The laws of physics can be deduced from the principle of least action, which
says that the action S is minimized.
The lagrangian L is the action per unit of time.
The lagrangian density Λ is the lagrangian per unit of volume.
The lagrangian density multiplied by a 4-dimensional volume of time-space
gives an action.
The fact that the action is minimized is expressed by
δ S = 0, which is equivalent to
d/dt (∂ L/∂ (dq/dt)) - ∂ L / ∂ q = 0.
Proof :
δ S = δ ∫t1t2 L dt = ∫t1t2 δ L dt
= ∫t1t2 ((∂ L/∂ q) δ q + (∂ L/∂ (dq/dt))
δ(dq/dt)) dt
∫t1t2 (∂ L/∂ (dq/dt)) δ(dq/dt) dt
= ∫t1t2 (∂ L/∂ (dq/dt)) d(δ q)/dt dt
= ∫t1t2 d/dt ((∂ L/∂ (dq/dt)) δ q) dt
- ∫t1t2 d/dt (∂ L/∂ (dq/dt)) δ q dt
because d/dt ((∂ L/∂ (dq/dt)) δ q) dt
= d/dt (∂ L/∂ (dq/dt)) δ q dt
+ (∂ L/∂ (dq/dt)) ((∂(δ q))/dt) dt
From δ S =
∫t1t2 ((∂ L/∂ q) δ q + (∂ L/∂ (dq/dt))
δ(dq/dt)) dt
and
∫t1t2 (∂ L/∂ (dq/dt)) δ(dq/dt)
= ∫t1t2 d/dt ((∂ L/∂ (dq/dt)) δ q) dt
- ∫t1t2 d/dt (∂ L/∂ (dq/dt)) δ q dt
follows :
δ S = [(∂ L/∂ (dq/dt)) δ q]t1t2 +
∫t1t2 (∂ L/∂ q - d/dt (∂ L/∂ (dq/dt)))
δ q dt
[(∂ L/∂ (dq/dt)) δ q]t1t2 = 0 because
δ q(t1) = δ q(t2)
δ S = 0 for all variation of δ q
Then d/dt (∂ L/∂ (dq/dt)) - ∂ L/∂ q = 0.
Other proof :
δ L = ∂ L/∂ q δ q + ∂ L/∂ (dq/dt)
δ (dq/dt) + ∂ L/∂ t δ t
δ t = 0 => δ L = ∂ L/∂ q δ q +
∂ L/∂(dq/dt) d/dt δ q
δ S = ∫ ∂ L/∂ q δ q dt
+ ∫ ∂ L/∂(dq/dt) d/dt δ q dt
∫t1t2 ∂ L/∂(dq/dt) d δ q
= [∂ L/∂(dq/dt) δ q]t1t2
- ∫t1t2 d/dt ∂ L/∂ q δ q dt
δ S = ∫ (∂ L/∂ q - d/dt ∂ L/∂(dq/dt))
δ q dt = 0
∂ L / ∂ q - d/dt ∂ L/∂(dq/dt) = 0
If we define the momentum p = ∂ L / ∂ (dq/dt), this equation can be written : dp/dt = ∂ L / ∂ q.
"Lagrangian mechanics works in n-dimensional configuration space, which includes all parameters, defining a static state of mechanical system (coordinates of particles, orientation of rigid body etc). A point x in this space draws a curve x(t) in evolution. For such curves a functional A[x(t)], called action, is introduced. Only those curves, on which the action reaches an extremum, correspond to real evolution.
Usually consideration is restricted to functionals of the form
A[x(t)] = ∫ dt L(x,dx/dt)
with Lagrangian L dependent only on coordinates and velocities. The condition of extremum for the action leads to Lagrange-Euler equations
d/dt (∂ L/∂(dx/dt)) - ∂ L/∂ x = 0
This is (normally) a system second order differential equations, with solutions uniquely defined by initial coordinates and velocities x(0), dx/dt (0).
In Hamiltonian mechanics the state of the system is described by a point (x,p) in 2n-dimensional phase space. The dynamics is defined by a function H(x,p), called Hamiltonian, via equations:
dx/dt = ∂ H / ∂ p , dp/dt = - ∂ H / ∂ x
Look here for alternative formulation of the same theory.
Transition from Lagrangian to Hamiltonian mechanics is performed by Legendre transformation. It defines the momenta and Hamiltonian as:
p(x,dx/dt) = ∂ L / ∂ (dx/dt) , H(x,p) = p dx/dt - L .
The Hamiltonian depends on coordinates and momenta, so one should express the velocities via momenta, inverting the definitions of momenta: dx/dt = dx/dt(x,p) and substitute the result into Hamiltonian. "
See also Poisson brackets and symplectic form - http://viswiz.gmd.de/~nikitin/course/node3.html
"A binary operation { , } , called Poisson brackets, is introduced for functions on the phase space:
{A, B} = ∂ A/∂ x ∂ B/∂ p - ∂ A/∂ p ∂ B/∂ x
In terms of Poisson brackets the Hamiltonian equations can be rewritten as
dx/dt = {x, H} , dp/dt = {p, H}.
Exercise: prove, that for any function A(t,x,p) the following identity (Liouville equation) is valid:
dA/dt = ∂ A/∂ t + {A, H}. "
To determine the laws of physics for some phenomenon, one can express the action, the lagrangian or the lagrangian density corresponding to this phenomenon and apply this equation to it.
Example : particle moving in a gravitational field in non-relativistic approximation (v<<c) :
Remark : mimimizing the action means maximizing the potential energy. It means that the trajectory is higher that the straightforward line, it is REPELLED by the mass. Generally one consider that the particles are ATTRACTED by masses. This paradox is because generally we consider the initial position and the initial speed as fixed, and the trajectory is lower than the straightforward line corresponding to this initial position and speed, but here the initial and final positions are fixed, and in this case the trajectory is higher than the straight line between these positions.
Conservation on energy deduced from the least action principle :
dL/dt = ∑\a ∂ L/∂ qa dqa/dt
+ ∂ L/∂(dqa/dt) d2 qa/dt2
∂ L/∂ qa = d/dt ∂ L/∂(dqa/dt)
dL/dt = ∑a d/dt ∂ L/∂(dqa/dt) dqa/dt
+ ∂L/∂(dqa/dt) d2 qa/dt2
= ∑a d/dt (∂ L/∂(dqa/dt) dqa/dt
d/dt (∑a ∂ L/∂(dqa/dt) dqa/dt - L) = 0
E = ∑a ∂ L/∂(dqa/dt) dqa/dt - L
dE/dt = 0
Path integral :
x(t1) -> x(t2)
ei/hbar S
Particules et lois de la physique, par Richard Feynman et Steven Weinberg
Conférence de Steven Weinberg : A la recherche des lois ultimes de la physique
Dans un univers mythique où les seules particules sont l'électron et le photon,
la densité de lagrangien est :
L = - psi bar (gamma^mu d/dx^mu + m) psi
- 1/4 (dA_nu/dx^mu - da_mu/dx^nu)^2
+ ieA_mu psi bar gamma^mu psi
- mu (dA_nu/dx^mu - dA_mu / dx_nu) psi bar sigma^mu^nu psi
- G psi bar psi psi bar psi + ...
avec :
Théorie des cordes :
métrique g_alpha_beta(sigma) de la surface décrivant l'évolution de la corde dans le temps
action I[x,g] = 1/2 intégrale (g(sigma))^1/2 g^alpha^beta(sigma)
dx^mu(sigma)/dsigma^alpha
dx^nu(sigma)/dsigma^beta d^2 sigma.
(1 0 0 0) (0 -1 0 0) (0 0 -1 0) (0 0 0 -1)Other possible metrics are (- + + +), or (+ + + +) with coordinates (ct ix iy iz) or (ict x y z).
The scalar product of two vectors uμ = (u0 u1 u2 u3) = (ut ux uy uz) and vν is gμν uμ vν = ∑μ=03 ∑ν=03 gμν uμ vν.
The electromagnetic field is a gauge field associated with U(1) which
is simply the circle.
It can also be viewed as gravitation in the fifth dimension of a 5 dimensional
space obtained by multiplying the 4 dimensional space-time by the circle U(1)
which is compactified to the Planck length. This is Kaluza-Klein theory.
The phase is a point on the circle U(1).
The potential Am = (At=V Ax Ay Az) is the gradient of the phase.
In the general case, the potential is Aam where a
is one of the compactified dimensions of the phase space, but in the case of
the electromagnetic field, the circle U(1) has only 1 dimension, so
this indice a is useless.
The field Fmn is dmAn-dnAm
where d is the partial derivation. It means that Fmn is the
dephasage of an infinitesimal loop in the plane mn.
S(t+dt,x) <--- -Ax(t+dt,x+dx/2) --- R(t+dt,x+dx) | ^ | | -At(t+dt/2,x) +At(t+dt/2,x+dx) | | t V | ^ P(t,x) ------- +Ax(t,x+dx/2) -----> Q(t,x+dx) | +--->x
In the general case the field is :
Fmna
= dmAna
- dnAma
+ fbcaAmbAnc
where m and n are dimensions of space-time (t,x,y,z), a, b, c are
dimensions of phase space (compactified dimensions) and fbca are the structure constants.
The field satisfies dmFnp + dpFmn + dnFpm = 0 or emnpq dnFpq = 0.
The potential Am is determined by the current density of the sources Jm = ρ vm (where ρ is the charge density and v the speed) by the relation : [] Am = μ0Jm with [] = 1/c d2/dt2 - d2/dx2 - d2/dy2 - d2/dz2.
The field is also directly determined from the sources by
dnFmn = -µ0Jm
A particle moving in an electromagnetic field is submitted to a force. f = e F v
( γ/c f.v ) (0 Ex/c Ey/c Ez/c) (γ c ) ( γ fx ) = e (Ex/c 0 Bz -By ) (γ vx) ( γ fy ) (Ey/c -Bz 0 Bx ) (γ vy) ( γ fz ) (Ez/c By -Bx 0 ) (γ vz)
With a metric (+ - - -), Fmn =
(0 Ex/c Ey/c Ez/c) (-Ex/c 0 -Bz By ) (-Ey/c Bz 0 -Bx ) (-Ez/c -By Bx 0 )
Fmn =
(0 -Ex/c -Ey/c -Ez/c) (Ex/c 0 -Bz By ) (Ey/c Bz 0 -Bx ) (Ez/c -By Bx 0 )The sign before E depends on the metrics (+---) or (-+++)
Notations :
g'_mn = ( g_mn - Φ A_m A_n - Φ A_m ) ( - Φ A_n - Φ ) g'^mn = ( g^mn - A^m ) ( - A^n - 1/Φ + A^2)
Conventions : mu' = { μ, m } where
E'^a'_mu' = ( e^a_μ 0 ) ( A_μ^k E^n_k E^n_m ) e'_a'^mu' = ( e_a^μ -e_a^ρ A_ρ^m ) ( 0 E^n_m )The metric is :
g'_mu'nu' = ( g_m_n + G_m_n A_μ^m A_ν^n A_μ^m G_m_n ) ( A_ν^n G_m_n G_mn ) g'^mu'nu' = ( g^μ^ν -A^μ^n ) ( -A^ν^m G^mn + A^ρ^m A_ρ_n) or more symetrically : g'^mu'nu' = ( g^μ^ν -A_ν^n g^μ^ν ) ( -A_μ^m g^μ^ν G^mn + g^μ^ν A_μ^m A_ν^n )
A charged particle moves in the fifth dimension with a speed proportional to
its charge divided by its mass.
Two particles which have the same charge move in the same direction in the
fifth dimension. According to special relativity, their mass increase
because of this speed, so the attractive force should be greater, but
from the point of view of these particles, they do not move.
The increasement of the attractive force is compensed by a repulsive force :
the electrostatic force.
According to the gauge field theory, the force fi = m ai which acts on a
particle of mass m, charge q and speed vj in an electromagnetic field Fij is : fi = q vj Fij.
According to Kaluza Klein theory, the particle follows geodesics of
5-dimensional space-time, which means that the acceleration ai = d2 x / dt2
= - Γijk vj vk, then fi = m ai = - m Γijk vj vk.
The affine connection Γ is :
Γijk = 1/2 gil (∂j glk + ∂k glj
- ∂l gjk)
Fij appears in Γiju and Γiuj where u is the compactified
Kaluza Klein dimension.
Action S = -mc ∫ ds
ds2 = c2dt2 - (dx2+dy2+dz2)
ds = c 2√(1-v2/c2)dt
L = 1/2 mv2 + e A.v - eV
where
S = - ∑ mc ∫ ds - ∑ e ∫ Amdxm
+ 1/2μ0 ∫t1t2dt
∫(E2/c2-B2)d τ
= - ∑ mc ∫ ds - ∑ e ∫ Amdxm
+ 1/2μ0 ∫t1t2dt
∫(-1/2 FmnFmn)d τ
L = -1/4 FmnFmn + AmJm
L = √(-g) (1/4 R - 1/2 gij(φ) ∂μφi ∂μφj -1/4 mIJ(φ) FμνI FμνJ - 1/8 εμνρσ aIJ(phi) FμνI FρσJ)
With metrics (- + + +) :
L = -1/4 Fμν Fμν + Jμ Aμ + Lmatter
L = -1/4 Fμν Fμν + ψ bar (γμ (∂μ
+ i e Aμ) + m) ψ
Jμ = ∂ L / ∂ Aμ = - i e ψ bar γμ ψ
V(t) = + i e ∫ d3 x (ψ bar(x,t) γμ ψ(x,t)) aμ(x,t)
+ Vcoul(t)
See also Lagrangian Theory of Fields and Particles
--> postscript only (10 pages)
This is a discussion of how to do particle and field Lagrangians. The particle's action principle seeks to maximise the proper time in a trajectory between two events in spacetime. The field's action principle seeks to minimise the integral over whatever Lorentz scalar can be constructed from the field, given its symmetry, and with the proviso that the resulting Hamiltonian be positive definite. The interaction arises from simple coupling, to the particle itself in the case of a scalar field, and through the particle's four velocity in the case of a vector field. We will see that the vector field coupled to the particle's motion in this way must give rise to an antisymmetric force field, which by symmetry of interaction forms the basis for the construction of Lorentz scalars. A very fundamental result is that like charges attract for scalar field interactions, while they repel for the vector field. The vector field, of course, is the basis for electromagnetism. An often repeated error concerning the particle's equation of motion in a scalar field is cleared up -- since the four velocity is a unit vector, the force must be projected orthogonal to the world line in order to have consistent dynamics. It is interesting to note that most of the properties of electrodynamics, including the repulsion or attratction of charges, are necessary consequences of linearity, masslessness, and the fact that it is a vector field.
Tips and tricks that you learn from this include how to decide what the signs should be for the free particle and field parts of the Lagrangian, how to manipulate the variation of the proper time, why the sign of the interaction part is arbitrary, how to convert back and forth between a Lagrangian and a Lagrangian field density and likewise between particles and ensemble densities, how to make to correspondence to the more common nonrelativistic model, and how to get all the factors of the speed of light, c, properly when you don't assume natural units with c equal to unity. These are all things I had trouble with and so I wanted to write them up together in a coherent whole. The result should be useful to any physicist or physics watcher interested in the basis for modern theory.
--> Back to the...
See http://www.uwm.edu/~norbury/gr/node37.html
In both special and general relativity we always seek covariant equations in which space and time are given equal status. The Euler-Lagrange equations (4.6) are clearly not covariant because special emphasis is placed on time via the dqi/dt and d/dt(∂ L/∂ (dqi/dt)) terms.
Let us replace the qi by a field φ ≡ φ (x) where x ≡ (t, x) . The generalized coordinate q has been replaced by the field variable φ and the discrete index i has been replaced by a continuously varying index x. In the next section we shall show how to derive the Euler-Lagrange equations from the action defined as
S ≡ ∫ L dt | (148) |
S ≡ ∫ L d4 x = ∫ L d3 x dt | (149) |
∂μ ∂ L / ∂ (∂ μ φ ) - ∂ L/∂ φ = 0 | (150) |
In analogy with the canonical momentum in equation (4.5) we define the covariant momentum density
Π μ ≡ ∂ L/∂ (∂μ φ ) | (151) |
∂μ Πμ = ∂ L/∂ φ | (152) |
Π ≡ Π0 = ∂ L/∂ dφ/dt | (153) |
Tμν ≡ Πμ ∂ν φ - gμν L | (154) |
Πα = ∂α φ | (155) |
Π = dφ/dt | (156) |
(p2 - m2 )φ = 0 | (157) |
The energy momentum tensor is
Therefore the Hamiltonian density is
H≡ T00=(dφ/dt)2-1/2(∂α
φ ∂α φ -m2 φ 2)
which becomes [31]
where we have relied upon the results of Section 3.4.1.