Analytical mechanics
EL equations
L = T - V
- Generalised momentum
- Generalised force
Cyclic coordinate
When the Lagrangian does not depend on a coordinate qk:
This is also generalised momentum.
q_k is thus a cyclic coordinate.
Energy conservation
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
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
each K_i(q) may be a function of all the q_i and need not be constant.
The quantity:
does not change with time and has the generic name of consered momentum.
Example:
For
Constraints
When the constraint has the form,
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
Dissipation forces
If not all the forces can be derivable from a potential, then EL equations can be written in the form,
Often the frictional force is proportional to the velocity of the particle. e.g.
where the summation is over the particles of the system.
Thus,
Or symbolically,
The generalized force is given by,
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
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
We may use it to find r(t) and theta(t) in terms of t:
or r(theta):
or Binet’s equation:
Gravitation
The motion of particles is like (earth-sun model):
Or in terms of specific angular momentum, h.
where the eccentricity
Kepler’s equation
Kepler’s equation
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.
From the geometrical definition of a ellipse and the above definition,
-
Orbits
Vis-viva equation:
-
-
Parabola (e = 1)
In polar coordinates,
For a left facing parabola,
, where a is the distance between the vertex and the focus of the parabola.-
Orbits
-
-
Hyperbola (e > 1)
-
Orbits
-
Waves
Energy density
In general.
For string waves:
KE:
PE:
The total energy per unit length is thus:
Further more, for a travelling wave, we can use the relationship
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
Again using
Sound waves
Taking a control small control volume that gets compressed,
It can then be seen that:
Using N2L:
It can then be shown that both
Due to collisions, heat flow is much slower than oscillation so the process is usually adiabatic and
Energy density
Since the potential energy is given by
Again for travelling waves:
Power
Same as strings but
Plane waves
where
For spherically symmetric waves, the general solution is:
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:
Furthermore, transverse tension must cancel.
Then defining the impedence
We obtain the transmission and reflection relations:
(Notice the analogy with elastic collisions, with impedence being analagous to mass)
We call each coefficient the reflection and transmission coefficients (
To find the resultant waves after collision, we makes use of the fact that the waves are travelling (like in method of characteristics).
(the wavelength is broadened by
Limiting cases of impedences
For non-negative impedences,
z2 > z1
The large force at the ring wrests the incoming wave down.
Since R is negative, it goes through a
At the extreme, the entire incident wave is reflected
z2 < z1
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
Fully transmitted. Can occur even when the string is inhomogenous by scaling tension and mass density equally.
Sound waves
For sound waves
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:
Identities
Length contraction
For distances along the direction of relative velocity.
Lorentz tranform
If S' is a coordinate system moving at speed v wrt S.
The inverse Lorentz transforms are given by the tranformation v-> v'.
where
Velocity addition
S' moves
This scenario is equivalent to A moving