Notes

Analytical mechanics

EL equations

L = T - V

ddt(Lx˙)=Lx

  • Generalised momentum

Lq˙=p

  • Generalised force

Lq=p˙

Cyclic coordinate

When the Lagrangian does not depend on a coordinate qk:

Lq˙k=C

This is also generalised momentum.

q_k is thus a cyclic coordinate.

Energy conservation

Et=Lt

E=(i=1NLq˙iqi˙)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,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 ϵ.

qiqi+ϵKi(q)

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

The quantity:

P(q,q˙)=iLq˙iKi(q)

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

Example:

For L=(m/2)(5x˙22x˙y˙+2y˙2)+C(2xy), K_x = 1 and K_y = 2:

P=Lx˙Kx+Ly˙Ky=m(3x˙+3y˙)

Constraints

When the constraint has the form,

f(r1,r2,,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

Qj=Uqj+ddtUq˙j

Dissipation forces

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

ddtLq˙jLqj=Qj

Often the frictional force is proportional to the velocity of the particle. e.g. Fx=kxvx. Thus we may derive them from Rayleigh’s dissipation function, F.

Missing or unrecognized delimiter for \left

where the summation is over the particles of the system.

Thus, Fxi=Fvxi.

Or symbolically,

F=vF

2F is the also the rate of energy dissipation due to friction.

The generalized force is given by,

Qj=Fq˙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

Veff=L22mr2+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

Feff=L2mr3V(r)

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

drdt=±2mEL22mr2V(r)

or r(theta):

(1r2drdθ)2=2mEL21r22mV(r)L2

or Binet’s equation:

(1+d2dθ2)u=mL2u2F(u)

d2udθ2=mL2Ueff(u)

Gravitation

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

1r=mαL2(1+ecosθ)

Or in terms of specific angular momentum, h.

r=h2/μ1+ecosθ

where the eccentricity e=1+2EL2mα2=1+2ϵh2μ2 , and α=GMm, μ=GM.

Kepler’s equation

r=a(1ecosψ)

Kepler’s equation

ωt=ψesinψ

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=1b2a2

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

    c=ae

    r=a(1e2)1+ecosθ

    • Orbits

      ϵ=μ2a

      h=μa(1e2)

      Vis-viva equation:

      v2=μ(2r1a)

  • Parabola (e = 1)

    In polar coordinates, r=p1+cosθ

    For a left facing parabola, y2=4ax, where a is the distance between the vertex and the focus of the parabola.

    r=2a1+cosθ

    • Orbits

      ϵ=0

      h=2aμ

  • Hyperbola (e > 1)

    x2a2y2b2=1

    e=1+b2a2

    r=a(e21)1+ecosθ

    rmin=μ2v04+b2μv02

    • Orbits

      ϵ=μ2a h=μa(e21)

Waves

Energy density

In general.

ϵA2

For string waves:

KE: ϵKE=12μ(ψt)2\
PE:

dW=TΔ=T12(ψx)2dx=dU

The total energy per unit length is thus:

ϵ(x,t)=μ2((ψt)2+v2(ψx)2)

Further more, for a travelling wave, we can use the relationship ψt=vψx to get:

ϵ(x,t)=μψt2

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 Tsinθ=Tψx. Power is thus:

P=Tψxψt

Again using ψt=vψx for a rightwards/leftwards travelling wave,

P=±ϵ(x,t)v

Sound waves

ψp=dP=BdVV

Taking a control small control volume that gets compressed,

dV=A[ψ(x+dx)ψ(x)]

It can then be seen that:

Bψx=ψp

Using N2L:

[ψp(x+dx)ψp(x)]A=ρAdxψtt

It can then be shown that both ψ and ψp obey the wave equation, though for sinusoidal equation for instance ψp leads psi by π/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=γp0.

Energy density

ϵKE=12ρ(ψt)2

Since the potential energy is given by pdV from \p0 to \p0+ψp, using our B we can show:

ϵPE=B2ψx2

Again for travelling waves:

ϵ=ρψt2

Power

Same as strings but Av instead of v.

Plane waves

ψ=Re{ekrωt}

where k2=ω2v2

For spherically symmetric waves, the general solution is:

ψ(r)=Arcos(krωt+ϕ)

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:

ψi(0,t)+ψr(0,t)=ψt(0,t)

Furthermore, transverse tension must cancel.

T1[ψxi(0,t)+ψxr(0,t)]=T2ψxt(0,t)

Then defining the impedence Z=Tv=Tμ, we get:

Z1ψi(0,t)Z1ψr(0,t)=Z2ψt(0,t)

We obtain the transmission and reflection relations:

ψr(0,t)=Z1Z2Z1+Z2ψi(0,t)

ψt(0,t)=2Z1Z1+Z2ψi(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).

ψt(x,t)=Tψi(v1v2x,t)

(the wavelength is broadened by v2/v1)

ψr(x,t)=Rψi(x,t)

Limiting cases of impedences

For non-negative impedences,

|R|1

0T2

z2 > z1

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

1R<0

0T<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<R1

1<T2

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=p0v=ρp0γ=ρvγ

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:

γ=11v2/c21

tB=γtA

Identities

γ21=γ2β2

γ(1β)=1β1+β

γ(1+β)=1+β1β

Length contraction

For distances along the direction of relative velocity.

l=lγ

Lorentz tranform

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

x=γ(x+vt)

t=γ(t+vx/c2)

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

(xct)=(γγβγβγ)(xct)

where β=v/c.

Velocity addition

S' moves v2 wrt to frame S. An object moves v1 wrt to frame S'. The velocity of the object wrt S is:

u=v1+v21+v1v2/c2

This scenario is equivalent to A moving v1 wrt C to the right and B moving v2 wrt C to the left, and we ask the velocity of A wrt to B.