A Primer on Geometric Mechanics [5pt] Variational ...isg › graphics › teaching › 2012 › gm_prime… · Variational mechanics Reduced variational principles: Euler-Poincar

A Primer on Geometric Mechanics

Variational principles andHamiltonian Mechanics

Alex L. Castro, PUC Rio de Janeiro

Henry O. Jacobs, CMS, Caltech

Christian Lessig, CMS, Caltech

Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 1 / 43

Outline

Overview

Variational mechanics

The Hamiltonian Picture

Bibliography

Overview

Course Outline


Outline

Overview



Bibliography


The principle of least actionFeynman’s lectures on Physics, vol. I – Lecture 19



Lagrange’s equations

Notation: q, q ∈ Rd and q(t) is a smooth path in Rd .Given a Lagrangian L(q, q), Lagrange’s equation of motion is

d

dt∇qL(q, q)−∇qL(q, q) = 0.

This equation is the Euler-Lagrange equation minimizing theaction integral (functional) S [q(t)] :=

∫ t1t0L(q(t), q(t))dt.



Euler-Lagrange equations

S(q(t) + h(t))− S(q(t)) =

=

∫ t1

t0

(L(q + h, q + h)− L(q, q)dt

=

∫(∂L

∂qh +

∂L

∂qh)dt + O(h2)

=

∫(− d

dt

∂L

∂q+∂L

∂q)hdt

+ boundary term + O(h2)

⇒ δS = 0⇒ − d

dt

∂L

∂q+∂L

∂q= 0



Hamilton’s principleMost famous action integral from classical mechanics isS =

∫(T − U)dt, where

T = kinetic energy

U =potential energy

For a particle of mass m in a constant gravitational field g k,

S =

∫ t2

t1

[1

2m(

dq

dt)2 −mgq],

where q is the height measured from ground level. E.-L. eqn.’s:

q = −g/m.

Hamilton stated his principle in 1834-35.Alex L. Castro (PUC-Rio) Variational principles and Hamiltonian Mechanics 8 / 43


Example/exercise:

Consider a particle moving in a constant force field(e.g. gravity near earth, g k) and starting at (x1, y1) (rest) anddescending to some lower point (x2, y2). Find the path thatallows the particle to accomplish the transit in the least possibletime.Hint. Compute the Euler-Lagrange equations for thetransit time functional given by

time =

∫ x2

x1

√(1 + y ′2)/2gxdx .

Can you describe the solution curves geometrically?



Calculus of variations

For us,

calculus of variations = calculus with functionals

A functional is a scalar field whose domain is a certain space offunctions (e.g. C k paths γ(t) on [0, 1] plus bdry. conditions).E.g. (calculus): arc length, area, time to travel etc.

∆s ≈√

∆x2 + ∆y 2

⇒ s =

∫ b

a

√1 + y ′(x)dx .



An important remark

The condition that q(t) be an extremal of a functional does notdepend on the choice of a coordinate system.For example, arc length of q(t) is given in different coordinatesby different formulas

s =

∫ t1

t0

√x21 + x22dt (cartesian),

s =

∫ t1

t0

√r 2 + r 2φ2dt (polar).

However, extremals are the same: straight lines in the plane.



Modern take on Variational Calculus

Hamilton’s principle has been generalized to variousnonlinear/curved contexts (e.g. constraints, optimal control, Liegroups (matrix groups), field theories etc.). Focus later onwill be on motion on Lie Groups (Henry Jacobs).

Dynamics on Lie groups:

I tops,

I fluids,

I plasma,

I Maxwell-Vlasov equations,

I Maxwell’s equations etc.



Variational Calculus on ManifoldsLet (Qn, gij) be a Riemannian manifold.

Let V(t) = q(t),A(t) = D

dtV(t) and

J(t) = Ddt

A(t). Examples offunctionals on a Riemannianmanifold:

1. S1 =∫ t1t0gq(t)(q(t), q(t))dt

⇒ E-L: D2qdt2

= 0(geodesic motion).

2. S2 =∫ t1t0gq(t)(q(t), q(t))dt

⇒E-L: DJ

dt− R(V,A)V = 0.



Motion on a potential field

We can generalize geodesic motion to include potentialsV : Q → R. The action functional is now

S =

∫ t1

t0

(1

2gq(t)(q(t), q(t))dt − V (q(t))dt

⇒ E.L. :D2

dtq(t)− gradV (q(t)) = 0.



Reduced variational principles: Euler-Poincare I

X reference configurationx spatial configuration

R(t) motion

R(t) = ddε|ε=0R(t + ε) gen. velocity

SO(3) = orthogonal matrices configuration spaceTSO(3) = tangent bundle state space



Reduced variational principles: Euler-Poincare II

x(t) = R(t)X,

X is a point on the reference configuration. Therefore,

x = RR−1x(t).

Exercise. RR−1 = ω is an anti-symmetric matrix.The kinetic energy of the body is:

L(R, R) = kinetic =

∫body

1

2dm x2

=1

2

∫body

m||Ω× X ||2dX =1

2〈IΩ,Ω〉 := l(Ω).



Reduced variational principles: Euler-Poincare IIITheorem (Poincare(1901-02): Geometric Mechanicsis born)Hamilton’s principle for rigid body actionδS = δ

∫ t1t0L(R, R)dt = 0 is equivalent to

δSred = δ

∫ t1

t0

l(Ω)dt = 0,

with Ω ∈ R3 and for variations of the form δΩ = Σ + Ω×Σ,and bdry. conditions Σ(a) = Σ(b) = 0.

How do they look like for the rigid body equation?Reduced Lagrange’s equations are called Euler-Poincareequations. Euler-Poincare equations occur for many systems:fluids, plasma dynamics etc.



What’s next? Lagrangian Reduction and other

bargains.

Kummer equations, Lagrange-Poincare equations etc.



Example/exercise: discrete variational mechanicsConsider the Lagrangian function L(q, q) and the action integral

S [q(t)] :=

∫ t1

t0

L(q(t), q(t))dt.

We replace the integral by a finite sum (discrete action)

Sdis[qn] =∑n

L(qn,qn+1 − qn

∆t)∆t

and find the local minimizer from the condition

∂

∂qnS [qn] = 0.

What numerical scheme do you obtain by explicitly evaluatingthe previous formula for a density L(q, q) = q2/2− V (q)? Thisderivation is a simple example of a simple discrete variationalprinciple.


Outline

Overview



Bibliography


The Legendre Transform; Hamiltonian mechanics I

Let y = f (x) be convex. Define

g(p) = maxx

(p x − f (x)).

Exercise. Experiment to compute the Legendre transform of aconvex function that’s a broken line.



The Legendre Transform; Hamiltonian mechanics IITake the Legendre transform w.r.t. v = q of L(q, v) and obtainH(q, p) called Hamiltonian function.After passing to the Hamiltonian side of the picture (on S-S.Chern’s word’s: “The sophisticated side.”) we obtain thatLagrange’s equations become:

q =∂H(q,p)

∂p,

p = −∂H(q,p)

∂q.

q generalized coordinatesq generalized velocities

p := ∂L∂q

generalized momentum∂L∂q

generalized force field



Canonical and non-canonical Hamiltonian

structures ILet z = (q,p)T ∈ Rd where d = no. of D.O.F. The space ofpositions and generalized momenta is called phase space. Itoften has the structure of a cotangent bundle.

Define J :=

[0d Id

−Id 0d

]. The block-matrices 0d, Id are d × d

matrices.A canonical Hamiltonian system is an O.D.E. system of the form

z = J∇zH(z).

For a mechanical system with Lagrangian L(q, v), theHamiltonian function is

H(q,p) = (p, v)− L(q, v)




structures II

and v = v(p) by the Legendre transform.




structures III

Example 1. H = (zTLz)/2⇒ z = (JL)z. Matrices of theform JL w/ L symmetric are called Hamiltonian matrices.They generate the algebra of infinitesimally symplectic matrices.(More about symplectic transformations later on.)Take for example the harmonic oscillator Hamiltonian

H = 12[q, p]

[ω2 00 1

] [qp

]= p2/2 + ω2q2/2.

Re-scaling: p = p√ω and q = q/

√ω we can re-write

Hamiltonian as,H = ω(p2 + q2)/2.




structures IV

In the new (canonical) coordinates:

d

dt

[qp

]=

[0 ω−ω 1

] [qp

]and exponentiating[

q(t)p(t)

]= exp(t

[0 ω−ω 1

])

[q0p0

]= Φωt

[q0p0

],

and Φωt is a rotation matrix in the plane.



Spectral structure of Hamiltonian matrices

λ eigenvalue ⇒ −λ, λ,−λ are also eigenvalues.Proof: JLv = λv⇒ JL(Jw) = λJw⇒ −L(Jw) = −λw, but

(LJ) = −(JL)T and therefore −λ is eigenvalue of thetransposed Hamiltonian matrix.



Examples I

(1) One D.O.F. problems.(2) Central forces.(3) Charged particle in a magnetic field (non-canonicalHamiltonian system):

ddt

[qp

]=

[0d Id

Id b

] [qp

], the matrix Jb is an example of a

non-canonical Hamiltonian structure.For this non canonical structure, the charged particleHamiltonian is written as:

H(q,p) =1

2m|p|2 − γ/|q|.



Examples II

(4) N-body problems.

Fij = −∂φij(rij)

∂rij

andH = 1

2

∑Ni=1 |pi

2|/mi +∑N−1

i=1

∑Nj=i+1 φ(rij) = T + U .

(5) Rigid-body motion, other types of Lie-Poissondynamics etc.



First Integrals and Poisson Brackets I

A first integral (or integral of motion) is a functionG : R2d → R such that G (z(t; z0) = G (z0). This physicalquantities are conserved along the trajectories (solution curves)of a Hamiltonian system

z = J∇zH(z).

First integrals usually lead to geometric reduction of theproblem: solution curves will live in G = constant.Problem: find a practical way to determine a function isan integral of motion.



First Integrals and Poisson Brackets IIUse the chain rule:

d

dt(G (z(t; z0)) = ∇zG (z(t; z0))T d

dtz(t; z0),

but ddt

z(t; z0) = J∇zH(z(t; z0)) and therefore

d

dt(G (z(t; z0)) = ∇zG (z(t; z0))TJ∇zH(z(t; z0)).

This leads us to introduce the following bilinear operation onscalar fields defined on phase space:

F ,G(z) := ∇zF (z)TJ∇zG (z) (Poisson bracket) .



First Integrals and Poisson Brackets III

1. antisymmetry F ,G = −G ,F.2. A fundamental property of the Poisson bracket is the

Jacobi identity:

H , F ,G = −G , H ,F − F , G ,H.

Exercise. Check that the components of the vectorm = q× p is a conserved quantity for the system with

Hamiltonian H(p,q) = |p|22− 1|q| .

The matrix J does not need to be constant. Non-canonicalstructures are very common all over physics and mechanics.Take for instance the following poisson structure in R3:

F (M),G (M)EP := 〈∇F (M),∇C (M)×∇G (M)〉,



First Integrals and Poisson Brackets IV

C = (M21 + M2

2 + M23 )/2 = |M|2/2.

Take H = 〈M, I−1M〉/2 (rigid body kinetic energy), therefore

M = ∇C ×∇H = M× I−1M.

Our non-constant Poisson structure is

JEP :=

0 −M3 M2

M3 0 M1

−M2 M1 0

,and the Poisson bracket becomes

F (M),G (M)EP = ∇FTJEP∇G.



First Integrals and Poisson Brackets V

The inertia tensor can be made diagonal by a orthogonalchange of basis, and

I = diag(I1, I2, I3),

I1 > I2 > I3.We need to check that surfaces H = const and C = constare invariant manifolds.



First Integrals and Poisson Brackets VIBy intersecting differentellipsoids

H =M2

1

I1+M2

2

I2+M2

3

I3= const

and the sphere

C = M21 + M2

2 + M23 = const

we obtain the reduced solutioncurves depicted in the bluesphere. Using reconstructionformulas we can compute theassociated motion in SO(3).

The famous picture from JEM’sbook cover.



Applications of Poisson structures

Lie-Poisson integrators, Lie-transformation methods inbifurcation theory, field theories, constrained Hamiltoniansystems etc.



Hamiltonian flows IFlows generated by Hamiltonian vector fields possess manyuseful geometric properties.The Hamiltonian vector field XH(z) = J∇H(z) generates a flowon the manifold M2d , often T ∗Q with coordinates (q,p).For example, consider a free particle moving in space

q = 0.

Its equations of motion in Hamiltonian form are

q = p,

p = 0,

and the corresponding Hamiltonian is Hpart(q,p) = |p|2/2.The flow map is

ΦtH(q0,p0) = (q0 + tp0, p0).



Hamiltonian flows II

The mapping ΦtH is a 1-parameter family of transformations of

R2d .Another useful example is the harmonic oscillator,Hosc = p2/2 + ω2q2/2. The flow generated by the Hamiltonianvector field in this case is

ΦtH(p0, q0) =

[cos(ωt) ω−1 sin(ωt)ω sin(ωt) sin(ωt)

] [q0p0

],

which is conjugate to a rotation matrix in the plane.An important property of Hamiltonian flows is that theyinfinitesimally preserve the symplectic (resp. Poisson) structure.



Hamiltonian flows III

This means that Φz(z)J(z)Φz(z)T = J(z)



Examples/exercises

1. A particle in a central field

L =1

2||q||2 − (−1/||q||),

and since m = 1, p = q.

2. A charged particle in a magnetic field

L =m

2||q||2 − (−γ/||q|| − 1

2B(q, q)),

B is an anti-symmetric matrix representing a constantmagnetic field. In this case, p = mq− 1

2B(q). Example

of a non-canonical hamiltonian system. More later.



The braquistocrhone problem: Pontryagin Principle

Our next goal is to connect the calculus of variations withHamiltonian mechanics.


Outline

Overview



Bibliography

Bibliography

Bibliography


Documents

A Primer on Geometric Mechanics [5pt] Variational ...isg › graphics › teaching › 2012 › gm_prime… · Variational mechanics Reduced variational principles: Euler-Poincar