Generators of infinitesimal transforms from Goldstein (1965)

[HeavyWater]

This is a two part question. I will write out the second part tomorrow.

I will be referring to pages 258-263 in Goldstein (1965) about infinitesimal transformations.
Eqn 8-66 specifies that δu=ε[u,G], where u is a scalar function and G is the generator of the transform. How do I find the Generators and how do I know when I have found all the generators? I know that the generators commute with the Hamiltonian BUT there may be several variables that commute with H. For example, (see 8-68), if q1and q2 are cyclic then I know that the momenta p1 and p2 are the generators. But the H may be cyclic in other variables that are not so obvious and how would I identify the generators in these cases?

 

[jambaugh]

As you are talking about canonical transformations, any function (of phase space variables p and q) generates a transformation, even ones which do not commute under the Poisson bracket with the Hamiltonian. Those that do commute with the Hamiltonian will generate symmetry transformations.

As to how you find all the symmetries of the Hamiltonian, that is the million dollar question and of course depends on the the form of the Hamiltonian. You can glean hints from the application, i.e. look for the usual suspects, translation symmetries (generated by the momenta), rotation symmetries, generated by anti-symmetric products of momenta and coordinates, etc. Noether's theorem will also apply so that the conserved quantity associated with a symmetry will generate that symmetry when expressed as a function on phase space. (Example angular momentum about z-axis).

For linear transformations on coordinate space (acting dualy on momentum space) your generators will be products of coordinates and momenta.
The matrix element mapping $q_1$ to $q_2$ will correspond to the generating function $G(p,q)=q_2 p_1$.

 

[HeavyWater]

Thank you jambaugh for an OUTSTANDING ANSWER! I need to think about your response and especially about Noether's Theorem (something I hear about but have never sat down and really concentrated on it). I will come back with a follow up question about those pesky infinitesimal transforms on Monday.

 

[HeavyWater]

...continuing with my question...and thank you for your help so far. I don't feel like I am understanding the use of the infinitesimal transformations and the Poisson Brackets as described by Goldstein (1965), p260,261. Here is an example; I am leveraging the work of Galvao and Teitelboim, Feb 1,1980, J. Math Phys. 21(7).

A simple Lagrangian for a free spinning particle (pseudo classical mechanics) is L=½mv*v + ½(i theta-dot)(theta). Where the thetas are a function of time and are real anti commuting variables. We find the momentum, p from Lagrange equations and π from a generalization of Lagrange equations. Since we have a free particle, the Hamiltonian is H= (p*p)/2m +(π*π)/2 and we can find δx, δp, δθ, and δπ, where we use p as the generator for the first two and π as the generator for the last two.

I am missing the physics--I don't appreciate (or understand) what these infinitesimal transforms are telling me. Feel free to ramble...

 

[HeavyWater]

...continuing with my question...and thank you for your help so far. I don't feel like I am understanding the use of the infinitesimal transformations and the Poisson Brackets as described by Goldstein (1965), p260,261. Here is an example; I am leveraging the work of Galvao and Teitelboim, Feb 1,1980, J. Math Phys. 21(7).

A simple Lagrangian for a free spinning particle (pseudo classical mechanics) is L=½mv*v + ½(i theta-dot)(theta). Where the thetas are a function of time and are real anti commuting variables. We find the momentum, p from Lagrange equations and π from a generalization of Lagrange equations. Since we have a free particle, the Hamiltonian is H= (p*p)/2m +(π*π)/2 and we can find δx, δp, δθ, and δπ, where we use p as the generator for the first two and π as the generator for the last two.

I am missing the physics--I don't appreciate (or understand) what these infinitesimal transforms are telling me. Feel free to ramble...