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AM Finalterm problem

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Analytical Mechanics

2026 Classical Mechanics Final Term

Problem 1 (50 points)

Based on the following effective action:

where is a momentum-space Berry connection, derive the Hamiltonian equation of motion and solve them to find the below expression

Missing \begin{cases} or extra \end{cases}\mathfrak{m}\frac{dx}{dt} = \hat{p} + eE \times \Theta + (\Theta \cdot \hat{p}) eB \\ \mathfrak{m}\frac{dp}{dt} = eE + e\hat{p} \times B + e^{2}(E \cdot B)\Theta \end{cases}$$ $$\mathfrak{m} = 1 + e\Theta \cdot B$$ Here, $\mathfrak{m}$ may be regarded as an effective mass for this particle. * **(Questions)** * Can you imagine a UV complete version of this effective action? In other words, figure out where this effective action comes from. *(Hint: Maybe google or AI with "chiral fermions". It is important to notice the momentum-linear dispersion, which implies that Poincaré invariance = Lorentz invariance + Translational invariance will play a key role.)* * Can you figure out the Lorentz symmetry in this particle dynamics (optional)? --- ## Problem 2 (50 points) Revisit this problem based on the symplectic geometry perspectives. Can you find the corresponding symplectic two-form and the symplectic vector field? Let me give you the answer. $$V^{7} = T(\mathbb{R}^{3}\backslash\{0\}) \times \mathbb{R} = (x, p, t) \quad \text{(6-dimensional phase space + 1 time dimension)}$$ $$\sigma = \omega - dh \wedge dt$$ $$\omega = \omega_{0} + \frac{e}{2}\epsilon_{ijk}B^{i}dx^{j} \wedge dx^{k}$$ $$\omega_{0} = dp_{i} \wedge dx^{i} - \frac{s}{2|p|^{3}}\epsilon^{ijk}p_{i}dp_{j} \wedge dp_{k}$$

h = |p| + e\phi

i_{X_{H}}\omega = dH \quad \text{or the following expression:}

\omega_{\alpha\beta}\dot{\xi}^{\beta} = \partial_{\alpha}h, \quad \text{where } \omega_{\alpha\beta} = \partial_{\alpha}u_{\beta} - \partial_{\beta}u_{\alpha}

You can't use 'macro parameter character #' in math mode (Question) * Clarify the symplectic structure. Show that this symplectic two-form is non-degenerate, i.e., given by $\det(\omega_{\alpha\beta}) \neq 0$. --- ## Problem 3 (100 points) Consider a spinning particle to follow relativity. To describe the spinning particle, we consider a 9-dimensional phase space, given by:

V^{9} = \left{ R, I, J \in \mathbb{R}^{3,1} ;\middle|; I_{\mu}I^{\mu} = J_{\mu}J^{\mu} = 0, ; I_{\mu}J^{\mu} = -1 \right}

You can't use 'macro parameter character #' in math mode Here, $I_{\mu} \& J_{\mu}$ are two independent null vectors, given above. To figure out that this is the 9-dimensional phase space, we introduce $P_{\mu} \& S_{\mu\nu}$ to replace $I_{\mu} \& J_{\mu}$ as follows: $$I_{\mu} \rightarrow P_{\mu} \quad \text{and} \quad S_{\mu\nu} = -s\epsilon_{\mu\nu\rho\sigma}P^{\rho}J^{\sigma}$$ Then, we obtain: $$V^{9} = \left\{ R, P \in \mathbb{R}^{3,1}, \; S \in \mathfrak{o}(3,1) \;\middle|\; P_{\mu}P^{\mu} = 0, \; S_{\mu\nu}P^{\nu} = 0, \; \frac{1}{2}S_{\mu\nu}S^{\mu\nu} = s^{2} \right\}$$ It is straightforward to check out that $P_{\mu} \& S_{\mu\nu}$ satisfy these constraints. Now, we introduce the following closed two-form: $$\sigma = -dP_{\mu} \wedge dR^{\mu} - \frac{1}{2s^{2}}dS_{\mu}^{\lambda} \wedge S_{\rho}^{\mu} dS_{\lambda}^{\rho}$$ Here, we do not consider electromagnetic fields. Identify the Hamiltonian vector field and obtain the following Hamiltonian equation of motion: $$\begin{cases} P_{\mu}\dot{R}^{\mu} = 0 \\ \dot{P}^{\mu} = 0 \\ \dot{S}^{\mu\nu} = P^{\mu}\dot{R}^{\nu} - P^{\nu}\dot{R}^{\mu} \end{cases}$$ It is quite surprising to realize that this symplectic two-form is identical to that of **Q. 2** if electromagnetic fields are neglected, i.e., $\omega_{0}$ in **Q. 2**. In other words, the symplectic two-form of **Q. 2** without electromagnetic fields is a symplectic reduction of the above symplectic two-form. Can you figure it out? See below. To obtain down-to-earth expressions, we put $R = (r, t)$, where $r$ and $t$ are the position and time coordinates in a chosen Lorentz frame. The two null-vectors are in turn $P = (p, |p|)$ and $J = (q, -|q|)$, where $p$ and $q$ are two (necessarily nonzero) 3-vectors which satisfy $p \cdot q + |p||q| = 1$. In these terms we have: $$s = s(p|q| + q|p|)$$ $$S_{ij} = \epsilon_{ijk}s^{k}$$ $$S_{j4} = s(p \times q)_{j} = (\hat{p} \times s)_{j}$$ * **(Question)** * Based on this coordinate representation, find the moment map that reduces the symplectic two-form in 9-dimensional phase space to that in 6-dimensional one. --- ## Problem 4 (100 points) Now, we introduce a non-minimal coupling of electromagnetic fields into this spin $1/2$ chiral fermion. Starting from the closed two-form: $$\sigma = -dP_{\mu} \wedge dR^{\mu} - \frac{1}{2s^{2}}dS_{\lambda}^{\mu} \wedge S_{\rho}^{\lambda} dS_{\mu}^{\rho} + \frac{1}{2}eF_{\mu\nu}dR^{\mu} \wedge dR^{\nu}$$ derive the corresponding equations of motion as follows: $$\begin{cases} \dot{R}^{\mu} = P^{\mu} + \frac{S^{\mu \nu} F_{\nu \rho} P^\rho}{\frac{1}{2}S \cdot F} \\ \dot{P}^{\mu} = -e F^{\mu}_{\nu} \dot{R}^{\nu} \\ \dot{S}^{\mu\nu} = P^{\mu}\dot{R}^{\nu} - P^{\nu}\dot{R}^{\mu} \end{cases}$$ Unfortunately, the above closed two-form is not completely satisfactory. We have to introduce the mass square constraint in the following way:

P_\mu P^\mu = -\frac{eg}{2}S \cdot F

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