Hamiltonian Formalism for the Vlasov-Maxwell System

Hamiltonian Formalism for the Vlasov-Maxwell System

Emmanuel Fr´ enod

^∗

December 6, 2010

Abstract - Keywords -

Contents

I Towards Hamiltonian description of a charged particle 2

1 Departure point (in terms of scientific knowledge) 3

2 Status of mathematical objects in game I 4

2.1 D’Alembert’s version of Newton’s Law . . . 4

2.2 Status of the forceIFand of the momentumIMin a given pointxofTxX . . . 4

2.3 Status of the forceFand of the momentum ˜M . . . 5

2.4 Status of the electric field . . . 6

2.5 Status of the magnetic field . . . 6

2.6 On Lorentz’s Force writing . . . 7

2.7 Remark on notation, Pushforward and Pullback . . . 8

2.8 Faraday’s Law from differential form point of view . . . 9

2.9 On Ampere’s Theorem compatibility with differential form viewpoint . . . 9

2.10 Status of the Current Density . . . 10

2.11 Ampere’s Theorem from differential form point of view . . . 10

2.12 Magnetic Field divergence free equation from differential form point of view . . . 10

2.13 Poisson’s Equation from differential form point of view . . . 10

2.14 Status of the Charge Density . . . 11

2.15 Maxwell System in the differential forms framework . . . 11

3 From Newton to Lagrange 11 3.1 Maxwell System with Electric and Magnetic Potentials . . . 11

3.2 On momentum choice . . . 12

3.2.1 Momentum possibly associated with a differential 1-form . . . 12

3.2.2 Time derivative of the momentum . . . 12

3.2.3 Pullback, Lie Derivative and Cartan’s Formula . . . 17

3.2.4 Another expression of the time-derivative of the momentum . . . 18

3.2.5 Pseudo-Force possibly associated with a differential 1-form . . . 20

3.3 A pertinent momentum . . . 20

3.4 The Lagrange Function . . . 21

3.5 The Lagrange’s formulation : Hamilton’s Least Action Principle . . . 22

∗Universit´e Europ´eenne de Bretagne, Lab-STICC (UMR CNRS 3192), Universit´e de Bretagne-Sud, Centre Yves Coppens, Campus de Tohannic, F-56017, Vannes

4 Towards Hamiltonian Formulation 23

4.1 Legendre’s Transform . . . 23

4.2 Hamiltonian Function definition . . . 23

4.3 On Hamilton’s Least Action Principle in Phase Space . . . 24

4.4 Hamiltonian shape of Dynamical System in Phase Space . . . 24

4.5 Usual Hamiltonian shape in coordinate systems of Dynamical System in Phase Space . . . 26

4.6 Expression of Hamiltonian Function I . . . 26

4.7 Expression of Hamiltonian Function II . . . 27

II Hamiltonian formulation of dynamics of a charged particle in an electromagnetic field 27

5 What does mean making ”Hamiltonian Mechanics”, or equivalently, ”Symplectic Ge- ometry” 27 5.1 Symplectic structure ofT^∗X . . . 28

5.1.1 Tangent and Cotangent Bundles . . . 28

5.1.2 Natural differential 1-formeγonT^∗X . . . 28

5.1.3 Natural differential 2-formeωonT^∗X . . . 29

5.1.4 Other differential 1-forme onT^∗X : γ^[A] . . . 30

5.1.5 Intrinsic expression of differential 1-formsγ^[A] in terms of Pullback . . . 30

5.1.6 On Pullback ofk-forms . . . 31

5.1.7 Expression of Pullback of 1-forms and 2-forms within coordinate systems whenN= 3 31 5.1.8 Other differential 2-forme onT^∗X : ω^[A] . . . 32

5.2 Differential 1-form representation by vector fields . . . 32

5.2.1 Representations of a 1-forms by vectors . . . 32

5.2.2 Representations of a differential 1-form by vector fields . . . 34

5.3 Remarks about notations . . . 35

5.4 Volume forms . . . 36

5.4.1 Building volume formsω^∧N andω^[A]∧N . . . 36

5.4.2 ω^∧N andω^[A]∧N are the same volume form . . . 37

5.5 Vector fields and differential (2N−1)-forms . . . 38

5.6 Divergence of a vector field . . . 39

5.7 Pure Geometrical Operators transforming differential 1-forms into differential (2N−1)-forms 40 5.8 Vector Field associated with a function . . . 41

6 Hamiltonian Formulation I 41 7 Hamiltonian Formulation II 42 7.1 The second Hamiltonian Formulation . . . 42

7.2 Equivalence of the two Hamiltonian formulation . . . 42

7.2.1 Link between the differentials of the two Hamiltonian Functions . . . 42

7.2.2 Link between the differentials of the two momentum trajectories . . . 44

8 Hodge Operators

εεε

and

µ µ

45 8.1 Non degenerated scalar product on 1-forms induced by the Lagrange Function . . . 45

8.2 Metric onX induced by the Lagrange Function . . . 45

Part I

Towards Hamiltonian description of a charged particle

1 Departure point (in terms of scientific knowledge)

The following is taken for granted.

In a flat space filled with vacuum, the most usual way to write the Maxwell equations is:

−1 c²

∂IE

∂t +∇ ×IB=µ0JI, (1.1)

∂IB

∂t +∇ ×IE= 0, (1.2)

∇ ·IE= 1 ε0

ρρ, (1.3)

∇ ·IB= 0. (1.4)

Equation (1.1) is called ”Ampere’s Theorem”, equation (1.2) ”Faraday’s Law” and (1.3) ”Poisson’s Equation”.

ε₀ is the ”Vacuum Electric Permittivity” andµ₀ the ”Vacuum Magnetic Permeability”. They are linked together by: µ₀ε₀= 1/c², where cis the ”Light Velocity in the Vacuum”.

System ((1.1) - (1.4)) models the time-space evolution of the ”Electric Field”IE=IE(t,x), and the

”Magnetic Field” IB= IB(t,x), seen by an observer in an inertial frame of reference. ρρ=ρρ(t,x) stands for the ”Charge Density” filling the space and JI=JI(t,x) for the ”Current Density” (seen in the inertial frame of reference).

If a particle of mass m, charge q and velocity (in the inertial frame of reference) v0 is situated in positionx0 at timet0, it feels the following ”Lorentz’s Force” .

IF(t0,x0,v0) =q(IE(t0,x0) +v0×IB(t0,x0)). (1.5) Hence applying ”Newton’s Law”, the considered particle follows a trajectory in the ”Position-Velocity Space” (X,V) = (X(t),V(t)) = (X(t;x0,v0, t0),V(t;x0,v0, t0)) which is solution to:

∂X

∂t =V, X(t0;x0,v0, t0) =x0, (1.6)

m∂V

∂t =IF(t,X,V), V(t0;x0,v0, t0) =v0, (1.7) or, in its expanded shapes:

∂X

∂t (t) =V(t), X(t0) =x0, (1.8)

m∂V

∂t (t) =IF(t,X(t),V(t)), V(t0) =v0, (1.9)

or

∂X

∂t(t;x₀,v₀, t₀) =V(t;x₀,v₀, t₀), X(t₀;x₀,v₀, t₀) =x₀, (1.10) m∂V

∂t (t;x0,v0, t0) =IF(t,X(t;x0,v0, t0),V(t;x0,v0, t0)), V(t0;x0,v0, t0) =v0. (1.11)

2 Status of mathematical objects in game I

Above, all the mathematical objects in game are the same. They are scalar or vector fields defined on the position space X, which is a subset of R^N or R^N itself, withN = 3 (or may be 2 or 1 in simplified models that will no be considered in the near sequel).

In view of generalizing the point of view, it is clever to question on their intrinsic status. For this, first, a weakened version of Newton’s Law saying: ”mass times acceleration of a particle equals the force acting on it” has to be considered.

2.1 D’Alembert’s version of Newton’s Law

This version which is sometimes called ”D’Alembert’s Principle” consists in considering in any point x of X all the ”Virtual Displacements” or ”Admissible Displacements” which are the elements of the tangent spaceT_xX ofX inx. Here, in any point x,T_xX andX may be considered as the same space.

It says: in any pointx ofX such that, for a given times, X(s;x0,v0, t0) =x, for any Admissible Displacement ν of T_xX the ”Virtual Work” of the forceIFwhich writesIF(t,X(s),V(s))·ν equals the ”Virtual Work” of the inertial force m∂V

∂t which writesm∂V

∂t (s)·ν

Defining the ”Momentum”IM=IM(t) =IM(t;x₀,v₀, t₀) associated with the trajectory (X(t;x₀,v₀, t₀), V(t;x0,v0, t0)), by

IM(t;x0,v0, t0) =mV(t;x0,v0, t0) (2.1)

”D’Alembert’s Principle” writes :

Principle 2.1 In any pointxofX such that, for a given times,X(s;x0,v0, t0) =x, the following equality holds for any ν of T_xX,

IF(s,X(s),V(s))−∂IM

∂t (s)

·ν = 0. (2.2)

Moreover, this equality characterizes trajectory X(.).

Remark 2.2 Definingm0 as mv0, it is relevant to use the following notation IM(t;x0,m0, t0) in place of IM(t;x0,v0, t0). From now, this notation will be used.

2.2 Status of the force IF and of the momentum IM in a given point x of T

_x

X

Once (2.2) is set, the force IF(s,X(s),V(s)) and the inertial force ∂IM

∂t (s) may be seen as objects acting on elements in T_xX via an inner product (here denoted by ”·”). Hence, intrinsically they they can be represented by elements denoted F(s,X(s),V(s)) and ∂M˜

∂t (s) of the cotangent space T_x^∗X ofX inx. This allows to avoid the inner product. Since ∂M˜

∂t (s) is inT_x^∗X, ˜M has to be in a manifold whose tangent space is T_x^∗X, then it is not irrelevant to consider that ˜Mis also in T_x^∗X. As a conclusion : In a point x of X such that, for a given time s, X(s;x0,v0, t0) = x the forceF(s,X(s;x0,v0, t0),V(s;x0,v0, t0)) and of the momentum ˜M(s;x0,v0, t0) may be considered as elements of the cotangent spaceT_x^∗X.

Once set this conclusion, the question of the definition of the momentum is asked. It has to be defined intrinsically and (2.1) has to be an expression of this intrinsic definition within a coordinates system inside which the inner product has its usual expression. Hence, a mappingM_x fromT_xX to T_x^∗X has to be built.

A way to build this mapping consists, for any v in T_xX, in definingM_x(v) as the unique linear form or 1-form such as hM_x(v),v/|v|i= sup{hM_x(v), νi, ν∈T_xX }andhM_x(v),vi=m|v|². Another way, which is equivalent, consists in considering the differentiable function ¯L_x:T_xX →R defined by

L¯_x(v) = 1

2m|v|², (2.3)

and in saying thatM_x(v) is the differential of ¯L_x in v, i.e.

M_x(v) =

d

_vL^¯_x. (2.4)

(Since ¯L_x : T_xX → R,

d

vL¯_x : T_v(T_xX) → R and since T_xX is a vector space, T_v(T_xX) may be identified with T_xX for anyvinT_xX.)

Remark 2.3 Notice that defining such a one-to-one linear applicationM_x fromT_xX ontoT_x^∗X in any x∈ X, in a way regular enough with respect toxand such that hM_x(ν), ν⁰i=hM_x(ν⁰), νifor anyν ∈T_xX and anyν⁰ ∈T_xX for anyx∈ X, defines a metric onX by consideringhM_x(ν), ν⁰ias the inner product ofν byν⁰.

With this mapping at hand, the momentum ˜M(t;x0,m0, t0) associated with trajectory (X(t;x0,v0, t0), V(t;x0,v0, t0)) is defined as

M(t;˜ x0,m0, t0) =M_{X(t;x0,v0,t0)}(V(t;x0,v0, t0)). (2.5) Remark 2.4 If in place of classical mechanics, the framework is special theory of relativity, then (2.1) is replaced by:

IM=

1−|V|² c²

−1/2

mV. (2.6)

Using as function ¯L_x:T_xX 7→R: ¯L_x(v) =mc²

1− r

1−|v|² c²

, setting againM_x(v) =

d

vL¯_x, the momentumIMdefined by (2.6) is associated with the following element ofT_x^∗X: ˜M=M_X(V).

Remark 2.5 Above, if, in place of choosing ¯L_x(v) = ¹₂m|v|², for any φ(x) depending only on the position variable ¯L_x(v) = ¹₂m|v|²−φ(x) is chosen,

d

_vL^¯_x is left unchanged and then setting again M_x(v) =

d

_vL^¯_x the definition of the momentum ˜M=M_X(V) is valid again.

2.3 Status of the force F and of the momentum M ˜

Firstly, reinterpreting the Lorentz’s Force expression (1.5), it may be deduced the following: From the fieldsIEandIBdefined onX at any timet, for any vector fieldWdefined onX, at any timet, a Lorentz’s Force Field is defined on X by

IF(t,x,W) =q(IE(t,x) +W×IB(t,x)), (2.7) for any xinX.

Remark 2.6 As W is here a vector field, it would be more correct to use an expanded notation W(x) in expression (2.7) to give:

IF(t,x,W(x)) =q(IE(t,x) +W(x)×IB(t,x)). (2.8)

As a conclusion : The force felt by a particle of chargeqbeing inx₀ with velocityv₀ is then the valueIF(t,x0,W(x0)) ofIFin x0, where the force field IF(t,x,W) is computed with a vector field Wwhich is such thatW(x0) =v0.

Secondly, in view of the conclusion of subsection 2.2, in any point x of X, IF and IM may be represented by linear forms or 1-forms onT_xX.

As a conclusion : Hence, for any time, the two vector fieldsIF(t, ., .) andIM(t) defined onX may be, more relevantly, represented by two differential 1-forms onX: F(t) and ˜M(t).

In the sequel, for anyn≤N, for a time-dependent differentialn-formK(t) defined onX, in anyx ofX, {K(t,x)} will denote then-form (T_xX)ⁿ 7→Rwhich is the value of the differentialn-form at x.

2.4 Status of the electric field

In view of Lorentz’s Force expression (2.7), it seems to be reasonable to consider the force and the Electric Field to be represented by objects of the same nature.

As a conclusion : Hence the Electric FieldIE(t, .) will be represented by a differential 1-formE(t) onX.

(As a consequence, the charge q have the status of a number or of a linear operator fromT_xX to T_xX in everyxofX.)

2.5 Status of the magnetic field

If, in anyxofX the tangent spaceT_xX is equipped with an inner product ”·” and a cross product

”×”, any differential 2-form (possibly time-dependant)B(t) defined onX can be represented by the vector field IB(t, .) onX whose value in anyxofX is such that

{B(t,x)}(ν, ν⁰) =IB(t,x)·(ν×ν⁰) = (IB(t,x)×ν)·ν⁰= (ν⁰×IB(t,x))·ν,

for anyν andν⁰ in T_xX. (2.9) Moreover, for any vector field W defined on X, the interior product

i

_WB(t) of B(t) by W and which results as the differential 1-form whose value in any xofX is:

i

WB(t,x) :ν7→ {B(t,x)}(W(x), ν), (2.10) for any ν in T_xX, is represented by the vector fieldIB(t, .)×W=−W×IB(t, .) (whose value in a pointxofX is−W(x)×IB(t,x)).

As a conclusion : Using these remarks, the Magnetic FieldIB(t, .) is represented by the differential 2-formB(t). B(t) andIB(t, .) are linked by (2.9).

Reinterpreting, once again, the expression of the Lorentz’s Force within the framework of differential forms, it may be deduced the following: From the differential 1-formE(t) and the differential 2-form

B(t) defined on X at any timet, for any vector fieldW defined onX, at any time t, a Lorentz’s Force differential 1-Form is defined onX by

F(t) =q(E(t)−

i

WB(t)). (2.11) As a conclusion: The object permitting to compute, in any x0 of X, the Virtual Work of the Lorentz’s Force for any Admissible Displacement of a particle of mass mbeing in x0 with velocity v0 is then the value{F(t,x0)}whereFis defined by (2.11) with a vector fieldWwhich is such that W(x0) =v0.

2.6 On Lorentz’s Force writing

The first way to write Lorentz’s Force is the one presented via formula (2.11), lines above it and interior product definition (2.10). There is another way to define the interior product, more compli- cated, but useful in the sequel and involving the tangent bundle TX which is defined, at any time t, byTX =∪x∈X(x,T_xX).

First, having a time-dependent differential 1-formA(t) defined onX, a function or differential 0-form

ι

◦A(t) may be defined onTX by setting

ι

◦^{A(t,x,v) =}

ι

^◦vA(t,x) ={A(t,x)}(v). (2.12) More generally, if K(t) is a time-dependent differential n-form defined on X, time-dependant dif- ferential (n−1)-form

ι

^◦K(t) may be defined on TX. To do this, it has to be noticed that, in any point (x,v) of TX, tangent space T_(x,v) TX

may be identified with T_xX ×T_xX, and making such an identification leads to the fact that any vector ofT_(x,v) TX

may be written as (ν, υ) with ν ∈ T_xX and υ ∈ T_xX. Then,

ι

^◦K(t) is defined by setting, for any ((ν1, υ1), . . . ,(νn−1, υn−1)) in

T(x,v) TXn−1

,

{

ι

^◦K(t,x,v)}((ν1, υ1), . . . ,(νn−1, υn−1)) ={

ι

^◦vK(t,x)}((ν1, υ1), . . . ,(νn−1, υn−1))

={K(t,x)}(v, ν1, . . . , νn−1). (2.13) Beside this, if K(t) is a differentialn-form onTX, for any (x,v) ofTX it defines an-form onT_xX (D_(x,v)Π)K

{(D_(x,v)Π)K(t)}(ν1, . . . , νn) ={K(t,x,v)}((ν1,0), . . . ,(νn,0)), (2.14) for any (ν1, . . . , νn) in (T_xX)ⁿ.

Then, the interior product

i

WB(t) may be defined using

i

WB(t,x) = D(x,W(x))Π

ι

^◦B

. (2.15)

and Lorentz’s Force may be written as F(t,x) =q

E(t)− D(x,W(x))Π

ι

^{◦B. (2.16)}

The interest of this shape is that it enables to write the force F(t,X(t)) in a given point of the trajectoryX(t), as an element ofT_X(t)^∗ X (considering the Position-Velocity trajectory (X(t),V(t))) by

F(t,X(t)) =q

E(X(t))− D(X(t),V(t))Π

ι

^◦B

. (2.17)

without introducing any vector fieldW.

2.7 Remark on notation, Pushforward and Pullback

The goal of this subsection is to justify the notation (D(x,v)Π)K(t) used in formula (2.14) and after.

Having the tangent bundle TX, the following projection may be defined:

Π : TX → X

(x,v) 7→ x. (2.18)

Once this is done, in any (x,v) ofTX, the differential

d

_(x,v)Π of Π is defined by

d

_(x,v)Π : T_(x,v) TX

→ T_xX (ν, υ) 7→ ν ,

(2.19) since, as already noticed,T_(x,v) TX

may be identified withT_xX ×T_xX, meaning that any vector ofT(x,v) TX

may be written as (ν, υ) withν∈T_xX andυ∈T_xX. The following application Π_∗(x,v): T_x^∗X → T^∗_(x,v) TX

π 7→ Π_∗(x,v)(π), (2.20)

may also be defined by setting, for any (ν, υ) inT(x,v) TX ,

hΠ∗(x,v)(π),(ν, υ)i=hπ,

d

_(x,v)Π((ν, υ))i=hπ, νi. (2.21) It is clear that this application is one-to-one onto its image Π_∗(x,v)(T_x^∗X). It may be noticed that

ker

d

_(x,v)Π

=

(ν, υ)∈T_(x,v) TX

, ν= 0 =

(0, υ), υ∈T_xX , (2.22) Π_∗(x,v)(T_x^∗X) =

π

∈T^∗_(x,v) TX

,∀(ν, υ)∈T_(x,v) TX

,h

π

,(ν, υ)i=h

π

,(ν,0)i , (2.23) and consequently, that

π

∈Π_∗(x,v)(T_x^∗X) if and only ifh

π

,(ν, υ)i= 0,∀(ν, υ)∈ker

d

_(x,v)Π

, (2.24)

(ν, υ)∈ker

d

_(x,v)Π

if and only ifh

π

,(ν, υ)i= 0,∀

π

∈Π_∗(x,v)(T_x^∗X). (2.25) Then, as there exits a natural projection

D_(x,v)^⊥ Π : T(x,v) TX

→ ker

d

_(x,v)Π (ν, υ) 7→ (0, υ),

(2.26) a second one may be defined:

D_(x,v)Π : T^∗_(x,v) TX

→ Π_∗(x,v)(T_x^∗X)

π

7→

π

−

π

◦ D_(x,v)^⊥ .

(2.27) In other words, (D_(x,v)Π)(

π

) is defined by h(D_(x,v)Π)(

π

),(ν, υ)i = h

π

,(ν,0)i for any (ν, υ) in T_(x,v) TX

.

Having (D_(x,v)Π) on hand, it is easy to see that (D_(x,v)Π) defined by (2.14), when it applies on 1-forms, is

D(x,v)Π : T^∗_(x,v) TX

→ T_x^∗X

π

7→ (Π_∗(x,v))⁻¹◦(D_(x,v)Π)

π

.

(2.28)

Indication : (

d

_(x,v)Π)(ν, υ) is called the ”Pushforward” of (ν, υ) by Π in (x,v) and Π_∗(x,v)(π) the ”Pullback” of πby Π in (x,v).

2.8 Faraday’s Law from differential form point of view

As, for any xof X, {B(t,x)} is a 2-form onT_xX, n∂B

∂t (t,x)o

is also a 2-form on T_xX. Hence, if the differential 2-form Bis regular enough, ∂B

∂t (t) is also a differential 2-form onX.

Beside this, ifEis the differential 1-form represented by the vector fieldIE(or, in other words, if in any xofX, the 1-form{E(t,x)} onT_xX, which is equipped with an inner product ”·” and a cross product ”×”, writes

{E(t,x)}= (IE(t,x))·, (2.29) then the exterior derivative dE of E (which is a differential 2-form if E is regular enough) is rep- resented by the vector field ∇ ×IE. (This means that, in anyx ofX, the following formula holds:

{dxE(t)}(ν, ν⁰) = (∇ ×IE)(t,x)·(ν×ν⁰) = ((∇ ×IE)(t,x)×ν)·ν⁰ for anyν andν⁰ inT_xX.) As a consequence of what is just said, Faraday’s Law (1.2) may be seen as the following differential 2-form equality:

∂B

∂t +

d

E= 0. (2.30)

2.9 On Ampere’s Theorem compatibility with differential form viewpoint

As, for any xof X,{E(t,x)}is a 1-form on T_xX, or equivalently is in T_x^∗X, n∂E

∂t(t,x)o

is also in T_x^∗X. Another way to see this consists in noticing that naturally, (x,{E(t,x)}) is in the cotangent bundleT^∗X. (By the way,T^∗X =∪_x∈X(x,T_x^∗X) has a manifold structure.) Then ∂(x,{E(t,x)})

∂t =

∂x

∂t,∂{E(t,x)}

∂t

=

0,n∂E(t,x)

∂t

ois inT(x,{E(t,x)}) T^∗X

; and this tangent space may be iden- tified with T_xX ×T_x^∗X.

Beside this, if the vector field IB onX represents the differential 2-form on X, ∇ ×IB cannot be clearly interpreted in terms of differential forms.

Then interpreting Ampere’s Theorem (1.1) in the framework of the differential forms is not as easy as for the Faraday’s Law.

Nonetheless, a vector field may represent a differential 1-form (see(2.29)) or a differential 2-form (see(2.9)). Hence rewriting Ampere’s Theorem as

−∂(ε₀IE)

∂t +∇ ×(µ₀⁻¹IB) =JI, (2.31) and considering the new vector fieldsIDandIHdefined fromIEandIBby:

ID=ε₀IE, IB=µ₀IH, (2.32)

the following is reached

−∂ID

∂t +∇ ×IH=JI, (2.33)

In (2.33)IDandIHare to be considered as vector fields representing respectively a differential 2-form D and a differential 1-form H. Consistently, in (2.32) the multiplications byε0 andµ0 have to be considered as operators transforming vector fields representing differential 1-forms into vector fields representing differential 2-forms.

2.10 Status of the Current Density

Watching equation (2.33), if, as announced,IDrepresents a differential 2-form andIHa differential 1-form then∇ ×IHrepresents a differential 2-form and thenJIhas to represent a differential 2-form.

The question is now: does this last assertion make sense? or may-be more: why does this last assertion make sense? To help the providing of an answer to this question, it has to be noticed that the essential role to be played by a current density is to be integrated on surfaces to give current fluxes, and that precisely, differential 2-forms may be integrated on surfaces.

Then, a current density may be represented by a differential 2-form J on X by setting that the current flux (associated with the considered current density) through any surface is the integral of Jover this surface.

2.11 Ampere’s Theorem from differential form point of view

Defining two ”Hodge Operators”

εεε

and

µ µ

that transform differential 1-forms into differential (N−1)- forms (differential 2-forms here sinceN = 3) and setting

D=

εεε

E, B=

µ µ

H (2.34)

Ampere’s Theorem may be interpreted in terms of differential forms exactly as Faraday’s Law may and then reads:

−∂D

∂t +

d

^{H=J, (2.35)}

(or

−∂(

εεε

E)

∂t +

d

(

µ µ

⁻¹B) =J). (2.36)

2.12 Magnetic Field divergence free equation from differential form point of view

if B is the differential 2-form represented by the vector fieldIB (or, in other words, if in any xof X, the 2-form{B(t,x)} onT_xX is defined by (2.9)) then the exterior derivative

d

B ofB(which is a differential 3-form if Bis regular enough) is represented by the function∇ ·IB. This means that in anyxofX, the following formula holds:

{

d

_xB(t)}(ν, ν⁰, ν⁰⁰) = (∇ ·IB(t,x))((ν×ν⁰)·ν⁰⁰) for anyν, ν⁰ andν⁰⁰ inT_xX (2.37) With this remark, equation (1.4) reads

d

^{B= 0. (2.38)}

2.13 Poisson’s Equation from differential form point of view

Rewriting equation (1.3) as

∇ ·(ε₀IE) =ρρ, (2.39) introducing the charge density differential 3-formρ, it can be read:

d

⁽

εεε

^{E) =ρ, (2.40)}

or

d

D=ρ. (2.41)

2.14 Status of the Charge Density

The role of the Charge Density is to be integrated over any tridimensional set in order to give charge contained within this set. Precisely, differential 3-forms may be integrated over tridimensional sets.

Then, it is consistent to represent a charge density by the differential 3-form ρ.

2.15 Maxwell System in the differential forms framework

Summarizing all the above versions of the equations of the Maxwell System, it gives

−∂D

∂t +

d

H=J, (2.42)

∂B

∂t +

d

^{E= 0, (2.43)}

d

D=ρ, (2.44)

d

^{B= 0. (2.45)}

with

D=

εεε

E, B=

µ µ

H. (2.46)

In ((2.42)-(2.46))EandHare 1-forms on the position spaceX;B,DandJare 2-forms onX; and;

ρ is a 3-form onX. Operators

εεε

^and

µ µ

are Hodge Operators mapping one-to-one the 1-forms on X onto the 2-forms onX.

3 From Newton to Lagrange

3.1 Maxwell System with Electric and Magnetic Potentials

As on X the forms with zero exterior derivatives are the exterior derivatives, from (2.45) it can be deduced that there exists a differential 1-formAonX such that

B=

d

^{A (3.1)}

Then, inserting this in (2.43) yields

d

^∂A

∂t +E

= 0. From this last equality, it can be deduced that there exists a 0-form Φ such that∂A

∂t +E

=−

d

^{Φ, or}

E=−

d

^Φ−∂A

∂t. (3.2)

Equations (2.42) and (2.44) yield:

−

∂

−

εεεd

Φ−

εεε

^∂A

∂t

∂t +

d

⁽

µ µ

⁻¹

d

^{A) =J (3.3)}

d

−

εεεd

Φ−

εεε

^∂A

∂t

=ρ, (3.4)

or

∂

εεε

^∂A

∂t

∂t +∂(

εεεd

Φ)

∂t +

d

(

µ µ

⁻¹

d

A) =J (3.5)

−

d

εεε

^∂A

∂t

−

d

(

εεεd

Φ) =ρ, (3.6)

(If Hodge operators

εεε

and

µ µ

depend neither on time nor on position, these last equations read

εεε

^∂2A

∂t² +

εεε

^∂(

d

^Φ)

∂t +

d

⁽

µ µ

⁻¹

d

^{A) =J, (3.7)}

−∂(

d

⁽

εεε

^A))

∂t −

d

(

εεεd

Φ) =ρ.) (3.8)

3.2 On momentum choice

3.2.1 Momentum possibly associated with a differential 1-form

Reinterpreting equality (2.2) with the differential form point of view, consists, once ˜M(t;x0,m0, t0) is defined from the Position-Velocity trajectory (X(t),V(t)) = (X(t;x0,v0, t0),V(t;x0,v0, t0)) by (2.5), (2.4) and (2.3), in considering that the Position-Velocity trajectory (X(t),V(t)) is solution to

n(F−∂M˜

∂t )(t,X(t))o

ν= 0, (3.9)

for all ν in T_X(t)X, where F(t) is defined, at any timet, from the differential 1-formE(t) and the differential 2-formB(t) by (2.11) with a vector field vwhich is such thatv((X(t)) =V(t).

But, if a regular differential 1-formAis defined onX, it is not forbidden to associate with Position- Velocity trajectory (X(t),V(t)) the following momentum

M(t) =M_X(t)^[A](V(t)), (3.10) with

M_x^[A](v) =M_x(v) +{qA(t,x)}=

d

_vL^¯_x+{qA(t,x)}, (3.11) where M_x is defined by (2.4) and ¯L_xby (2.3).

Remark 3.1 Following remark 2.3, replacingM_x byM_x^[A] may be seen as replacing the metric on X by another operator onT_xX ×T_xX which is no more an inner product.

3.2.2 Time derivative of the momentum Considering TX =∪_x∈X(x,T_xX), at any timet,

BM^[A]: (x,v)7→ x,M_x^[A](v)

= x,M^[A](x,v)

, (3.12)

may be seen as a one-to-one mapping fromTX ontoT^∗X. (BM^[A]could be called the bundlization ofM_x^[A] andM^[A](x,v) is the tangent space’s componant of this bundlization in (x,v).) The value

d

_(x,v)M^[A] of its differential

d

M^[A] in (x,v) maps T_(x,v) TX

onto T

(x,Mx^[A](v)) T^∗X

. Tangent space T_(x,v) TX

may be identified with T_xX ×T_xX and T_(x,M[A]

x (v)) T^∗X

with T_xX ×T_x^∗X. Looking at (X(.),V(.)) as a trajectory onTX, it is mapped to a trajectory (X(.),M(.)) onT^∗X, and

its tangent vector∂X

∂t (t),∂V

∂t (t)

in (X(t),V(t)) is mapped to the tangent vector∂X

∂t (t),∂M

∂t (t) of trajectory (X(.),M(.)) in (X(t),M(t)) by the differential

d

(X(t),V(t))M^[A] to which the time partial derivative of{qA(t,x)}has to be added. In other words,

∂X

∂t (t),∂M

∂t (t)

=

d

(X(t),V(t))(BM^[A])∂X

∂t (t),∂V

∂t (t) +

0,n q∂A

∂t (t,X(t))o

. (3.13)

To compute

d

(X(t),V(t))(BM^[A]) the following chart of a neighborhoodB(X(t)) ofX(t) onX is used C: B(X(t)) → R^N

x 7→ q= (q₁,q₂,q₃). (3.14)

OnR^N stands the canonical frame (e_q1,e_q2,e_q3). Then (3.15) may be rewritten asC(x) =q₁e_q1+ q₂e_q2+q₃e_q3. For everyx∈ X, the following chart is used onT_xX:

d

xC: T_xX → R^N (=T_qR^N)

v 7→ q˙ = ( ˙q₁,q˙₂,q˙₃). (3.15) Taking the same frame as previously but naming it (∂q1, ∂q2, ∂q3), (3.15) reads also

d

_xC(v) =

˙

q₁∂_q1+ ˙q₂∂_q2+ ˙q₃∂_q3

Another way to formulate this consists in saying that the following chart in the following neighbor- hood ∪_x∈B(X(t))(x,T_xX) of (X(t),V(t)) on TX is used:

(C,

d

xC) : ∪_x∈B((X(t))(x,T_xX) → R^N×R^N

(x,v) 7→ (q,q) = ((q˙ ₁,q₂,q₃),( ˙q₁,q˙₂,q˙₃)), (3.16) which also reads : (C,

d

_xC)(x,v) =q1eq1+q2eq2+q3eq3+ ˙q1∂q1+ ˙q2∂q2+ ˙q3∂q3.

Beside this, sinceC:B(X(t))→R^N it may be seen asNor 3 regular functions or differential 0-forms (C1,C2,C3), the exterior derivatives or differentials

d

xC1=

d

_xC1,

d

xC2=

d

_xC2,

d

xC3=

d

_xC3 may be considered in every x∈ X. They define a frame on T_x^∗X which is denoted (dq₁, dq₂, dq₃). (dq₁ is nothing but v7→q˙₁.) In this frame any 1-formm has the following coordinatesp= (p₁,p₂,p₃) = p₁dq₁+p₂dq₂+p₃dq₃. Then, the following chart is built onT_x^∗X:

DxC: T_x^∗X → R^N (=T_q^∗R^N)

m 7→ p= (p1,p2,p3). (3.17)

Another way to formulate this consists in following, in a simpler manner, the way followed in subsection 2.7 while defining (D_(x,v)Π) from Π_∗(x,v). Indeed,C_∗xmay be defined as

C_∗x: R^N (=T_q^∗R^N) → T_x^∗X, (3.18) in defining the Pullback C_∗x(p) ofpinx, by setting for anyν inT_xX,

hC_∗x(p), νi=hC_∗x(p1,p2,p3), νi=hC_∗x(p1dq1+p2dq2+p3dq3), νi

=hp1dq1+p2dq2+p3dq3,

d

_xC(ν)i=hp1dq1+p2dq2+p3dq3,( ˙q1,q˙2,q˙3)i

=hp1dq1+p2dq2+p3dq3,q˙1∂q₁+ ˙q2∂q₂+ ˙q3∂q₃i=p1q˙1+p2q˙2+p3q˙3. (3.19)

Easyly, C_∗x is one-to-one onto T_x^∗X and DxC = (C_∗x)⁻¹. This may of course be taken as the defininition ofDxC.

OnceDxCis properly introduced, it is possible to consider that the following chart, in neighborhood

∪_x∈B(X(t))(x,T_x^∗X) of (X(t),M(t)) on T^∗X, is used:

(C,DxC) : ∪_x∈B((X(t))(x,T_x^∗X) → R^N×R^N

(x,m) 7→ (q,p) = ((q₁,q₂,q₃),(p₁,p₂,p₃)), (3.20) or (C,DxC)(x,m) =q₁e_q1+q₂e_q2+q₃e_q3+p₁dq₁+p₂dq₂+p₃dq₃.

Using those charts, ˘A₁(t,q), ˘A₂(t,q) and ˘A₃(t,q) are functions onR^N which are the coordinates ofA, they are defined to be such that

(C,DxC)(x,A(t,x)) =q1eq₁+q2eq₂+q3eq₃+ ˘A1(t,q)dq1+ ˘A2(t,q)dq2+ ˘A3(t,q)dq3, (3.21) and ˘Ais the associated vector, i.e. ˘A= ( ˘A1,A˘2,A˘3). Function ˘L_q is the expression of ¯L_x within the coordinate system. It is such that

L¯_x(v) = ˘L_q( ˙q), with (q,q) = (C,˙

d

_xC)(x,v), (3.22) MappingBM˘^[A]which maps one-to-oneR^N×R^N ontoR^N×R^N is the expression ofBM^[A] within the coordinate systems. It is such that

(C,DxC)(BM^[A](x,v))

=BM˘^[A](q,q) = (q,˙ M˘_q^[A]( ˙q)) = (q,M˘^[A](q,q)),˙ with (q,q) = (C,˙

d

xC)(x,v). (3.23) MappingBM˘^[A] may be expressed as

BM˘^[A](q,q) =˙ q1eq₁+q2eq₂+q3eq₃

+ ∂L˘_q

∂q˙₁( ˙q) +qA˘1(q)

!

dq1+ ∂L˘_q

∂q˙₂( ˙q) +qA˘2(q)

!

dq2+ ∂L˘_q

∂q˙₃( ˙q) +qA˘3(q)

!

dq3. (3.24)

Matrix ∇(q,˙q)(BM˘^[A]) is an expression of

d

_(q,˙q)(BM˘^[A]) which is an expression of

d

_(x,v)(BM^[A]) within the coordinates. This means, differentiating (3.23), that

d

_(BM^[A]_(x,v))(C,DxC)

d

_(x,v)(BM^[A])

=

d

_(q,˙q)(BM˘^[A])

d

_(x,v)(C,

d

_xC)

, (3.25)

and matrix∇(q,˙q)(BM˘^[A]) has the following expression:

∇(q,˙q)(BM˘^[A]) =



























I_RN 0



















∂ _∂L˘

q

∂q˙1+qA˘1

∂q₁

∂ _∂L˘

q

∂q˙1+qA˘1

∂q₂

∂ _∂L˘

q

∂˙q1+qA˘1

∂q₃

∂ _∂L˘

q

∂q˙2+qA˘₂

∂q1

∂ _∂L˘

q

∂q˙2+qA˘₂

∂q2

∂ _∂L˘

q

∂˙q2+qA˘₂

∂q3

∂ ∂L˘q

∂q˙3+qA˘₃

∂q₁

∂ ∂L˘q

∂q˙3+qA˘₃

∂q₂

∂ ∂L˘q

∂˙q3+qA˘₃

∂q₃































∂²L˘_q

∂q1∂q˙1

∂²L˘_q

∂q2∂q˙1

∂²L˘_q

∂q3∂q˙1

∂²L˘_q

∂q₁∂q˙₂

∂²L˘_q

∂q₂∂q˙₂

∂²L˘_q

∂q₃∂q˙₂

∂²L˘_q

∂q1∂q˙3

∂²L˘_q

∂q2∂q˙3

∂²L˘_q

∂q3∂q˙3







































=





I_RN 0

∇qM˘^[A] ∇q˙M˘^[A]



. (3.26) Defining trajectoriesQ(t) =C(X(t)), ˙Q(t) = (

d

X(t)C)(V(t)) andP(t) = (DX(t)C)(M(t)) within the coordinate systems, equation (3.13) may be translated into

∂Q

∂t (t),∂P

∂t(t)

=

d

_(Q(t),Q(t))˙ (BM˘^[A])∂Q

∂t (t),∂Q˙

∂t (t) +

0,n q∂A˘

∂t(t,Q(t))o ,

= ∇_(Q(t),Q(t))˙ (BM˘^[A])∂Q

∂t(t),∂Q˙

∂t(t) +

0,n q∂A˘

∂t (t,Q(t))o .

(3.27)

IntroducingBM^{[ ]}: (x,v)7→ x,M_x(v)

= x,M^{[ ]}(x,v)

mappingTX toT^∗X, whose differential

d

_(x,v)(BM^{[ ]}) is represented, within the coordinate systems, by

∇_(q,˙q)(BM˘^{[ ]}) =



























I_RN 0



















∂ _∂L˘

q

∂˙q1

∂q1

∂ _∂L˘

q

∂q˙1

∂q2

∂ _∂L˘

q

∂q˙1

∂q3

∂ _∂L˘

q

∂˙q2

∂q1

∂ _∂L˘

q

∂q˙2

∂q2

∂ _∂L˘

q

∂q˙2

∂q3

∂ ∂L˘q

∂˙q3

∂q₁

∂ ∂L˘q

∂q˙3

∂q₂

∂ ∂L˘q

∂q˙3

∂q₃































∂²L˘_q

∂q1∂q˙1

∂²L˘_q

∂q2∂q˙1

∂²L˘_q

∂q3∂q˙1

∂²L˘_q

∂q₁∂q˙₂

∂²L˘_q

∂q₂∂q˙₂

∂²L˘_q

∂q₃∂q˙₂

∂²L˘_q

∂q₁∂q˙₃

∂²L˘_q

∂q₂∂q˙₃

∂²L˘_q

∂q₃∂q˙₃







































=





I_RN 0

∇qM˘^{[ ]} ∇q˙M˘^{[ ]}



, (3.28) (which means this matrix is a representation of

d

_(x,v)(BM^{[ ]}) defined as

d

_(BM^{[ ]}_(x,v))(C,DxC)

d

_(x,v)(BM^{[ ]})

=

d

_(q,˙q)(BM˘^{[ ]})

d

_(x,v)(C,

d

_xC)

) (3.29)

equality (3.26) reads

∇_(q,˙q)(BM˘^[A]) =∇_(q,q)˙ (BM˘^{[ ]}) +



















0 0

q













∂A˘₁

∂q1

∂A˘₁

∂q2

∂A˘₁

∂q3

∂A˘2

∂q₁

∂A˘2

∂q₂

∂A˘2

∂q₃

∂A˘₃

∂q₁

∂A˘₃

∂q₂

∂A˘₃

∂q₃











 0



















. (3.30)

Another shape will now be given to (3.27). Since ˘A(t,q) = ˘A1(t,q)dq1+ ˘A2(t,q)dq2+ ˘A3(t,q)dq3,

d

_qA^˘ = ∂A˘3

∂q2

−∂A˘2

∂q3

!

dq2∧dq3+ ∂A˘3

∂q1

−∂A˘1

∂q3

!

dq1∧dq3+ ∂A˘2

∂q1

−∂A˘1

∂q2

!

dq1∧dq2,

(3.31) which is the expression of

d

xA in the coordinate system associated with frame (dq2∧dq3, dq1∧ dq3, dq1∧dq2), then the expression of differential 1-form

i

W(

d

A), which is the interior product of

d

Aby any vector fieldW, has the following expression within the coordinate system associated with frame (dq₁, dq₂, dq₃):

i

W˘(

d

A) =^˘ ∂A˘₃

∂q2

−∂A˘₂

∂q3

!

( ˘W2dq3−W˘ 3dq2) + ∂A˘₃

∂q1

−∂A˘₁

∂q3

!

( ˘W1dq3−W˘3dq1)

+ ∂A˘₂

∂q1

−∂A˘₁

∂q2

!

( ˘W₁dq₂−W˘ ₂dq₁) = "

∂A˘₁

∂q2

−∂A˘₂

∂q1

# W˘₂+

"

∂A˘₁

∂q3

−∂A˘₃

∂q1

# W˘₃

! dq₁

+ "

∂A˘2

∂q₃ −∂A˘3

∂q₂

# W˘ ₃+

"

∂A˘2

∂q₁ −∂A˘1

∂q₂

# W˘ ₁

! dq₂

+ "

∂A˘3

∂q₁ −∂A˘1

∂q₃

# W˘ 1+

"

∂A˘3

∂q₂ −∂A˘2

∂q₃

# W˘ 2

!

dq3. (3.32) On the other hand,

i

WAhas the following expression:

i

W˘A˘ = ˘A1(t,q) ˘W1(q) + ˘A2(t,q) ˘W2(q) + ˘A3(t,q) ˘W3(q), (3.33) and, as a consequence, if ˘W is independent of q, the exterior derivative

d

_x(

i

_WA) of

i

_WA in x writes, within the coordinate system,

d

q(

i

W^˘A) =˘ ∂A˘1

∂q₁

W˘₁+∂A˘2

∂q₁

W˘₂+∂A˘3

∂q₁ W˘ ₃

! dq₁

+ ∂A˘1

∂q₂

W˘ 1+∂A˘2

∂q₂

W˘ 2+∂A˘3

∂q₂ W˘3

!

dq2+ ∂A˘1

∂q₃

W˘ 1+∂A˘2

∂q₃

W˘ 2+∂A˘3

∂q₃ W˘ 3

!

dq3. (3.34)

Coupling (3.27) (with (3.24) or (3.30)) on the one hand, and, on the other hand, (3.32) and (3.33), the following equality holds:

∂Q

∂t (t),∂P

∂t(t)

= 0,n q∂A˘

∂t(t,Q(t))o

!

+ ∇(q,q)˙ (BM˘_q^{[ ]}) ∂Q

∂t (t),∂Q˙

∂t (t)

!

+ 0,

q

i

_Ve^˘(

d

^{A)(t,˘ Q(t)) +0,q}

d

⁽

i

_Ve^{˘A)(t,˘ Q(t)) , (3.35)}

where ˘

Ve is a the vector field which is such that ˘

V(X(t)) =e V(t) and whose coordinates

d

_xC(˘ V(x))e are independant ofxin a neighborhood ofX(t).