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Final Answers
© 2000-2014   Gérard P. Michon, Ph.D.

Analytical Mechanics
Rational Mechanics  &  Field Theory

 Joseph-Louis Lagrange  
 1736-1813  Jean-le-Rond d'Alembert 
 1717-1783  Pierre-Louis Moreau de Maupertuis 
 1698-1759  Isaac Newton  
 1643-1727  Pierre de Fermat 

Related articles on this site:

Related Links (Outside this Site)

Principle of Virtual Work
Lagrangian and Hamiltonian Mechanics  by  Kevin S. Brown.
Young-Laplace equation (1805 & 1806)  as derived by Gauss (1830).
Analytical Mechanics and Classical Field Theory  by  Mats Wallin  (KTH)

D'Alembert's Principle   |   Analytical Mechanics   |   Lagrangian Mechanics   |   Hamiltonian Mechanics
Canonical quantization (and "second quantization")

DMOZ: Lagrangian and Hamiltonian Mechanics

Video :   Classical Field Theory  by  Leonard Susskind1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
Principle of Minimal Action and Kinetic Energy



Analytical Mechanics  is a mathematical reformulation of Newtonian mechanics.  Its physical content is the same, but its mathematical structure has interesting features.  It can also describe the relativistic dynamics of particles and fields.

Originally, the mathematical formalism may have been invented merely to unify the patterns which appear in the solutions of basic problems in Newtonian mechanics.  However, it allows a deeper understanding of the structure of classical mechanics and also of physical laws in general.  The most fundamental part of this structure has served as the basis for the current formalism of  quantum mechanics developed by P.A.M. Dirac and others:  the quantum commutators were directly inspired by  Poisson brackets...


Analytical Mechanics

 Pierre de Fermat 
 1601-1665 (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).

 Willebrord Snell 
 Rene Descartes 

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...

   Snell's Law

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   |   Duo by Russ & Eli   (courtesy of PhysicsSongs.org).

 Pierre-Louis Moreau de 
 Maupertuis (1698-1759) (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).

 Come back later, we're
 still working on this one...

 Jean-le-Rond d'Alembert 
 1717-1783 (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

 Come back later, we're
 still working on this one...

(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
Vinculum Vinculum
vi pi

H  +  L     =     p . v     =     å i   p i  v i

 Albert Einstein 
 1879-1955 (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) :

p   =   m v
Ö 1 - v2/c2

This relation does indeed summarize the three relations:

 L    =    px    L    =    py    L    =    px
Vinculum Vinculum Vinculum
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   Ö 1 - v2 / 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 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 :

H   =   E   =     m c2
Ö 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 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 :

v   =   p / m
Ö 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   Ö  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)   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 with electric charge  q  is governed by the following Lagrangian.  We'll 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 :
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
Ö 1 - v2/c2

Note that  ( H/c , p )   is a  4-vector.  Let's see how this  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
Vinculum Vinculum Vinculum Vinculum Vinculum
 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
Vinculum Vinculum
 x d t

Let's expand each of those two terms...  The above expression of  L  yields:

 L       =       (   -    f    +     Ax   vx   +     Ay   vy   +     Az   vz   )
Vinculum Vinculum Vinculum Vinculum Vinculum
 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       =       (    Ax    +     Ax   vx   +     Ax   vy   +     Ax   vz   )
Vinculum Vinculum Vinculum Vinculum Vinculum
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 QED

We may also express  H  as an explicit function of the canonical momentum:

Hamiltonian of a charged particle :
H   =   q f  +  m c 2   Ö  1 + (p-qA)2 / (mc)2

 Joseph-Louis Lagrange 
 1736-1813 (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).

 Come back later, we're
 still working on this one...

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 
Vinculum Vinculum Vinculum
q i v i  t

The definitions of the momenta  make the term in the bracket cancel with the last one.  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
Vinculum Vinculum Vinculum Vinculum
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 
Vinculum Vinculum Vinculum
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
Vinculum Vinculum Vinculum Vinculum
p i d t q i d t

Our previous expression also gives the total derivative of  H  [each bracket vanishes since, clearly,  dqdpi = dpdq].

d H    =    -   L
Vinculum Vinculum
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.

   Simeon Poisson
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
Vinculum Vinculum Vinculum Vinculum Vinculum
d t  q i d t  p i d t
     =     å i      A     H    -     A     H
Vinculum Vinculum Vinculum Vinculum
 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
Vinculum Vinculum Vinculum Vinculum
 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

   Joseph Liouville
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).

 Come back later, we're
 still working on this one...

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

  Emmy Noether
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

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)   |   Noether's biography
MathPages   |   Wikipedia   |   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 ( yy0y1y2y3 )
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 ( ymy  m y )

[ Einstein's summation convention is used here. ]

The equivalent of the Euler-Lagrange equation is:

 L     =    m  (   L  )
Vinculum Vinculum
y ym

The role of the conjugate momenta is played by a 4-vector density field  p :

pm     =      L

The component  p0 / c  can be denoted  py  and called  field momentum.

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