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AV-differential geometry and Lagrangian Mechanics
Paweł Urbanski
Faculty of Physics
University of Warsaw
Tenerife, 14.09.2006 – p. 1/13
Motivation
Frame independent (intrinsic) formulation of some physicaltheories requires affine objects.
Tenerife, 14.09.2006 – p. 2/13
Motivation
Frame independent (intrinsic) formulation of some physicaltheories requires affine objects.
• The phase space for relativistic charged particle is not thecotangent bundle, but an affine bundle, modelled on thecotangent bundle (Weistein, Sternberg, Tulczyjew).
Tenerife, 14.09.2006 – p. 2/13
Motivation
Frame independent (intrinsic) formulation of some physicaltheories requires affine objects.
• The phase space for relativistic charged particle is not thecotangent bundle, but an affine bundle, modelled on thecotangent bundle (Weistein, Sternberg, Tulczyjew).
• Hamiltonian for a time-dependent system is not a function,but a section of an affine bundle over the phase manifold(Mangiarotti, Martínez, Popescu, Sarlet, Sardanashvili, ...).
Tenerife, 14.09.2006 – p. 2/13
Motivation
Frame independent (intrinsic) formulation of some physicaltheories requires affine objects.
• The phase space for relativistic charged particle is not thecotangent bundle, but an affine bundle, modelled on thecotangent bundle (Weistein, Sternberg, Tulczyjew).
• Hamiltonian for a time-dependent system is not a function,but a section of an affine bundle over the phase manifold(Mangiarotti, Martínez, Popescu, Sarlet, Sardanashvili, ...).
• Frame independent Lagrangian in Newtonian mechanics isan affine object (Duval, Tulczyjew, ...).
Tenerife, 14.09.2006 – p. 2/13
What is AV-differential geometry?
Differential geometry of affine values(AV-differential geometry) is, roughlyspeaking, the differential geometry builton the set of sections of one-dimensionalaffine bundle ζ : Z → M modelled onM × R, instead of just functions on M .
Tenerife, 14.09.2006 – p. 3/13
What is AV-differential geometry?
The bundle Z will be called a bundle of affine values.
Z is modelled on M × R, so we can add reals in each fiber of Z.
Z is an (R,+)-principal bundle.
Tenerife, 14.09.2006 – p. 4/13
Affine analog of the cotangent bundle T∗M
We define an equivalence relation in the set of pairs of (m, σ),where m ∈ M and σ is a section of Z.
(m, σ), (m′, σ′) are equivalent if m = m′ and d(σ − σ′)(m) = 0,where we have identified the difference of sections of Z with afunction on M .
Tenerife, 14.09.2006 – p. 5/13
Affine analog of the cotangent bundle T∗M
We define an equivalence relation in the set of pairs of (m, σ),where m ∈ M and σ is a section of Z.
(m, σ), (m′, σ′) are equivalent if m = m′ and d(σ − σ′)(m) = 0,where we have identified the difference of sections of Z with afunction on M .
The equivalence class of (m, σ) is denoted by dσ(m). The set ofequivalence classes is denoted by PZ and called the phasebundle for Z.
Tenerife, 14.09.2006 – p. 5/13
Affine analog of the cotangent bundle T∗M
We define an equivalence relation in the set of pairs of (m, σ),where m ∈ M and σ is a section of Z.
(m, σ), (m′, σ′) are equivalent if m = m′ and d(σ − σ′)(m) = 0,where we have identified the difference of sections of Z with afunction on M .
The equivalence class of (m, σ) is denoted by dσ(m). The set ofequivalence classes is denoted by PZ and called the phasebundle for Z.
Pζ : PZ → M : dσ(m) 7→ m
is an affine bundle modelled on the cotangent bundleT∗M → M .
Tenerife, 14.09.2006 – p. 5/13
Other examples of affine constructions
A, B - affine spaces modelled on a vector space V .A × B ∋ (a, b), (a′, b′) are equivalent if a − a′ = b′ − a′.Equivalence class is the affine sum a⊞b.A⊞B is an affine space modelled on V .
Tenerife, 14.09.2006 – p. 6/13
Other examples of affine constructions
A, B - affine spaces modelled on a vector space V .A × B ∋ (a, b), (a′, b′) are equivalent if a − a′ = b′ − a′.Equivalence class is the affine sum a⊞b.A⊞B is an affine space modelled on V .
Similarly, we have A⊟B. In particular, A⊟A = V .
Tenerife, 14.09.2006 – p. 6/13
Other examples of affine constructions
A, B - affine spaces modelled on a vector space V .A × B ∋ (a, b), (a′, b′) are equivalent if a − a′ = b′ − a′.Equivalence class is the affine sum a⊞b.A⊞B is an affine space modelled on V .
Similarly, we have A⊟B. In particular, A⊟A = V .
c = γ([a, b]) - 1-dimensional, oriented cell in M .ϕ, ϕ′ - sections of PZ are equivalent if
∫
c
(ϕ − ϕ′) = 0.
Equivalence class is the integral∫cϕ.
Tenerife, 14.09.2006 – p. 6/13
Other examples of affine constructions
A, B - affine spaces modelled on a vector space V .A × B ∋ (a, b), (a′, b′) are equivalent if a − a′ = b′ − a′.Equivalence class is the affine sum a⊞b.A⊞B is an affine space modelled on V .
Similarly, we have A⊟B. In particular, A⊟A = V .
c = γ([a, b]) - 1-dimensional, oriented cell in M .ϕ, ϕ′ - sections of PZ are equivalent if
∫
c
(ϕ − ϕ′) = 0.
Equivalence class is the integral∫cϕ.
We have∫cϕ ∈ Zγ(b)⊟Zγ(a).
Tenerife, 14.09.2006 – p. 6/13
Duality
Let A be an affine space modelled on V ∗. V is a vector space.
Tenerife, 14.09.2006 – p. 7/13
Duality
Let A be an affine space modelled on V ∗. V is a vector space.
Standard: f ∈ V ∗ is a linear function on V , i.e. a linear sectionof the trivial bundle V × R → V .
Tenerife, 14.09.2006 – p. 7/13
Duality
Let A be an affine space modelled on V ∗. V is a vector space.
Standard: f ∈ V ∗ is a linear function on V , i.e. a linear sectionof the trivial bundle V × R → V .
AV: a ∈ A is a linear section of a bundle τ : A†→ V .
Tenerife, 14.09.2006 – p. 7/13
Duality
Let A be an affine space modelled on V ∗. V is a vector space.
Standard: f ∈ V ∗ is a linear function on V , i.e. a linear sectionof the trivial bundle V × R → V .
AV: a ∈ A is a linear section of a bundle τ : A†→ V .
As A† we can take the vector space of all affine functions on A.τ(f) is the linear part of f .
Tenerife, 14.09.2006 – p. 7/13
Duality
Let A be an affine space modelled on V ∗. V is a vector space.
Standard: f ∈ V ∗ is a linear function on V , i.e. a linear sectionof the trivial bundle V × R → V .
AV: a ∈ A is a linear section of a bundle τ : A†→ V .
As A† we can take the vector space of all affine functions on A.τ(f) is the linear part of f .
A† is a special vector space, i.e. a vector space with adistinguished, non-zero vector (constant function 1).We call τ : A†
→ V the bundle of affine values for affineco-vectors in A.
Tenerife, 14.09.2006 – p. 7/13
Duality
Let A be an affine space modelled on V ∗. V is a vector space.
Standard: f ∈ V ∗ is a linear function on V , i.e. a linear sectionof the trivial bundle V × R → V .
AV: a ∈ A is a linear section of a bundle τ : A†→ V .
As A† we can take the vector space of all affine functions on A.τ(f) is the linear part of f .
A† is a special vector space, i.e. a vector space with adistinguished, non-zero vector (constant function 1).We call τ : A†
→ V the bundle of affine values for affineco-vectors in A.
There is full duality between affine spaces and special vectorspaces.
Tenerife, 14.09.2006 – p. 7/13
Example
A = PmZ
The AV-bundle for A can be identified with TzZ for any ζ(z) = mwith τ = Tζ restricted to TzZ.
Tenerife, 14.09.2006 – p. 8/13
Example
A = PmZ
The AV-bundle for A can be identified with TzZ for any ζ(z) = mwith τ = Tζ restricted to TzZ.
For a ∈ PmZ we take a representative (σ, m) such that σ(m) = z.We put a(v) = Tσ(v).
Tenerife, 14.09.2006 – p. 8/13
Example
A = PmZ
The AV-bundle for A can be identified with TzZ for any ζ(z) = mwith τ = Tζ restricted to TzZ.
For a ∈ PmZ we take a representative (σ, m) such that σ(m) = z.We put a(v) = Tσ(v).
For the whole bundle, (PZ)† = TZ, where TZ = TZ/R
Tenerife, 14.09.2006 – p. 8/13
References
First appearances:
W.M. Tulczyjew, P. Urbanski, S. Zakrzewski, A pseudocategoryof principal bundles, Atti Accad. Sci. Torino, 122 (1988), 66–72
W.M. Tulczyjew, P. Urbanski, An affine framework for thedynamics of charged particles, Atti Accad. Sci. Torino Suppl. n.2, 126 1992, 257–265.
Tenerife, 14.09.2006 – p. 9/13
References
First appearances:
W.M. Tulczyjew, P. Urbanski, S. Zakrzewski, A pseudocategoryof principal bundles, Atti Accad. Sci. Torino, 122 (1988), 66–72
W.M. Tulczyjew, P. Urbanski, An affine framework for thedynamics of charged particles, Atti Accad. Sci. Torino Suppl. n.2, 126 1992, 257–265.
For more recent references look in
K. Grabowska, J. Grabowski and P. Urbanski: AV-differentialgeometry: Poisson and Jacobi structures, J. Geom. Phys. 52(2004) no. 4, 398-446.
P. Urbanski, Affine framework for analytical mechanics, in"Classical and Quantum Integrability", Grabowski, J., Marmo, G.,Urbanski, P. (eds.), Banach Center Publications, vol. 59 (2003),257–279.
and references there.
Tenerife, 14.09.2006 – p. 9/13
Lagrangian and action
Lagrangian is a section λ of the AV-bundle Tζ : TZ → TM
Tenerife, 14.09.2006 – p. 10/13
Lagrangian and action
Lagrangian is a section λ of the AV-bundle Tζ : TZ → TM
An example is given by a section ϕ of the phase bundle PZ:for each m ∈ M , ϕ(m) corresponds to a linear section of theAV-bundle Tζ : TmZ → TmM . We denote this section by iTϕ(m).
Tenerife, 14.09.2006 – p. 10/13
Lagrangian and action
Lagrangian is a section λ of the AV-bundle Tζ : TZ → TM
An example is given by a section ϕ of the phase bundle PZ:for each m ∈ M , ϕ(m) corresponds to a linear section of theAV-bundle Tζ : TmZ → TmM . We denote this section by iTϕ(m).The action of λ along a curve γ : [a, b] → M is defined by the
formula∫ b
a
λ ◦ γ =
∫ b
a
(λ ◦ γ − iTϕ ◦ γ) +
∫
γ([a,b])ϕ .
It does not depend on the choice of ϕ.
Tenerife, 14.09.2006 – p. 10/13
Lagrangian and action
Lagrangian is a section λ of the AV-bundle Tζ : TZ → TM
An example is given by a section ϕ of the phase bundle PZ:for each m ∈ M , ϕ(m) corresponds to a linear section of theAV-bundle Tζ : TmZ → TmM . We denote this section by iTϕ(m).The action of λ along a curve γ : [a, b] → M is defined by the
formula∫ b
a
λ ◦ γ =
∫ b
a
(λ ◦ γ − iTϕ ◦ γ) +
∫
γ([a,b])ϕ .
It does not depend on the choice of ϕ.
∫ b
a
λ ◦ γ ∈ Zγ(b)⊟Zγ(a)
Tenerife, 14.09.2006 – p. 10/13
Euler-Lagrange equation. Standard case
The basis for the representation of the differential of the actionfunctional is the decomposition
dL = ((τ12 )∗dL − dT(iF dL)) + dT(iF dL) (∗)
Tenerife, 14.09.2006 – p. 11/13
Euler-Lagrange equation. Standard case
The basis for the representation of the differential of the actionfunctional is the decomposition
dL = ((τ12 )∗dL − dT(iF dL)) + dT(iF dL) (∗)
Where τ12 is the canonical projection τ1
2 : T2M → TM ,
Tenerife, 14.09.2006 – p. 11/13
Euler-Lagrange equation. Standard case
The basis for the representation of the differential of the actionfunctional is the decomposition
dL = ((τ12 )∗dL − dT(iF dL)) + dT(iF dL) (∗)
Where τ12 is the canonical projection τ1
2 : T2M → TM ,
iF is a derivation associated with the vertical (1,1) tensor F(vector valued 1-form) on TM . Essentially, it is the verticalderivative of L.
Tenerife, 14.09.2006 – p. 11/13
Euler-Lagrange equation. Standard case
The basis for the representation of the differential of the actionfunctional is the decomposition
dL = ((τ12 )∗dL − dT(iF dL)) + dT(iF dL) (∗)
Where τ12 is the canonical projection τ1
2 : T2M → TM ,
iF is a derivation associated with the vertical (1,1) tensor F(vector valued 1-form) on TM . Essentially, it is the verticalderivative of L.dT = diT + iTd is the ’total time derivative’.
Tenerife, 14.09.2006 – p. 11/13
Euler-Lagrange equation. Standard case
The basis for the representation of the differential of the actionfunctional is the decomposition
dL = ((τ12 )∗dL − dT(iF dL)) + dT(iF dL) (∗)
Where τ12 is the canonical projection τ1
2 : T2M → TM ,
iF is a derivation associated with the vertical (1,1) tensor F(vector valued 1-form) on TM . Essentially, it is the verticalderivative of L.dT = diT + iTd is the ’total time derivative’.
The first component in (∗) is a 1-form on T2M , vertical with
respect to projection T2M → M .
It can be considered a mapping T2M → T
∗M
Tenerife, 14.09.2006 – p. 11/13
Euler-Lagrange equation. Affine case
• The AV-bundle for (τ12 )∗dλ is
(Tτ12 )∗TTZ = T(τ1
2 )∗TZ = T(Tτ01 )∗TZ = TT(τ0
1 )∗Z
(pull-back commutes with the exterior derivative and τ12
coincides with Tτ01 ).
Tenerife, 14.09.2006 – p. 12/13
Euler-Lagrange equation. Affine case
• The AV-bundle for (τ12 )∗dλ is
(Tτ12 )∗TTZ = T(τ1
2 )∗TZ = T(Tτ01 )∗TZ = TT(τ0
1 )∗Z
(pull-back commutes with the exterior derivative and τ12
coincides with Tτ01 ).
• The AV-bundle for iF dλ is (Tτ01 )∗TZ = T(τ0
1 )∗Z.
Tenerife, 14.09.2006 – p. 12/13
Euler-Lagrange equation. Affine case
• The AV-bundle for (τ12 )∗dλ is
(Tτ12 )∗TTZ = T(τ1
2 )∗TZ = T(Tτ01 )∗TZ = TT(τ0
1 )∗Z
(pull-back commutes with the exterior derivative and τ12
coincides with Tτ01 ).
• The AV-bundle for iF dλ is (Tτ01 )∗TZ = T(τ0
1 )∗Z.
• dT = iTd + diT and the first term gives an ordinary 1-form.Hence the AV-bundle for dT iF dλ is just the AV-bundle ford iT iF dλ, i.e. TT(τ0
1 )∗Z
Tenerife, 14.09.2006 – p. 12/13
Euler-Lagrange equation. Affine case
• The AV-bundle for (τ12 )∗dλ is
(Tτ12 )∗TTZ = T(τ1
2 )∗TZ = T(Tτ01 )∗TZ = TT(τ0
1 )∗Z
(pull-back commutes with the exterior derivative and τ12
coincides with Tτ01 ).
• The AV-bundle for iF dλ is (Tτ01 )∗TZ = T(τ0
1 )∗Z.
• dT = iTd + diT and the first term gives an ordinary 1-form.Hence the AV-bundle for dT iF dλ is just the AV-bundle ford iT iF dλ, i.e. TT(τ0
1 )∗Z
• The AV-bundle for ((τ12 )∗dλ − dT(iF dλ)) is trivial.
Tenerife, 14.09.2006 – p. 12/13
As in the standard case, the form ((τ12 )∗dL − dT(iF dL)) is
semi-basic and can be interpreted as a mapping T2M → T
∗M .
Tenerife, 14.09.2006 – p. 13/13
As in the standard case, the form ((τ12 )∗dL − dT(iF dL)) is
semi-basic and can be interpreted as a mapping T2M → T
∗M .
Also iF dλ is semi-basic and can be interpreted as a mappingTM → PZ.
Tenerife, 14.09.2006 – p. 13/13
As in the standard case, the form ((τ12 )∗dL − dT(iF dL)) is
semi-basic and can be interpreted as a mapping T2M → T
∗M .
Also iF dλ is semi-basic and can be interpreted as a mappingTM → PZ.
Forces are co-vectors on M , momenta are affine co-vectors,elements of PZ.
Tenerife, 14.09.2006 – p. 13/13