Analytical Mechanics is a mathematical reformulation of Newtonianmechanics. Its physical content is the same, but its mathematical structurehas interesting features. It can also describe therelativistic dynamics of particles and fields.
Analytical Mechanics
(2008-09-04) 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 postulating that lightalways 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 shorterdistance between two points than any neighboring ray which would bounce in some otherway. Finally,Snell's law of refractioncan also be derived from Fermat's principleif we only assume that the celerity of lightin a medium is inversely proportional to its index of refraction (n).
n1 sin1 = n2 sin2
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 directcomparisons of the celerities of light in air and water (along the linesAragohad suggested in 1838). Fermat was vindicated.
(2008-09-04) Introducing a quantity which is minimized in Newtonian motion.
(2005-07-11) A synthetic statement equivalent to Newton's laws.
D'Alembert's principle is also known as the principle of virtual work.
(2008-09-03) The entire state of a classical system is described by its phase.
Newton's law (F = ma) involves the second-order derivativesof 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 bythe generalized positions (qi) and generalized momenta (pi) of all its "particles".
Those can be, respectively, the cartesian coordinates of the positions of theparticles and the corresponding cartesian coordinates of their momenta, but theyneed not be... For example, if particles are constrained to move onthe surface of a sphere, their positions would be determined by theirlatitudes and longitudes.
Newton's laws are such that the above description effectively encodes deterministicallynot only the future but also the the past of the present system (seeLaplace's Demon). Thus, the system evolves along definite trajectories where the aforementionedpoints are called phases. The term evokes a stage of development which may or may not be part ofa repeating cycle.
If the phase of a system is described by N positions (q) and N momenta (p) then its evolution is governedby 2N first-order differential equations. Loosely speaking, N equations would be needed tospecify that the p's are essentially the first order derivatives of the q's and N other first-order differential equations would expressNewton's laws by stating that forces are the first-order derivativesof momenta.
The positional information alone is traditionally called configuration. The complete description of a phase thus entails specifyingboth the configuration and the first derivatives of all configurationconstituents.
(2008-09-03) 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 complicatedbut they remain one-to-one.
The one-to-one correspondence between momenta and velocities impliesthat either type of variables can be used to describe a pointinphase space.
In general, those relations can bespecified by introducing two special (scalar) functions of the phase; the Lagrangian (L) and the Hamiltonian (H) which areLegendre transformsof each other with respect to the two sets of variables which formrespectively the vectors v and p (as positions q are held constant) :
Those theoretical definitions make the following practical relations hold :
L
= pi
H
= vi
vi
pi
H + L = p . v = i p i v i
(2008-09-04) Practical definitions of the Lagrangian.
In classical mechanics, a Lagrangian consists of a kinetic term (T) and a potential term (U).
L = T U
T is normally a function of some quadratic expression of the velocities, involving ageneralized tensor of inertiaJ :
T = ½ f (v*Jv )
(2008-09-04) In a steady force field, the Hamiltonian is conserved.
Clearly, thedifferentialof the Hamiltonian H = p.v - L is:
dH =
i
p i dv i + v i dp i
L
dq i
L
dv i
L
dt
q i
v i
t
Thedefinitions of the momenta make the first and fourth termsin the square bracket cancel each other. Furthermore, since L is assumed to satisfy a principle of least action, the third term can be modified byusing 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 =
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 [because each square bracket vanishes since, clearly, qi' pi' = pi' qi' ].
d H
=
L
dt
t
In a steady force field, the right-hand-side vanishes. So, the Hamiltonian H (the total mechanical energy) remains constant throughout the motion.
That resembles a special case of Noether's theorem: If the force laws ofa system are time-invariant, then its total energy (H = E) is conserved. So, Hamiltonian energy is the conserved quantity linked to time-symmetry.
(2009-07-06) An abstract synthetic view of classical mechanics.
La vie n'est bonne qu'à deux choses, àfaire des mathématiques et à les professer. Siméon DenisPoisson (1781-1840) X1798
Consider how a steady function A (A/t = 0) of the canonical Hamiltonian variables pi and qi evolves with timebecause of the actual motion itself:
dA
=
i
A
dq i
+
A
dp i
dt
q i
dt
p i
dt
=
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
=
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
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 asJacobi's identity) is characteristic of a Lie algebra. With just anticommutativity. linearity and the product rule, the followingequations fully specify the relationship of the Poisson brackets with thecanonical variables:
qi , pj = ij (this is to say: 1 if i=j and 0 otherwise) qi , qj = pi , pj = 0
With the above relations,the original differential definition of the brackets can be retrievedfor polynomial functions of the canonical variables (and other smooth enoughfunctions, by continuity). : First retrieve the following relations, by induction on the degree of the polynomial A.
qi , A = A/pi pi , A = A/qi
(2008-09-04) Density in phase is conserved in Hamiltonian space.
Thequantum counterpart of Liouville's theoremis the unitarity of evolution with time (the norm of aketdoesn't change as it evolves). Liouville's theorem is construed as the statement that information is conserved.
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 theclassical fields discussed below. It also applies to thequantum counterparts of those...
A simple proof is givenelsewhere on this sitein 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 thatits Lagrangian L(q,v,t) is unchanged (to the first order in ) if each component q i is replaced by:
q i + h i
It's understood that, for any constant ,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
i
h i
L
vi
The conservation of [Hamiltonian] energy for a system with a steadyLagrangian (i.e., a Lagrangian which does not explicitely depend on time) is often construed as a special case of Noether's theorem, although it'sestablished differently.
Electric charge is the conserved quantityobtained for a Lagrangian which is invariant under multiplicationof all its arguments by a (complex) phase.
In the case of a scalarfield with complex values,the following real 4-vector field turns out to be thecurrent density associated with the field (the time-component of that is the charge density ).
i/2 [ * * ]
Thus, loosely speaking, charge has the same type of algebraic expressionas angular momentum and ends up being quantizedfor the same reasons.
(2008-09-04) Momenta, Lagrangians and Hamiltonians in Special Relativity.
One good way to see what the concepts of analytical mechanics entail is toapply them in the simple context of a relativistic point-mass.
Our starting point will be theexpression 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 theprevioussection are just the ordinary three-dimensional velocity and relativistic momentum ofthat point (of rest-mass m) :
p =
mv
1 -v2/c2
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 :
L = m c 2
1v2 / c2
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 derivativeof 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 :
H = E =
m c2
1 -v2/c2
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 oftotal mechanical energy (which is conserved).
The Hamiltonian transforms like a mechanical energy (the time-component ofan energy-momentum 4-vector) whereas the Lagrangian transforms like aquantity of heat. Those two things are very different physical beasts...
We may also invert our first relation and express v as a function of p :
v =
p / m
1 +p2/(mc)2
An explicit expression of H in terms of p can be desirable:
Hamiltonian of a free point-mass :
H = m c 2
1 +p2 / (mc)2
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) Lagrangian for the Lorentz force and associatedcanonical momentum.
In theelectromagnetic fieldspecified by aquadripotential (/c, A) a point-mass with electric charge q is governed bythe following Lagrangian. We'll prove that this expression is valid by showing that it givesthe correct formula for the Lorentz force exerted on the particle by the field.
Lagrangian of a charged particle :
L = q (A.v ) m c 2
1v2 / c2
This yields : p =
qA +
mv
1 -v2/c2
H = p.v L =
q +
m c2
1 -v2/c2
Note that ( H/c ,p ) is a 4-vector. Let's see how this encodes the familiar expression of the LorentzforceF exerted by the electromagnetic field on the particle. We first examine the coordinates along the x-axis only.
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 momentumwhose (total) derivative with respect to time is the applied force (namely, the Lorentz force whose expressionwe are aiming to retrieve). Therefore:
Fx =
L
q
dAx
x
dt
Let's expand each of those two terms... The above expression of L yields:
L
=
q
+
Ax
vx +
Ay
vy +
Az
vz
x
x
x
x
x
Generically dA = tA dt +xA dx +yA dy +zA dz. Therefore:
q
dAx
=
q
Ax
+
Ax
vx +
Ax
vy +
Ax
vz
dt
t
x
y
z
Subtracting those two expansions, we see that Fx is the x-coordinate of:
q [ (-gradA/t ) + vrot A ]
As F and this vector have identical projections alongany arbitrary x-axis, they must be equal and we have indeed retrievedthe vectorial expression of the force F = q [ E + vB ] with thecorrectexpressions of the electromagnetic fields E and B in terms of the potentials and A.
We may also express H as an explicit function of the canonical momentum:
Hamiltonian of a charged particle :
H = q + m c 2
1 + (p-qA)2 / (mc)2
(2008-09-06) 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 , the values of the field at every point of 4D-space play the roleof the q coordinates and its partial spacetime derivativesplay the role of the velocities v. The Lagrangian itself would be equivalent to the integral over 3D-space ofthe following Lagrangian density L whose integral overspace and time will thus play the role of an action which must be stationary :
L = L ( , 0 , 1 , 2 , 3 ) where =
Note the subtle point in notation: is just the name of a particular argumentof the Lagrangian function. It makes perfect sense to consider the partial derivativeof L with respect to . It makes no more sense to consider a derivative"with respect to" 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 is assumed to be a scalar itself, then one example of a proper L would be of the form:
