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Please give the thermodynamic identity for $U$ .

In thermodynamics, the internal energy $U$ of a system is related to its entropy $S$ , volume $V$ , and the number of particles $N$ by the fundamental thermodynamic identity:

$dU = TdS - PdV + \mu dN$

Where:

$\cdot$ $dU$ is the change in internal energy

$\cdot$ $T$ is the temperature

$\cdot$ $P$ is the pressure

$\cdot$ $\mu$ is the chemical potential

$\cdot$ $dS$ is the change in entropy

$\cdot$ $dV$ is the change in volume

$\cdot$ $dN$ is the change in the number of particles

This identity shows how the internal energy of a system changes when its entropy, volume, and number of particles change.

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Thermodynamics Summary

Internal Energy $dU = TdS - PdV + \mu dN$

Helmholtz Free Energy $dF = -SdT - PdV$

Gibbs Free Energy $dG = -SdT + VdP + \mu dN$

Thermodynamic identities describe the relationship between thermodynamic properties.

Partition function $Z = \sum_i e^{-\beta E_i}$ (with $\beta = \frac{1}{kT}$ )

Describes the sum of all possible energy states of a system.

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Classical Mechanics Exam

29/05/2025Student: ___________________

Question 1

Consider a pendulum consisting of a mass $m$ attached to a massless rod of length $L$ , swinging in a vertical plane. The pivot point is attached to a horizontal spring with spring constant $k$ . Using the angle $\theta$ from the vertical as your generalized coordinate, write down the Lagrangian, the generalized momentum $p_\theta$ , and the Hamiltonian:

$\mathcal{H} = p_\theta \dot{\theta} - \mathcal{L}$

Find Hamilton's equations and use them to determine the angular acceleration $\ddot{\theta}$ .

Question 2

A charged particle of mass $m$ and charge $q$ is moving in a uniform magnetic field $\mathbf{B} = B_0\hat{k}$ and a uniform electric field $\mathbf{E} = E_0\hat{i}$ .

(a) Using the $x$ -coordinate as the generalized coordinate, write down the Lagrangian, the generalized momentum $p_x$ , and the Hamiltonian $\mathcal{H}$ .

(b) Derive Hamilton's equations and use them to determine the equations of motion for the particle.

(c) Discuss the motion in terms of the Lorentz force and consider the case where $E_0 = 0$ .

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