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Analytical Mechanics
(2008-09-04) Fermat's Principle of Least Time
(c. 1655)
A unifying principle for the geometrical propagation of light.
Around 1655,
Pierre de Fermat (1601-1665)
justified all laws of the newly developed science of geometrical optics by
stating that light always travels between two points in the
least possible amount of time.
This does say that light travels in a straight line through an homogeneous medium.
Less trivially, the
law
of reflection
(whereby the angle of reflection is equal to the angle of incidence)
can be justified by saying that a ray of light which bounces in this way traces a shorter
distance between two points than any neighboring ray which would bounce in some other
way. Finally,
Snell's law of refraction
can also be derived from Fermat's principle
if we only assume that the celerity of light
in a medium is inversely proportional to its index of refraction (n).
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n1 sin q1
=
n2 sin q2
The law of refraction was discovered by
Thomas
Harriot in July 1601 and independently by
Snell
(1621) and
Descartes
(1637) who was the first to publish it.
The Dutchman
Christiaan
Huygens (1678) was instrumental in attributing the
Law to his countryman, Snell...
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In a medium whose index of refraction is n,
the celerity of light is thus
c / n.
That relation remained controversial among scientists until 1850, when
Fizeau and Foucault carried out direct
comparisons of the celerities of light in air and water
(along the lines
Arago
had suggested in 1838). Fermat was vindicated.
Before that time, some physicists had wrongly
argued that light could be accelerated
when passing from a rarer to a denser medium!
Snell's Law Song
(2002 lyrics
by Marian McKenzie & Walter Fox Smith) in MP3 format:
Solo by Walter F. Smith
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Duo by Russ & Eli
(courtesy of PhysicsSongs.org).
(2008-09-04) Maupertuis Principle of
Least Action (1744)
Introducing a quantity which is minimized in Newtonian motion.
-
Adding up the masses of all bodies multiplied by their respective
speeds
and the distances they travel yields the quantity called action,
which is always the least possible in any natural motion.
Pierre-Louis Moreau de Maupertuis.
"Sur les lois du mouvement " (1746).
(2005-07-11) D'Alembert's Principle: Virtual Work
A synthetic statement equivalent to Newton's laws.
D'Alembert's principle is also known as the principle of virtual work.
(2008-09-03) Phase Space
The entire state of a classical system is described by its phase.
Newton's law (F = ma) involves the second-order derivatives
of the positions of the particles which compose a system.
The entire future evolution is thus determined if the positions
and their first-order derivatives are known.
Equivalently, the present situation of a classical system is fully described by
the generalized positions (qi) and generalized momenta
(pi) of all its "particles".
Those can be, respectively, the cartesian coordinates of the positions of the
particles and the corresponding cartesian coordinates of their momenta, but they
need not be... For example, if particles are constrained to move on
the surface of a sphere, their positions would be determined by their
latitudes and longitudes.
Newton's laws are such that the above description effectively encodes deterministically
not only the future but also the the past of the present system
(see Laplace's Demon).
Thus, the system evolves along definite trajectories where the aforementioned
points are called phases.
The term evokes a stage of development which may or may not be part of
a repeating cycle.
If the phase of a system is described by N positions (q)
and N momenta (p) then its evolution is governed
by 2N first-order differential equations.
Loosely speaking, N equations would be needed to
specify that the p's
are essentially the first order derivatives of the q's
and N other first-order differential equations would express
Newton's laws by stating that forces are the first-order derivatives
of momenta.
The positional information alone is traditionally called configuration.
The complete description of a phase thus entails specifying
both the configuration and the first derivatives of all configuration
constituents.
(2008-09-03) Velocities and Momenta
Two equivalent ways to specify first-order derivatives in phase space.
For a classical point-mass, the (vectorial) momentum p
is proportional to its (vectorial) velocity v.
The coefficient of proportionality (m) is the
mass of the particle.
In more general contexts
(e.g., Special Relativity)
the relations between momenta and velocities can be more complicated
but they remain one-to-one.
The one-to-one correspondence between momenta and velocities implies
that either type of variables can be used to describe a point
in phase space.
In general, those relations can be
specified by introducing two special (scalar) functions of the
phase;
the Lagrangian (L) and the Hamiltonian
(H)
which are Legendre transforms
of each other with respect to the two sets of variables which form
respectively the vectors v and p
(as positions q are held constant) :
H(p) = maxv { p.v
- L(v) }
and
L(v) = maxp { p.v
- H(p) }
Those theoretical definitions make the following practical relations hold :
| ¶ L |
= pi |
|
¶ H |
= vi |
 |
|
| ¶ vi |
¶ pi |
H + L = p . v
=
å i
p i v i
(2008-09-04) Relativistic Point-Mass
Momenta, Lagrangians and Hamiltonians in
Special Relativity.
One good way to see what the concepts of analytical mechanics entail is to
apply them in the simple context of a relativistic point-mass.
Our starting point will be the
expression of momentun known from the 4-vector formalism of
Special Relativity.
For a single point in free space, the abstract vectors
v and p introduced in the previous
section are just the ordinary three-dimensional velocity and relativistic momentum of
that point (of rest-mass m) :
This relation does indeed summarize the three relations:
| ¶ L |
= px |
|
¶ L |
= py |
|
¶ L |
= px |
|
|
|
| ¶ vx |
¶ vy |
¶ vz |
Provided we define L as follows (up to an irrelevant additive constant) :
Lagrangian of a free point-mass :
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| L =
- m c 2 |
Ö |
1 - v2 / c2 |
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A justification for this formula (without an exra constant)
is that it makes the action a Lorentz-invariant scalar.
Indeed, the observed action is the product of the observed Lagrangian L
by the observer's time. So, L ought to be proportional to the derivative
of the particle's proper time with respect to the observer's time.
Computing the Hamiltonian as H = p.v - L,
we obtain a familiar expression :
That would be simply E = m c2 if we were to define "m"
as the relativistic mass instead of the invariant
rest mass.
Indeed, the Hamiltonian corresponds to the usual concept of
total mechanical energy (which is conserved).
The Hamiltonian transforms like a mechanical energy
(the time-component of
an energy-momentum 4-vector) whereas the Lagrangian transforms like a
quantity of heat.
Those two things are very different physical beasts, indeed...
We may also invert our first relation and express v
as a function of p :
An explicit expression of
H in terms of p can be desirable:
Hamiltonian of a free point-mass :
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| H =
m c 2 |
Ö |
1 + p2 / (mc)2 |
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The next section shows how the canonical momenta
may differ substantially from the dynamical momenta
whose time-derivatives are the applied forces.
(2008-09-08) Relativistic Charge in an Electromagnetic Field
Lagrangian for the Lorentz force and associated canonical momentum.
In the electromagnetic field
specified by a quadripotential
(f/c, A)
a point-mass endowed with an electric charge q is governed by
the following Lagrangian.
We shall presently prove that this expression is valid by showing that it gives
the correct formula for the Lorentz force
exerted on the particle by the field.
Lagrangian of a charged particle :
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|
| L =
q ( A.v - f )
- m c 2 |
Ö |
1 - v2 / c2 |
|
| This yields : p = |
q A + |
m v |
|
| |
| |
| Ö |
1 - v2/c2 |
|
| H = p.v
- L = |
q f + |
m c2 |
|
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| |
| Ö |
1 - v2/c2 |
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Note that ( H/c , p ) is a 4-vector.
Let's see how the above actually encodes
the familiar expression of the Lorentz
force F
exerted by the electromagnetic field on the particle.
We first examine the coordinates along the x-axis only.
For a stationary Lagrangian,
the relevant Euler-Lagrange
equation is:
| ¶ L |
= |
d |
( |
¶ L |
) |
= |
d px |
= q |
d Ax |
+ Fx |
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|
|
|
| ¶ x |
d t |
¶ vx |
d t |
d t |
The last equality illustrates the distinction between the canonical
momentum p
(whose components are partial derivatives of the Lagrangian L
with respect to the velocities) and the dynamical momentum
whose (total) derivative with respect to time is the applied force
(namely, the Lorentz force whose expression
we are aiming to retrieve). Therefore:
| Fx = |
¶ L |
- q |
d Ax |
|
|
| ¶ x |
d t |
Let's expand each of those two terms... The above expression of L yields:
| ¶ L |
= |
q |
( |
- |
¶ f |
+ |
¶ Ax |
vx + |
¶ Ay |
vy + |
¶ Az |
vz |
) |
|
|
|
|
|
| ¶ x |
¶ x |
¶ x |
¶ x |
¶ x |
The generic expression dA =
¶t A dt +
¶x A dx +
¶y A dy +
¶z A dz gives:
| q |
d Ax |
= |
q |
( |
¶ Ax |
+ |
¶ Ax |
vx + |
¶ Ax |
vy + |
¶ Ax |
vz |
) |
|
|
|
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|
| d t |
¶ t |
¶ x |
¶ y |
¶ z |
Subtracting those two expansions, we can see that Fx is the x-coordinate of:
q [ (- grad f
-
¶A/¶t ) + v ´ rot A ]
As F and this vector have identical projections along
any arbitrary x-axis, they must be equal and we have indeed retrieved
the vectorial expression of the force
F = q [ E + v´B ] with the correct
expressions of the electromagnetic fields
E and B in terms of the potentials
f and A.
We may also express H
as an explicit function of the canonical momentum:
Hamiltonian of a charged particle :
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| H = q f +
m c 2 |
Ö |
1 + (p-qA)2 / (mc)2 |
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(2008-09-04) Lagrangian Mechanics (1788)
Practical definitions of the Lagrangian.
In classical mechanics, a Lagrangian consists of a kinetic term
(T) and a potential term (U).
L = T - U
The symbol T is related to the french term "travail" (work)
which was introduced in 1829 by the French physicist
Gaspard-Gustave Coriolis, who
proved
that the change in kinetic energy is always equal to the "work done"
by all applied forces (including friction).
On the other hand, U is only defined in the case of conservative
forces.
This ideal Lagrangian description is appropriate for particle physics
but it's no good at certain
observation scales (a concept advocated by
C.E. Guye)
where friction becomes relevant,
including ordinary human scale or long-term celestial mechanics
(involving "secular terms").
T is normally a function of some
quadratic expression of the velocities, involving a
generalized
tensor of inertia J :
T = ½ f ( v* J v )
As usual, we use a star (*) to denote the transpose (or rhe
transpose conjugate in the complex realm)
of any vectorial quantity.
Thus, the above just states that the argument of
f is some homogeneous quadratic polynomial of the velocity
components. J is a symmetric matrix
(in the complex realm, J
would be Hermitian, in order to make the argument of
f real).
Wikipedia :
Lagrangian
(2008-09-04) Hamiltonian Mechanics (1833)
In a
steady force field, the Hamiltonian is conserved.
Clearly, the differential
of the Hamiltonian H = p.v - L is:
| dH = |
å i |
[ |
p i dv i +
v i dp i
- |
¶ L |
dq i - |
¶ L |
dv i |
] |
- |
¶ L |
dt |
|
|
|
| ¶ q i |
¶ v i |
¶ t |
The definitions of the momenta make
the first and last bracketed terms cancel.
Furthermore, since L is assumed to satisfy a
principle of least action, the third term can be modified by
using the relevant Euler-Lagrange equation, namely:
| ¶ L |
= |
d |
( |
¶ L |
) |
= |
dpi |
|
|
|
|
| ¶ q i |
dt |
¶ vi |
dt |
Using this and the relation vi =
dq i / dt
the equation becomes:
| dH = |
å i |
[ |
dq i |
dp i - |
dp i |
dq i |
] |
- |
¶ L |
dt |
|
|
|
| dt |
dt |
¶ t |
The partial derivatives of H appear here as coefficients of the differentials:
Fundamental Equations of Hamiltonian Mechanics :
| ¶ H |
= |
d q i |
|
¶ H |
= - |
d p i |
|
|
|
|
| ¶ p i |
d t |
¶ q i |
d t |
|
Our previous expression also gives the total derivative of H
[each bracket vanishes since, clearly,
dqi dpi = dpi dqi ].
| d H |
= - |
¶ L |
|
|
| d t |
¶ t |
In a steady
force field, the right-hand-side vanishes.
So, the Hamiltonian H
(the total mechanical energy)
remains constant throughout the motion.
This can also be stated as a special case of
Noether's theorem:
If the force laws of a system are time-invariant, then its total
energy (H = E) is conserved.
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Siméon Poisson |
(2009-07-06) Poisson Brackets (1809)
An abstract synthetic view of classical mechanics.
La vie n'est bonne qu'à deux choses,
à
faire des mathématiques et à les professer.
Siméon Denis
Poisson
(1781-1840) X1798
Consider how a steady function A
(¶A/¶t = 0)
of the canonical Hamiltonian variables
pi and qi evolves with time
because of the actual motion itself:
| d A |
= |
å i |
¶ A |
|
d q i |
+ |
¶ A |
|
d p i |
|
|
|
|
|
| d t |
¶ q i |
d t |
¶ p i |
d t |
| |
= |
å i |
¶ A |
|
¶ H |
- |
¶ A |
|
¶ H |
|
|
|
|
| ¶ q i |
¶ p i |
¶ p i |
¶ q i |
| |
= |
{ A , H } |
The compact notation introduced in the last line is what is known as a
Poisson bracket
(French: Crochet de Poisson ).
The general definition is:
Poisson Brackets
| { A , B } |
= |
å i |
¶ A |
|
¶ B |
- |
¶ A |
|
¶ B |
|
|
|
|
| ¶ q i |
¶ p i |
¶ p i |
¶ q i |
|
The quantity A is a
constant of the motion if and only if
{ A , H } = 0
On September 25, 1925,
P.A.M. Dirac identified
Poisson brackets as the classical counterparts of quantum
commutators
(whose importance he had just discovered himself).
That was a Sunday and the library was closed.
So, Dirac had to wait until the next morning to verify his hunch, with
the help of what Jacobi had
written on the subject in 1841.
Poisson brackets share the following properties with quantum commutators:
{ A , B }
=
-
{ B , A } (anticommutativity)
{ A , u B + v C }
=
u { A , B }
+ v { A , C } (linearity)
{ A , BC }
=
{ A , B } C
+
B { A , C }
(product rule)
0 =
{ A , {B,C} }
+
{ B , {C,A} }
+
{ C , {A,B} }
The last relation
(known as Jacobi's identity)
is characteristic of a Lie algebra.
With just anticommutativity. linearity and the product rule, the following
equations fully specify the relationship of the Poisson brackets with the
canonical variables:
{ qi , pj }
= dij
(this is to say: 1 if i=j and 0 otherwise)
{ qi , qj }
=
{ pi , pj }
= 0
With all the above relations,
the original differential definition of the brackets can be retrieved
for polynomial functions of the canonical variables (and other smooth enough
functions, by continuity).
HINT: First retrieve the following relations
by induction on the degree of the polynomial A:
{ qi , A }
=
¶A/¶pi
{ pi , A }
=
- ¶A/¶qi
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Joseph Liouville |
(2008-09-04) Liouville's Theorem
Density in phase
is conserved in Hamiltonian space.
The quantum counterpart of Liouville's theorem
is the unitarity
of evolution with time (the norm of a ket
doesn't change as it evolves).
Liouville's theorem can also be construed as the statement that
information is conserved.
Joseph Liouville
(1809-1882; X1825). Liouville's measure.
Josiah Willard Gibbs
(1839-1903). "Density in phase".
Ludwig Boltzmann (1844-1906).
Preservation
of Phase Volume &
Liouville's
Theorem for non-Hamiltonian systems by Mark Tuckerman
Application to Electron Orbits
in the Earth's Magnetic Field by W. F. G. Swann (1933)
Liouville's theorem
and the velocity field of galaxies by
Arthur
Rex Rivolo (1985)
Wikipedia :
Liouville's
Theorem
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Emmy Noether |
(2003-05-26) Noether's Theorem (1915)
The German mathematician Emmy Noether (1882-1935)
established this deep result (Noether's Theorem) in 1915:
For every continuous symmetry of the laws of physics,
there's a conservation law,
and vice versa.
The theorem is true of any physical theory based on a Lagrangian formalism,
including discrete classical systems of finitely many particles and the
classical fields discussed below.
It also applies to the quantum counterparts of those...
A simple proof is given elsewhere on this site
in the basic case of a classical system with finitely many degrees of freedom...
Formally, a continuous symmetry of such a system is expressed by stating that
its Lagrangian L(q,v,t)
is unchanged (to the first order in
e ) if each component
q i is replaced by:
q i +
e h i
It's understood that, for any constant e,
the [new] component
vi remains equal to the time-derivative of the [new] component
q i ...
If such a symmetry exists, then Noether's theorem
states that the following quantity is a constant of motion :
Conserved Noether charge
| å i |
h i |
¶ L |
|
| ¶ vi |
|
The conservation of [Hamiltonian] energy for a system with a steady
Lagrangian (i.e., a Lagrangian which does not explicitely depend on time)
is often construed as a special case of Noether's theorem, although it's
established differently.
Electric charge is the conserved quantity
obtained for a Lagrangian which is invariant under multiplication
of all its arguments by a (complex) phase.
In the case of a scalar field with complex values,
the following real 4-vector field turns out to be the
current density associated with the field
(the time-component of that is the charge density ).
i/2 [
y ¶my*
-
¶my y* ]
Thus, loosely speaking, charge has the same type of algebraic expression
as angular momentum and ends up being quantized
for the same reasons.
E. Noether's 1918 paper,
translated by M.A. Tavel (1971)
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Noether's
biography
MathPages
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Wikipedia
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Noether's
Discovery |
Noether's Theorem
in a Nutshell by John Baez
(2008-09-06) Classical Field Theory
& Lagrangian Density
Let the q-coordinates be the values of a field at different points in space.
For the above discrete mechanical systems,
the value of the Lagrangian L was a function of time (t)
the N position coordinates (q)
and their derivatives (v).
For a continuous field
y,
the values of the field at every point of 4D-space play the role
of the q
coordinates and its partial spacetime derivatives
play the role of the velocities v.
The Lagrangian itself would be equivalent to the integral over 3D-space of
the following Lagrangian density L whose integral over
space and time will thus play the role of an
action which must be stationary :
L =
L ( y ,
y0 ,
y1 ,
y2 ,
y3 )
where ym
= ¶m y
Note the subtle point in notation:
ym
is just the name of a particular argument
of the Lagrangian function.
It makes perfect sense to consider the partial derivative
of L with respect to
ym.
It makes no more sense to consider a derivative
"with respect to"
¶m y
than it would to speak of the derivative of
f (x) "with respect to" 2
to denote the value of d f /dx
when x = 2.
A proper Lagrangian density must be a relativistic scalar.
So, if y is assumed to be a
scalar itself, then one example of a proper L
would be of the form:
L = f ( y ,
¶my
¶ m y )
[ Einstein's summation convention
is used here. ]
The equivalent of the Euler-Lagrange equation is:
| ¶ L |
= ¶m |
( |
¶ L |
) |
|
|
| ¶ y |
¶ ym |
|
The role of the conjugate momenta is played by a 4-vector density field
p :
| pm
= |
¶ L |
|
| ¶ ym |
The component p0 / c
can be denoted py
and called field momentum.
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