Title: ETH_Sem5 / AdvQC / 03_Classical_Mech

Q: What is the relativity principle of Galilei?
A: 1. All inertial frames are equivalent; none can be favored over another.
2. Newton's laws of motion have the same form in all inertial frames of reference (invariance in form = covariance).
\[(t,x_i)\text{ in IS } \xrightarrow[\text{transformation}]{\text{Galilei}} (t\strich,x_i\strich)\text{ in IS}\strich\]
The transformation is given by a constant temporal shift \(t_0\), constant spatial shifts \(\bm{b_i}\), uniform spatial shifts \(\vel t\), and a constant spatial rotation described by the orthogonal transformation \(R\), \(RR^T = \bm{1}\):
\[\pos\strich=R\pos-\vel t-\Bfield\]

Q: What is the Lagrange Formalism? What is the action and the principle of stationary action?
A: Any mechanical system of \(N\) particles in \(\mathbb{R}^3\), which is subject to \(R\) constraints, features in general \(f=3N-R\) degrees of freedom represented by generalized coordinates \(q_1, q_2, \dots, q_f\) independent of each other and compatible with all constraints.
The Lagrangian is given by the difference of kinetic and potential energy:
\[L(q,\dq,t) = T(q,\dq,t) - U(q,\dq,t)\]
The time integral over the Lagrangian defines the action \(S\):
\[S=S[q]=\int_{t_1}^{t_2}\mathrm{d} t\ L(q,\dq,t)\]
The principle of stationary action (Hamiltonian principle) states, that the actual path \(q(t)\) taken between the fixed start and end points \(q_1\) and \(q_2\) is stationary with respect to \(\delta q\):
\[\delta S=\delta S[q]=\delta \int_{t_1}^{t_2}\mathrm{d} t\ L(q,\dq,t) = 0\]

Q: Write down the Euler-Lagrange equations! What is the gauge transformation?
A: Applying a Taylor series expansion of the Lagrangian around the actual path leads to the Euler-Lagrange eqs.
\[\dd{}{t}\pdv{L}{\dq_i}-\pdv{L}{q_i}=0 \qquad \forall i=1,2,\dots,f \]
with the momentum given by
\[p_i=\pdv{L}{\dq_i}\]
Lagrangians are said to be equivalent to each other if they give rise to the same Euler-Lagrange eqs. Because of the Hamiltonian principle, two Lagrangians \(L\) and \(L\strich\) are equivalent and yield the same equations of motion, if
\[{L\strich(q,\dq,t) = L(q,\dq,t) +\dd{t}F(q,t)}\]

Q: What is the Hamiltonian?
A: A Legendre transformation of the Lagrangian leads to the Hamiltonian \(H\)
\[H=H(q,p,t)=\sum_{i=1}^f p_i\dq_i(q,p,t)-L(q,\dq(q,p,t),t)\]
The \(2f\) variables \(q\) and \(p\) are called canonical variables of the system and specify the state of the system completely. If
1. constraints on Cartesian coordinates do not explicitly depend on time and
2. the potential is independent of the velocities,
the Hamiltonian equals the energy:
\[H=T+U\]

Q: What are Poisson brackets? How are they related to the Hamiltonian?
A: The Poisson bracket between two arbitrary phase space (means that they depend on \(q\), \(p\) and \(t\)) functions \(u = u(q,p,t)\) and \(v = v(q,p,t)\) is defined as
\[\pb{u}{v} = \sum_{i=1}^f \left(\pdv{u}{q_i}\pdv{v}{p_i}-\pdv{u}{p_i}\pdv{v}{q_i}\right)\]
The Poisson brackets for canonical variables \(q\) and \(p\) reads
\[\begin{array}{llll} \pb{q_i}{q_j} = 0, & \pb{p_i}{p_j} = 0, & \pb{q_i}{p_j} = \delta_{ij}, & \forall i,j=1,2,\dots,f \end{array}\]
The e.o.m. are yielded with the total time derivative of \(u\):
\[\begin{aligned} \dot{u}=\dv{u}{t}=\sum_{i=1}^{f}\left(\pdv{u}{q_i}\dq_i+\pdv{u}{p_i}\dot{p}_i\right)+\pdv{u}{t} \\ \thus {\dv{u}{t} = \pb{u}{H}+\pdv{u}{t}} \end{aligned}\]
Any phase space function \(u = u(q,p)\), which does not explicitly depend on time, will be conserved if
\[\pb{u}{H}=0 \means u \text{ is a constant of motion}\]