Notes

Analytical mechanics

EL equations

L = T - V

\[ \dv{t}\left(\pdv{L}{\dot x}\right) = \pdv{L}{x} \]

• Generalised momentum

\[ \pdv{L}{\dot q} = p \]

• Generalised force

\[ \pdv{L}{q} = \dot{p} \]

Cyclic coordinate

When the Lagrangian does not depend on a coordinate qk:

\[ \pdv{L}{\dot q_k} = C \]

This is also generalised momentum.

q_k is thus a cyclic coordinate.

Energy conservation

\[ \pdv{E}{t} = -\pdv{L}{t} \]

\[ E = \left(\sum_{i=1}^N \pdv{L}{\dot q_i}\dot{q_i}\right) - L \]

If L has no time dependence, then E is conserved.

E is indeed total energy when the new coordinate q is not related to the old coordinate x with a time dependence \(x = x(q,t)\) or qdot dependence \(x = x(q,\dot{q})\).

Noether’s theorem

For each symmetry of the Lagrangian, there is a conserved quantity.

Let the Lagrangian be invariant to first order in the small number \(\epsilon\).

\[ q_i \to q_i + \epsilon K_i(q) \]

each K_i(q) may be a function of all the q_i and need not be constant.

The quantity:

\[ P(q,\dot{q}) = \sum_i \pdv{L}{\dot q_i}K_i(q) \]

does not change with time and has the generic name of consered momentum.

Example:

For \(L = (m/2)(5\dot{x}^2 - 2\dot{x}\dot{y} + 2\dot{y}^2) + C(2x-y)\), K_x = 1 and K_y = 2:

\[ P = \pdv{L}{\dot{x}}K_x + \pdv{L}{\dot y}K_y = m(3\dot{x} + 3\dot{y}) \]

Constraints

When the constraint has the form,

\[ f(\bm{r_1},\bm{r_2},\ldots,t) = 0 \]

then it is said to be holonomic. Otherwise they are nonholonomic.

Equations of constraints which explicitly contain time as a variable are called rheonomous while those which don’t are called scleronomous.

Velocity-dependent potentials

\[ Q_j = -\pdv{U}{q_j} + \dv{}{t} \pdv{U}{\dot{q}_j} \]

Dissipation forces

If not all the forces can be derivable from a potential, then EL equations can be written in the form,

\[ \dv{}{t} \pdv{L}{\dot{q}_j} - \pdv{L}{q_j} = Q_j \]

Often the frictional force is proportional to the velocity of the particle. e.g. \(F_{x} = -k_xv_x\). Thus we may derive them from Rayleigh’s dissipation function, \(\mathcal{F}\).

\[ \mathcal{F} = \frac12 \sum_i \left \left(k_x v^2_{ix} + k_y v^2_{iy} + k_z v^2_{iz}\right) \]

where the summation is over the particles of the system.

Thus, \(F_{x_i} = - \pdv{\mathcal{F}}{v_{x_i}}\).

Or symbolically,

\[ \bm{F} = -\grad_v\mathcal{F} \]

\(2\mathcal{F}\) is the also the rate of energy dissipation due to friction.

The generalized force is given by,

\[ Q_j = -\pdv{\mathcal{F}}{\dot{q}_j} \]

Central forces

Only depends of radius from source. Motion takes place in a plane.

Effective potential

Since L is conserved, express w as L/mr^2

\[ V_{eff} = \frac{L^2}{2mr^2} + V( r) \]

the first term on RHS is often called the L barrier. This does not hold if v(r) goes to -infty faster than -1/r^2

\[ F_{eff} = \frac{L^2}{mr^3} - V'( r) \]

We may use it to find r(t) and theta(t) in terms of t:

\[ \dv{r}{t} = \pm \sqrt{\frac{2}{m}} \sqrt{E - \frac{L^2}{2mr^2} - V( r)} \]

or r(theta):

\[ \left(\frac{1}{r^2}\dv{r}{\theta}\right)^2 = \frac{2mE}{L^2} - \frac{1}{r^2} - \frac{2mV( r)}{L^2} \]

or Binet’s equation:

\[ \left(1+\dv[2]{}{\theta}\right)u = -\frac{m}{L^2u^2}F(u) \]

\[ \dv[2]{u}{\theta} = -\frac{m}{L^2}U_{eff}'(u) \]

Gravitation

The motion of particles is like (earth-sun model):

\[ \frac{1}{r} = \frac{m\alpha}{L^2}(1+e\cos\theta) \]

Or in terms of specific angular momentum, h.

\[ r = \frac{h^2 / \mu}{1+e\cos\theta} \]

where the eccentricity \(e = \sqrt{1 + \frac{2EL^2}{m\alpha^2}} = \sqrt{1+\frac{2\epsilon h^2}{\mu^2}}\) , and \(\alpha = GMm\), \(\mu = GM\).

Kepler’s equation

\[ r = a(1-e\cos\psi) \]

Kepler’s equation

\[ \omega t =\psi - e \sin\psi \]

Eccentricity and orbits

• Circle (e = 0)

• Ellipse (0 < e < 1)

  WLOG, let a be the semi-major axis and b be the semi-minor axis. c is the distance of a focus to the centre.

  \[ e = \sqrt{1 - \frac{b^2}{a^2}} \]

  From the geometrical definition of a ellipse and the above definition,

  \[ c = ae \]

  \[ r = \frac{a(1-e^2)}{1+e\cos\theta} \]

  • Orbits

    \[ \epsilon = -\frac{\mu}{2a} \]

    \[ h = \sqrt{\mu a (1-e^2)} \]

    Vis-viva equation:

    \[ v^2 = \mu\left( \frac{2}{r} - \frac{1}{a} \right) \]

• Parabola (e = 1)

  In polar coordinates, \(r = \frac{p}{1+\cos\theta}\)

  For a left facing parabola, \(y^2 = -4ax\), where a is the distance between the vertex and the focus of the parabola.

  \[ r = \frac{2a}{1+\cos\theta} \]

  • Orbits

    \[ \epsilon = 0 \]

    \[ h = \sqrt{2a\mu} \]

• Hyperbola (e > 1)

  \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

  \[ e = \sqrt{1+\frac{b^2}{a^2}} \]

  \[ r = \frac{a(e^2-1)}{1+e\cos\theta} \]

  \[ r_{min} = \sqrt{\frac{\mu^2}{v_0^4} + b^2} - \frac{\mu}{v_0^2} \]

  • Orbits

    \[ \epsilon = \frac{\mu}{2a} \] \[ h = \sqrt{\mu a (e^2 - 1)} \]

Waves

Energy density

In general.

\[ \epsilon \propto A^2 \]

For string waves:

KE: \[ \epsilon_{KE} = \frac12 \mu \left(\pdv{\psi}{t}\right)^2 \]\
PE:

\[ \dd{W} = T\Delta = T \frac12 \left(\pdv{\psi}{x}\right)^2 \dd{x} = \dd{U} \]

The total energy per unit length is thus:

\[ \epsilon(x,t) = \frac{\mu}{2} \left(\left(\pdv{\psi}{t}\right)^2 + v^2\left(\pdv{\psi}{x}\right)^2 \right) \]

Further more, for a travelling wave, we can use the relationship \(\psi_t = \mp v \psi_x\) to get:

\[ \epsilon(x,t) = \mu \psi_t^2 \]

Note that this implies that segments of string the possess the greatest KE also have the greatest PE.

Power

Power is similarly proportional to the square of the amplitude.

For string waves, note that the transverse (vertical) force on a point Q on the string on the left of Q is \(-T\sin\theta = -T\psi_x\). Power is thus:

\[ P = -T\psi_x\psi_t \]

Again using \(\psi_t = \mp v \psi_x\) for a rightwards/leftwards travelling wave,

\[ P = \pm \epsilon(x,t) v \]

Sound waves

\[ \psi^p = \dd{P} = -B\frac{\dd{V}}{V} \]

Taking a control small control volume that gets compressed,

\[ \dd{V} = A[\psi(x+\dd{x}) - \psi(x)] \]

It can then be seen that:

\[ -B\psi_x = \psi^p \]

Using N2L:

\[ -[\psi^p(x+\dd{x}) - \psi^p(x)] A = \rho A \dd{x} \psi_{tt} \]

It can then be shown that both \(\psi\) and \(\psi^p\) obey the wave equation, though for sinusoidal equation for instance \(\psi^p\) leads \(psi\) by \(\pi/2\) in the direction of propogation (From the second last equation).

Due to collisions, heat flow is much slower than oscillation so the process is usually adiabatic and \(B =\gamma p_0\).

Energy density

\[ \epsilon_{KE} = \frac12 \rho (\psi_t)^2 \]

Since the potential energy is given by \(-\int p \dd{V}\) from \(\p_0\) to \(\p_0 +\psi^p\), using our \(B\) we can show:

\[ \epsilon_{PE} = \frac{B}{2} \psi_x^2 \]

Again for travelling waves:

\[ \epsilon = \rho \psi_t^2 \]

Power

Same as strings but \(Av\) instead of \(v\).

Plane waves

\(\psi = \Re\left\{ e^{\bm{k}\cdot\bm{r}-\omega t}\right\}\)

where \(\norm{\bm{k}}^2 = { \omega^2 \over v^2 }\)

For spherically symmetric waves, the general solution is:

\[ \psi( r) = \frac{A}{r} \cos(kr-\omega t + \phi) \]

Wave at boundary

A wave is traveling towards a boundary from the left, which results in a reflected and transmitted wave. We impose a continuity constraint:

\[ \psi^i(0,t) + \psi^r(0,t) = \psi^t(0,t) \]

Furthermore, transverse tension must cancel.

\[ T_1[ \psi^i_x(0,t) + \psi^r_x(0,t) ] = T_2 \psi_x^t(0,t) \]

Then defining the impedence \(Z = \frac{T}{v} = \sqrt{T\mu}\), we get:

\[ Z_1 \psi^i(0,t) - Z_1 \psi^r(0,t) = Z_2 \psi^t(0,t) \]

We obtain the transmission and reflection relations:

\[ \psi^r(0,t) = \frac{Z_1 - Z_2}{Z_1 + Z_2}\psi^i(0,t) \]

\[ \psi^t(0,t) = \frac{2Z_1}{Z_1 + Z_2} \psi^i(0,t) \]

(Notice the analogy with elastic collisions, with impedence being analagous to mass)

We call each coefficient the reflection and transmission coefficients (\(R\) and \(T\)) respectively, where \(T = 1 + R\).

To find the resultant waves after collision, we makes use of the fact that the waves are travelling (like in method of characteristics).

\[ \psi^t(x,t) = T\psi^i\left(\frac{v_1}{v_2}x,t\right) \]

(the wavelength is broadened by \(v_2/v_1\))

\[ \psi^r(x,t) = R\psi^i(-x,t) \]

Limiting cases of impedences

For non-negative impedences,

\[ \abs{R} \leq 1 \]

\[ 0 \leq T \leq 2 \]

z2 > z1

The large force at the ring wrests the incoming wave down.

\[ -1 \leq R < 0 \]

\[ 0 \leq T < 1 \]

Since R is negative, it goes through a $π$-radian phase shift.

At the extreme, the entire incident wave is reflected

z2 < z1

\[ 0 < R \leq 1 \]

\[ 1 < T \leq 2 \]

There is no longer a phase shift. At the extreme, the displacement at the origin is twice the displacement than what would have been produced by incident wave alone.

z2=z1

\[ T = 1, R=0 \]

Fully transmitted. Can occur even when the string is inhomogenous by scaling tension and mass density equally.

Sound waves

For sound waves

\(Z = \frac{p_0}{v} = \sqrt{\frac{\rho p_0}{\gamma}} = \frac{\rho v}{\gamma}\)

Special relativity

Loss of simultaneity

Last car first: The clock on the back of a moving train is ahead of the front clock.

Time dilation

Lorentz factor:

\[ \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \geq 1 \]

\[ t_B = \gamma t_A \]

Identities

\[ \gamma^2 - 1 = \gamma^2\beta^2 \]

\[ \gamma(1-\beta) = \sqrt{\frac{1-\beta}{1+\beta}} \]

\[ \gamma(1 + \beta) = \sqrt{\frac{1+\beta}{1-\beta}} \]

Length contraction

For distances along the direction of relative velocity.

\[ l = \frac{l'}{\gamma} \]

Lorentz tranform

If S' is a coordinate system moving at speed v wrt S.

\[ x = \gamma(x' + vt') \]

\[ t = \gamma(t' + vx'/c^2) \]

The inverse Lorentz transforms are given by the tranformation v-> v'.

\[ \mqty(x \\ ct) = \mqty(\gamma & \gamma \beta \\ \gamma\beta & \gamma ) \mqty(x' \\ ct') \]

where \(\beta = v/c\).

Velocity addition

S' moves \(v_2\) wrt to frame S. An object moves \(v_1\) wrt to frame S'. The velocity of the object wrt S is:

\[ u = \frac{v_1 + v_2}{1+v_1v_2/c^2} \]

This scenario is equivalent to A moving \(v_1\) wrt C to the right and B moving \(v_2\) wrt C to the left, and we ask the velocity of A wrt to B.