B.7 Chapters 14 to 16

B.7 Chapters 14 to 16

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie bracket

Problem 1:

Consider a particle moving in two dimensions, with the Hamiltonian function

$$h=\frac{1}{2m}((p_1-B{q}_2{)}^2+p_2^2)$$

• Find the vector field $X_h$ associated to this function.

• Show that the quantities

$$p_1\mathrm{and}{p}_2-B{q}_1$$

are conserved.

• Write down Hamilton’s equations for this system and find the general solutions for the trajectories $(q(t),p(t))$ .

This system describes a particle moving in a plane, experiencing a magnetic field orthogonal to the plane. You should find that the trajectories are circles in the plane, with a frequency called the Larmor frequency.

Problem 2:

Consider the action of the group $SO(3)$ on phase space ${\mathbf{R}}^6$ by simultaneously rotating position and momentum vectors.

• For the three basis elements $l_j$ of $\mathfrak{so}(3)$ , show that the moment map gives functions ${\mu }_{l_j}$ that are just the components of the angular momentum.

• Show that the maps

$$l_j\in \mathfrak{so}(3)\to {\mu }_{l_j}$$

give a Lie algebra homomorphism from so(3) to the Lie algebra of functions on phase space (with Lie bracket on such functions the Poisson bracket).

Problem 3:

In the same context as problem 2, compute the Poisson brackets

$$\{{\mu }_{l_j},q_k\}$$

between the angular momentum functions ${\mu }_{l_j}$ and the configuration space coordinates $q_j$ . Compare this calculation to the calculation of

$${\pi }^{\prime }(l_j){\mathbf{e}}_k$$

for $\pi$ the spin-1 representation of $SO(3)$ on ${\mathbf{R}}^3$ (the vector representation).

Problem 4:

Consider the symplectic group $Sp(2d,\mathbf{R})$ of linear transformations of phase space ${\mathbf{R}}^{2d}$ that preserve Ω.

• Consider the group of linear transformations of phase space ${\mathbf{R}}^{2d}$ that act in the same way on positions and momenta, preserving the standard inner products on position and momentum space. Show that this group is a subgroup of $Sp(2d,\mathbf{R})$ , isomorphic to $O(d)$

• Using the identification between $Sp(2d,\mathbf{R})$ and matrices satisfying equation 16.10, which matrices give the subgroup above?

• Again in terms of matrices, what is the Lie algebra of this subgroup?

• Identifying the Lie algebra of $Sp(2d,\mathbf{R})$ with quadratic functions of the coordinates and momenta, which such quadratic functions are in the Lie algebra of the $SO(d)$ subgroup?

• Consider the function

$$\frac{1}{2}\sum _{j=1}^d(q_j^2+p_j^2)$$

What matrix does this correspond to as an element of the Lie algebra of $Sp(2d,\mathbf{R})?$ Show that one gets an $SO(2)$ subgroup of Sp(2d, $R$) by taking exponentials of this matrix. Is this $SO(2)$ a subgroup of the $SO(d)$ above?

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正文：英文 · OCR 机器稿 · 待校对

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原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

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