The Hamilton-Pontryagin Principle and the Hamel Equations · 2011-12-29 · The Hamilton-Pontryagin Principle and the Hamel Equations KennethBall withDmitryZenkov Department of Mathematics,

The Hamilton-Pontryagin Principle and the HamelEquations

Kenneth Ball

with Dmitry Zenkov

Department of Mathematics, North Carolina State University

SIAM SEAS 2011

Charlotte, March 2011

Kenneth Ball (NCSU) The HP Principle & the Hamel Equations Charlotte, March 2011 1 / 20

Introduction

Goals

Provide a brief exposition to the principle of least action on theHamilton-Pontryagin space.

(Re)introduce the Hamel equations of motion and compare them to theEuler-Lagrange equations.

Illustrate our constraint-free derivation of the Hamel equations utilizing theprinciple of least action on the Hamilton-Pontryagin bundle.


Introduction

Goals





Introduction

Goals





The Hamilton-Pontryagin principle The HP bundle

Concatenated velocity and momentum space

Suppose our configuration space Q is an n-dimensional differentiablemanifold with local coordinates in an open neighborhood of q ∈Q defined as

q 7→ (q i )ni=1

The coordinate chart induces a frame and coframe on T Q and T ∗Q so thatvectors and covectors over q may be written as

v = v i ∂

∂q i∈ TqQ and p = pi d q i ∈ T ∗

q Q.




Suppose our configuration space Q is an n-dimensional differentiablemanifold with local coordinates in an open neighborhood of q ∈Q defined as

q 7→ (q i )ni=1

The coordinate chart induces a frame and coframe on T Q and T ∗Q so thatvectors and covectors over q may be written as

v = v i ∂

∂q i∈ TqQ and p = pi d q i ∈ T ∗

q Q.




We define the Hamilton-Pontryagin bundle as a “concatenation” (or directsummation) of velocity and momentum space

HP = TQ ⊕T ∗Q.

Then, at each q ∈Q, the fiber HPq is just the Cartesian product of the vectorspaces TqQ and T ∗

q Q,

HPq = TqQ ×T ∗q Q.




We define the Hamilton-Pontryagin bundle as a “concatenation” (or directsummation) of velocity and momentum space

HP = TQ ⊕T ∗Q.

Then, at each q ∈Q, the fiber HPq is just the Cartesian product of the vectorspaces TqQ and T ∗

q Q,

HPq = TqQ ×T ∗q Q.




q

TqQ

⊕T∗

q Q

= HPq


The Hamilton-Pontryagin principle Variational principle

The action on HP

Consider the set of smooth curves on the 3n-dimensional differentiablemanifold HP . Such a curve, parameterized by time t , may be represented inlocal coordinates as

t 7→(q i (t ), v j (t ), pk (t )

)n

i , j ,k=1.

Note that v(t ) corresponds to the observed dynamic velocity of such asystem, which is not a priori the time derivative q(t ) denoted by q(t ). Weshall see that the covectors p(t ) will both impose the condition that q = vand correspond to the momentum of the system. [5]Given a Lagrangian function L : TQ →R. We define an action function S as

S(q, v, p) =∫ t1

t0

[L(q(t ), v(t ))+⟨

p(t ), q(t )− v(t )⟩]

d t .



The action on HP


t 7→(q i (t ), v j (t ), pk (t )

)n

i , j ,k=1.


S(q, v, p) =∫ t1

t0

[L(q(t ), v(t ))+⟨

p(t ), q(t )− v(t )⟩]

d t .



The action on HP


t 7→(q i (t ), v j (t ), pk (t )

)n

i , j ,k=1.


S(q, v, p) =∫ t1

t0

[L(q(t ), v(t ))+⟨

p(t ), q(t )− v(t )⟩]

d t .



The principle of least action

Variation of curves (q(t ), v(t ), p(t )) allows us to consider all possibledynamical paths of a system in terms of perturbations, so that a new path(q(t ), v(t ), p(t )) might be described in terms of the original as

(q i (t ), v j (t ), pk (t )) = (q i (t )+δq i (t ), v j (t )+δv j (t ), pk (t )+δpk (t ))

We require that the position curve q(t ) be fixed at the boundary

δq(t0) = δq(t1) = 0

q0

q1

q(t )

q(t )

δq(t )

We observe that at each t , for some choice of pertubation, δq,δv and δp arelinearly independent.Kenneth Ball (NCSU) The HP Principle & the Hamel Equations Charlotte, March 2011 7 / 20






δq(t0) = δq(t1) = 0

q0

q1

q(t )

q(t )

δq(t )







δq(t0) = δq(t1) = 0

q0

q1

q(t )

q(t )

δq(t )




We seek paths (q(t ), v(t ), p(t )) that extremize S. For such paths (q, v, p),δS = 0 up to first order in (δq,δv,δp) where

δS = S(q +δq, v +δv, p +δp)−S(q, v, p).

Following through with the variational derivative, we find

δS =∫ t1

t0

[⟨∂L

∂q− p,δq

⟩+

⟨∂L

∂v−p,δv

⟩+⟨

δp, q − v⟩]

d t = 0.

By linear independence of δq, δv and δp, we see that if a curve t 7→ (q, v, p)extremizes S, it must satisfy the following equations.

∂L

∂q= p,

∂L

∂v= p and q = v.

This is exactly the Euler-Lagrange equation – assuming the Legendretransform is invertible – coupled with the condition that the velocity v be thetime derivative of position (a phenomenon certainly consistent withobservation).Kenneth Ball (NCSU) The HP Principle & the Hamel Equations Charlotte, March 2011 8 / 20






δS =∫ t1

t0

[⟨∂L

∂q− p,δq

⟩+

⟨∂L

∂v−p,δv

⟩+⟨

δp, q − v⟩]

d t = 0.


∂L

∂q= p,

∂L

∂v= p and q = v.







δS =∫ t1

t0

[⟨∂L

∂q− p,δq

⟩+

⟨∂L

∂v−p,δv

⟩+⟨

δp, q − v⟩]

d t = 0.


∂L

∂q= p,

∂L

∂v= p and q = v.







δS =∫ t1

t0

[⟨∂L

∂q− p,δq

⟩+

⟨∂L

∂v−p,δv

⟩+⟨

δp, q − v⟩]

d t = 0.


∂L

∂q= p,

∂L

∂v= p and q = v.


The Hamel equations A coordinate independent frame

The Hamel frame and coframe

We will define a Hamel frame to be a smooth section of the frame bundleover Q. The Hamel frame is a geometric object that is independent of ourchoice of local coordinates. However, given local coordinates (q i )n

i=1, we maydefine the Hamel frame in terms of the induced coordinate frame:

ui (q) =ψ ji (q)

∂

∂q j.

where ψ(q) ∈GLn(R) for all q ∈Q.Likewise, we will define a Hamel coframe as the set of covectors dual to theHamel frame,

ui (q) =φij (q)d q j

where φ=ψ−1 so that

ui (q)u j (q) = δij .



The Hamel frame and coframe

We will define a Hamel frame to be a smooth section of the frame bundleover Q. The Hamel frame is a geometric object that is independent of ourchoice of local coordinates. However, given local coordinates (q i )n

i=1, we maydefine the Hamel frame in terms of the induced coordinate frame:

ui (q) =ψ ji (q)

∂

∂q j.

where ψ(q) ∈GLn(R) for all q ∈Q.Likewise, we will define a Hamel coframe as the set of covectors dual to theHamel frame,

ui (q) =φij (q)d q j

where φ=ψ−1 so that

ui (q)u j (q) = δij .



The Hamel frame

∂∂q2

∂∂q1

u1(q0)u2(q0) u1(q1)u2(q1)

Tq0Q Tq1Q



Vectors and covectors in the Hamel frame

Consider a vector v ∈ TqQ. In the local coordinate frame, v has components(v i )n

i=1 such that

v = v i ∂

∂q i.

We define a new set of components (η j )nj=1 for the same velocity vector v ,

this time with respect to the Hamel basis:

v = η j u j (q) = η jψij (q)

∂

∂q i= v i ∂

∂q i.

Likewise, we will define a new set of components corresponding to thecovector p ∈ T ∗

q Q and the time derivative of the path, q ∈ TqQ with respectto the Hamel coframe and frame respectively, so that

p =µ j u j (q) =µ jφji (q)d q i = pi d q i

q = ξ j u j (q) = ξ jψij (q)

∂

∂q i= q i ∂

∂q i.





i=1 such that

v = v i ∂

∂q i.




∂

∂q i= v i ∂

∂q i.





∂

∂q i= q i ∂

∂q i.





i=1 such that

v = v i ∂

∂q i.




∂

∂q i= v i ∂

∂q i.





∂

∂q i= q i ∂

∂q i.


The Hamel equations The equations of motion

The Hamel equations

Consider the Lagrangian function of our mechanical system, defined in termsof the local coordinate-induced frame:

L : T Q →R, L = L(q i , q j ).

We may define the function ` : TQ →R as L with respect to the change ofvelocity coordinates to the Hamel frame:

`(qk ,ξi ) = L(qk ,ξiψji (q)) = L(qk , q j ).



The Hamel equations

Consider the Lagrangian function of our mechanical system, defined in termsof the local coordinate-induced frame:

L : T Q →R, L = L(q i , q j ).

We may define the function ` : TQ →R as L with respect to the change ofvelocity coordinates to the Hamel frame:

`(qk ,ξi ) = L(qk ,ξiψji (q)) = L(qk , q j ).



The Hamel Equations

At each q ∈Q, let the the constants cki j (q) correspond to the components of

the Jacobi-Lie bracket of the vector fields in the Hamel frame, that is

[ui (q),u j (q)] = [ui (q),u j (q)]k uk (q) = cki j (q).

These components may also be written with respect to ψ and φ in coordinateform as

cki j (q) =φk

m

(ψr

i

∂ψmj

∂qr −ψrj

∂ψmi

∂qr

)The Hamel equations, defined in coordinates as

d

d t

∂`

∂ξ j= u j [`]+ ck

i j∂`

∂ξkξi

are equivalent to the Euler-Lagrange equations

d

d t

∂L

∂q i= ∂L

∂q i.



The Hamel Equations





cki j (q) =φk

m

(ψr

i

∂ψmj

∂qr −ψrj

∂ψmi

∂qr


d

d t

∂`

∂ξ j= u j [`]+ ck

i j∂`

∂ξkξi


d

d t

∂L

∂q i= ∂L

∂q i.



The Hamel Equations





cki j (q) =φk

m

(ψr

i

∂ψmj

∂qr −ψrj

∂ψmi

∂qr


d

d t

∂`

∂ξ j= u j [`]+ ck

i j∂`

∂ξkξi


d

d t

∂L

∂q i= ∂L

∂q i.


The HP principle and the Hamel equations The HP variational principle wrt the Hamel frame

The HP action with respect to the Hamel frame

We would like to redefine the action S in terms of components in the Hamelframe and coframe. We define s as an action the same set of smooth curveson HP as

S(q i , v j , pk ) = s(q i ,η j ,µk ) =∫ t1

t0

[`(q i ,η j )+⟨p, q − v⟩

]d t

=∫ t1

t0

[`(q i ,η j )+µk (ξk −ηk )

]d t

.Again, we would like to find paths in HP that extremize the action s, so wetake the variational derivative as it was defined before:

δs = 0 =∫ t1

t0

[∂`

∂q iδq i + ∂`

∂η jδη j +δ(

µk (ξk −ηk ))]

d t

=∫ t1

t0

[∂`

∂q iδq i + ∂`

∂η jδη j +δµk (ξk −ηk )+µkδξ

k −µkδηk]

d t .



The HP action with respect to the Hamel frame

We would like to redefine the action S in terms of components in the Hamelframe and coframe. We define s as an action the same set of smooth curveson HP as

S(q i , v j , pk ) = s(q i ,η j ,µk ) =∫ t1

t0

[`(q i ,η j )+⟨p, q − v⟩

]d t

=∫ t1

t0

[`(q i ,η j )+µk (ξk −ηk )

]d t

.Again, we would like to find paths in HP that extremize the action s, so wetake the variational derivative as it was defined before:

δs = 0 =∫ t1

t0

[∂`

∂q iδq i + ∂`

∂η jδη j +δ(

µk (ξk −ηk ))]

d t

=∫ t1

t0

[∂`

∂q iδq i + ∂`

∂η jδη j +δµk (ξk −ηk )+µkδξ

k −µkδηk]

d t .



Linear independence of variations

Our assumption of linear independence of the 3n vectors and covectorsdescring variations in HP carries over to their counterparts in the Hamelframe. That is, δq i ,δv j and δpk are linearly independent if and only ifδq i ,δη j and δµk are linearly independent.Also, since, ξk =φk

j q j , we find that

δξk = δ(φk

j q j)=φk

j ,i q jδq i +φkj

d

d tδq j .

Then the term µkδξk in our action may be rewritten as∫ t1

t0

µkδξk d t =

∫ t1

t0

[µkφ

kj ,i q jδq i +µkφ

kj

d

d tδq j

]d t

=∫ t1

t0

[µkφ

kj ,iξ

mψjmδq i +p j

d

d tδq j

]d t

=∫ t1

t0

[µkφ

kj ,iξ

mψjmδq i − p jδq j

]d t +p jδq j

∣∣∣t1

t0






δξk = δ(φk

j q j)=φk

j ,i q jδq i +φkj

d

d tδq j .


t0

µkδξk d t =

∫ t1

t0

[µkφ


kj

d

d tδq j

]d t

=∫ t1

t0

[µkφ

kj ,iξ

mψjmδq i +p j

d

d tδq j

]d t

=∫ t1

t0

[µkφ

kj ,iξ


]d t +p jδq j

∣∣∣t1

t0






δξk = δ(φk

j q j)=φk

j ,i q jδq i +φkj

d

d tδq j .


t0

µkδξk d t =

∫ t1

t0

[µkφ


kj

d

d tδq j

]d t

=∫ t1

t0

[µkφ

kj ,iξ

mψjmδq i +p j

d

d tδq j

]d t

=∫ t1

t0

[µkφ

kj ,iξ


]d t +p jδq j

∣∣∣t1

t0


The HP principle and the Hamel equations Restating the Hamel equations

The Hamel equations

Variations are defined as zero at the boundary, so the term we integrated outis zero. The variation of the action now reads

δs =∫ t1

t0

[(∂`

∂q i+µkξ

mψjmφ

kj ,i − pi

)δq i

+(∂`

∂η j−µ j

)δη j

+ δµk

(ξk −ηk

)]d t

From this point, it is a matter of index gymnastics to show that the aboveequations may be rewritten as

δs =∫ t1

t0

[(∂`

∂q iψi

j − µ j +µkξiψr

i ψmj

(φk

r,m −φkm,r

))φ

jsδq s

+(∂`

∂η j−µ j

)δη j

+ δµk

(ξk −ηk

)]d t



The Hamel equations

Variations are defined as zero at the boundary, so the term we integrated outis zero. The variation of the action now reads

δs =∫ t1

t0

[(∂`

∂q i+µkξ

mψjmφ

kj ,i − pi

)δq i

+(∂`

∂η j−µ j

)δη j

+ δµk

(ξk −ηk

)]d t

From this point, it is a matter of index gymnastics to show that the aboveequations may be rewritten as

δs =∫ t1

t0

[(∂`

∂q iψi

j − µ j +µkξiψr

i ψmj

(φk

r,m −φkm,r

))φ

jsδq s

+(∂`

∂η j−µ j

)δη j

+ δµk

(ξk −ηk

)]d t



The Hamel equations

By linear independence of the variations δq, δη and δp the above equationresults in the following three conditions, analogous to those derived earlier

µ j = u j [`]+µkξi ck

i j (q), µ j = ∂`

∂η jand ξk = ηk

This is, again, the system of Hamel equations along with the Legendretransform (defined in the Hamel coframe) coupled with the condition thatdynamical velocity correspond to the time derivative of position (with respectto components in the Hamel frame).



The Hamel equations

By linear independence of the variations δq, δη and δp the above equationresults in the following three conditions, analogous to those derived earlier

µ j = u j [`]+µkξi ck

i j (q), µ j = ∂`

∂η jand ξk = ηk

This is, again, the system of Hamel equations along with the Legendretransform (defined in the Hamel coframe) coupled with the condition thatdynamical velocity correspond to the time derivative of position (with respectto components in the Hamel frame).


Concluding remarks Summary

Summary and roadmap for future work

This derivation of the Hamel equations from an action on theHamilton-Pontryagin bundle is significant because we did not imposeconstraints on our variations. Previous derivations [1] of the Hamel equationshave considered variations of paths in Q, and constrained velocity variationsso that

δv = w + [v, w]q where w i =ψijδq j ,

whereas in our derivation, we consider variations of paths in HP and weneedn’t impose any constraints upon our variations.Bou-Rabee and Marsden [2] had success studying integrators on rigidbody-like mechanical systems using a discrete version of theHamilton-Pontryagin principle.Kobilarov et al. [5] also utilized a discrete version of this principle coupledwith the Lagrange D’Lambert principle to study nonholonomic systems withsymmetry.We expect that a discrete version of this Hamilton-Pontryagin principle willbe useful in studying integrators of more general systems.Kenneth Ball (NCSU) The HP Principle & the Hamel Equations Charlotte, March 2011 18 / 20

Concluding remarks Bibliography (very nice)

Bibliography

1 Bloch, A. M., J. E. Marsden, and D. V. Zenkov [2009], Quasivelocities andSymmetries in Nonholonomic Systems, Dynamical Systems: An InternationalJournal 24, 187–222.

2 Bou-Rabee, N. and J. E. Marsden [2009], Hamilton–Pontryagin Integrators on LieGroups Part I: Introduction and Structure-Preserving Properties, Foundations ofComputational Mathematics 9Ê, 197–219.

3 Euler, L. [1752], Decouverte d’un nouveau principe de Mecanique, Mémoires del’académie des sciences de Berlin 6, 185–217.

4 Hamel, G. [1904], Die Lagrange–Eulersche Gleichungen der Mechanik, Z. Math.Phys. 50, 1–57.

5 Kobilarov, M., J. E. Marsden and G. Sukhatme [2010], Geometric Discretization ofNonholonomic Systems with Symmetries, Discrete and Continuous DynamicalSystems Series S 3, 1, 61–84.

6 Yoshimura, H. and J. E. Marsden [2006], Dirac Structures in Lagrangian Mechanics.Part I: Implicit Lagrangian Systems, Journal of Geometry and Physics 57, 133–156.

7 Yoshimura, H. and J. E. Marsden [2006], Dirac Structures in Lagrangian Mechanics.Part II: Variational Structures, Journal of Geometry and Physics 57, 209–250.


Concluding remarks Errata

Errata

Apparantly, HP = TQ ⊕T ∗Q can be referred to as the Whitney sum bundle ofthe manifold Q. See Yoshimura and Marsden, Nov 12, 2007.I totally misspelled D’Alembert.


Documents

The Hamilton-Pontryagin Principle and the Hamel Equations · 2011-12-29 · The Hamilton-Pontryagin Principle and the Hamel Equations KennethBall withDmitryZenkov Department of Mathematics,