Hamiltonian Noether's theorem in classical mechanics

[bolbteppa]

How does one think about, and apply, in the classical mechanical Hamiltonian formalism?

From the Lagrangian perspective, Noether's theorem (in 1-D) states that the quantity

$$\sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \frac{d y_i}{dx})} \frac{\partial y_i^*}{\partial \varepsilon} - \left[\sum_{j=1}^n \frac{\partial \mathcal{L}}{\partial ( \frac{d y_j}{dx})} \frac{d y_j }{\partial x} - \mathcal{L}\right]\frac{\partial x^*}{\partial \varepsilon}$$

is conserved if the Lagrangian $\mathcal{L}(x,y_i,y_i')$ is invariant under a continuous one-parameter group of infinitesimal transformations of the form

$$T(x,y_i,\varepsilon) = (x^*,y_i^*) = (x^*(x,y_i,\varepsilon),y_i^*(x,y_i,\varepsilon)).$$

From the action perspective, Noether's theorem states the equality of the 1-forms:

$$\mathcal{L}(x,y_i,y_i')dx = \sum_{j=1}^n p_i d y_j - \mathcal{H}dx = \mathcal{L}(x^*,y_i^*,y_i'^*)dx^* = \sum_{i=1}^n p_i d y_i^* - \mathcal{H}dx^*$$

which can be used to determine (additive) symmetries nicely.

How do I use this formalism to understand the Hamiltonian Noether theorem in a general context? I'll usually see a claim that $dA/dt = [H,A]$ is the Hamiltonian Noether's theorem, and I can't make sense of this in the context of my description of Noether above. This appears to derive the Poisson brackets as part of Noether from what I've developed above, but I can't make much sense of it to be honest I'm sure the answer is supposed to link the local Lie algebra tangent vector structure to the global Lie group transformation in the Lagrangian, but saying that in words is one thing, in math it's another, thanks.

 

[eljose79]

I would approach this question by first understanding the basic principles of the classical mechanical Hamiltonian formalism. This includes understanding the Hamiltonian function, Hamilton's equations, and the concept of phase space.

Next, I would familiarize myself with Noether's theorem, which is a fundamental principle in classical mechanics that relates symmetries of a physical system to conserved quantities. In this case, Noether's theorem states that for every continuous symmetry of a physical system, there exists a corresponding conserved quantity.

To apply this in the Hamiltonian formalism, I would use the Hamiltonian Noether theorem, which states that for every continuous symmetry of a physical system, there exists a corresponding generator of the symmetry that satisfies the Hamiltonian equations. This generator is known as the Noether charge and it can be used to determine the conserved quantity associated with the symmetry.

To understand this in a general context, I would start by considering a simple example, such as a system with translational symmetry. In this case, the generator of the symmetry is the momentum, and the conserved quantity is the total energy of the system. From this example, I would then move on to more complex systems and use the Hamiltonian Noether theorem to determine the conserved quantities associated with different symmetries.

In terms of the math, I would use the Poisson bracket, which is a mathematical operation that relates the Hamiltonian function to other functions of the system. This bracket is defined as the commutator of the Hamiltonian function with another function, and it is related to the time derivative of that function. This is where the equation dA/dt = [H,A] comes from, as it is a direct result of the Hamiltonian Noether theorem.

In summary, to think about and apply the classical mechanical Hamiltonian formalism, it is important to understand the principles of Hamiltonian mechanics and Noether's theorem. Using these concepts, we can determine conserved quantities associated with symmetries of a physical system, and use the Hamiltonian Noether theorem to relate these quantities to the Hamiltonian function. This allows us to better understand the behavior of physical systems and make predictions about their evolution over time.