender and Stefan Boettcher have a book called Advanced Mathematical Methods for Scientists and Engineers, which I believe has the answer. I'm sure you can find it on Amazon. The authors give detailed instructions on how to calculate symmetries for more general potentials in their book, so it should be helpful in your case.

 

[thepatientmental]

The Lie symmetry of this potential is rotational symmetry. This can be seen by the presence of the trigonometric functions in the potential, which indicate that the potential is invariant under rotations around the origin. This is further supported by the subsidiary conditions, which require the variables to sum up to zero, indicating that the potential is symmetric under rotations in all three dimensions.

To find the continuous symmetry of this potential, one can use the method of infinitesimal generators. This involves taking a series expansion around an arbitrary point and writing down the infinitesimal generators by solving a linear algebra problem. The commutators can then be calculated to determine the continuous symmetry of the potential.

Alternatively, one can use a computer program to calculate the symmetry. This would involve inputting the explicit formula for the potential and using the program to determine the rotational symmetry. This method may be easier and more efficient, especially for more complex potential functions.

In conclusion, the Lie symmetry of this potential is rotational symmetry, which is evident from the presence of trigonometric functions and the subsidiary conditions. The method of infinitesimal generators or a computer program can be used to determine the continuous symmetry of the potential.