Hamels Formalism and Variational Integrators Dmitry Zenkov - - PowerPoint PPT Presentation

Hamel’s Formalism and Variational Integrators

Dmitry Zenkov Department of Mathematics, North Carolina State University with Ken Ball Nonholonomic Mechanics and Optimal Control Institute Henri Poincaré Paris, November 25–28, 2014 Supported by the NSF Grants DMS-0908995 and DMS-1211454

Key Concepts

• Ideal constraints and reaction forces

(Hermann, d’Alembert, Euler, Lagrange, Fourier, Laplace...)

• Euler–Lagrange equations and variational principles

(Euler, Lagrange, Hamilton, Poincaré, Boltzmann, Hamel...)

• Geometric and structure-preserving integration

(Veselov, Marsden, Leok, Hairer, Lubich, Wanner...)

Lagrangian Mechanics with Constraints

• Lagrangian (kinetic minus potential energy) L : TQ → R on the

configuration space Q

• Ideal velocity constraints

a(q), ˙ q

• = 0, for each q ∈ Q
• Holonomic or nonholonomic
• Can be replaced with reaction forces, work along the constrained directions

vanishes

• Define the constrained submanifold and a projection onto this submanifold
• Euler–Lagrange equations (Lagrange [1788])

d dt ∂L ∂ ˙ q − ∂L ∂q = λa

• Represent the dynamics in a covariant (coordinate-independent) form
• Equivalent to the Lagrange–d’Alembert variational principle

Discrete Hamilton’s Principle and Variational Integrators

• Discrete analogue of the continuous-time phase flow:

Q ×Q ∋ (qk−1,qk) → (qk,qk+1) ∈ Q ×Q

• Discrete Lagrangian: Ld : Q ×Q → R;

Ld(qk,qk+1) ≈ h L(q(t), ˙ q(t))dt, usually Ld(qk,qk+1) = hL qk+qk+1

2

, qk+1−qk

h

• Discrete Hamilton’s principle: Trajectory q0,q1,...,qN is defined by

δ

N−1

• k=0

Ld(qk,qk+1) = 0,

where

δq0 = δqN = 0

• Discrete Euler–Lagrange equations

D1Ld(qk,qk+1)+D2Ld(qk−1,qk) = 0

• Lagrangian symplectic form preservation
• Momentum preservation for systems with symmetry
• Good long-term numerical behavior and no numerical dissipation

Constraint Discretization

• Constraint preservation – important in multibody systems
• Holonomic constraints:

f (q) = 0 (continuous-time), then f (qk) = 0

• Not so obvious for velocity/nonholonomic constraints
• Discrete Lagrange–d’Alembert principle

(Cortés and Martínez [2001], Fedorov and Zenkov [2005], McLachlan and Perlmutter [2006], Iglesias, Marrero, Martín de Diego, and Martínez [2008], Lynch and Zenkov [2009])

• Discrete constraints: A submanifold of Q ×Q
• A projection onto this submanifold
• Both may be defined in a number of ways, leading to a variety of discrete

Lagrange–d’Alembert principles

Constraint Discretization

• Constraint preservation – important in multibody systems
• Holonomic constraints:

f (q) = 0 (continuous-time), then f (qk) = 0

• Not so obvious for velocity/nonholonomic constraints
• Discrete Lagrange–d’Alembert principle

(Cortés and Martínez [2001], Fedorov and Zenkov [2005], McLachlan and Perlmutter [2006], Iglesias, Marrero, Martín de Diego, and Martínez [2008], Lynch and Zenkov [2009])

• Discrete constraints: A submanifold of Q ×Q
• A projection onto this submanifold
• Both may be defined in a number of ways, leading to a variety of discrete

Lagrange–d’Alembert principles

Nonholonomic Integrators

• A trajectory of the contact point of a discrete balanced Chaplygin sleigh

(a platform supported by a skate) may spiral down to a point, which is not what the continuous-time model predicts

• Anticipated trajectories for proper discrete constraint

Nonholonomic Integrators

• A trajectory of the contact point of a discrete balanced Chaplygin sleigh

(a platform supported by a skate) may spiral down to a point, which is not what the continuous-time model predicts

• Anticipated trajectories for proper discrete constraint

Equilibria and Structural Stability

• Loss of the concept of ideal constraints after discretization, leading to

a loss of structural stability

• Possible changes in the dimension and stability of manifolds of relative

equilibria

• g t : M → M – phase flow of a continuous-time system of interest
• h – time step
• An exact integrator: A discrete dynamical system generated by g h : M → M
• Real-life integrators are perturbations of ideal integrators

Equilibria and Structural Stability

• Loss of the concept of ideal constraints after discretization, leading to

a loss of structural stability

• Possible changes in the dimension and stability of manifolds of relative

equilibria

• g t : M → M – phase flow of a continuous-time system of interest
• h – time step
• An exact integrator: A discrete dynamical system generated by g h : M → M
• Real-life integrators are perturbations of ideal integrators

Equilibria and Structural Stability

Equilibria and Structural Stability

• Preservation of the manifold of relative equilibria and their stability type
• Importance for long-term numerical integration
• Preservation of α- and ω-limit sets
• Otherwise, structural instability of an integrator
• Utilize Hamel’s formalism

Hamel’s Equations

• u1(q),...,un(q), n = dimQ – independent vector fields on Q
• Velocity components ξ =
• ξ1,...,ξn

∈ Rn relative to u1(q),...,un(q): ˙ q = ˙ qi∂qi = ξiui(q)

• Lagrangian as a function of (q,ξ): l(q,ξ) := L
• q,ξiui(q)
• Dynamics (Euler, Lagrange, Poincaré, Boltzmann, Hamel...):

d dt ∂l ∂ξj = ca

i j (q) ∂l

∂ξa ξi +u j [l]

• ck

i j (q) – structure functions:

• ui (q),uj (q)
• (q) = ca

i j (q)ua(q)

• uj [l] – directional derivatives
• Special case: Euler–Poincaré equations

Systems with Velocity Constraints

• For a suitable frame, the constraints read ξm+1 = ··· = ξn = 0
• Constrained Hamel equations

d dt ∂l ∂ξj = ca

i j (q) ∂l

∂ξa ξi +u j [l], i, j = 1,...,m, a = 1,...,n,

coupled with ˙

q = ξiui(q)

• Systematic way of representing dynamics in redundant coordinates for

holonomic systems

• No Lagrange multipliers!!

Planar Pendulum

• Planar pendulum in redundant coordinates: L = 1

2( ˙

x2 + ˙ y2)− y, x2 + y2 = 1

• Differential-algebraic equations
• Lack of constraint preservation!!
• 1
• 0.5

0.5 1

• 1
• 0.75
• 0.5
• 0.25

0.25 0.5 0.75

Variational Principles

Hamilton’s Principle

The curve

• q(t),ξ(t)

satisfies the Hamel equations d dt ∂l ∂ξj = ca

i j (q) ∂l

∂ξa ξi +u j [l], a,i, j = 1,...,n

if and only if

δ b

a

l(q,ξ)dt = 0,

where

δq(t) = ζj (t)u j (q(t)), ζ(a) = ζ(b) = 0, and δξa(t) = ˙ ζa(t)+ca

i j (q(t))ξi(t)ζj (t)

Lagrange–d’Alembert Principle

Constrained variations: ζm+1 = ··· = ζn = 0 Constraints ξm+1 = ··· = ξn = 0, imposed after taking the variations

Variational Principles

Hamilton’s Principle

The curve

• q(t),ξ(t)

satisfies the Hamel equations d dt ∂l ∂ξj = ca

i j (q) ∂l

∂ξa ξi +u j [l], a,i, j = 1,...,n

if and only if

δ b

a

l(q,ξ)dt = 0,

where

δq(t) = ζj (t)u j (q(t)), ζ(a) = ζ(b) = 0, and δξa(t) = ˙ ζa(t)+ca

i j (q(t))ξi(t)ζj (t)

Lagrange–d’Alembert Principle

Constrained variations: ζm+1 = ··· = ζn = 0 Constraints ξm+1 = ··· = ξn = 0, imposed after taking the variations

Discrete Hamel’s Equations

• Phase space: T Rn
• Discrete Lagrangian:

ld(qk+1/2,ξk,k+1) := hl(qk+1/2,ξk,k+1)

• l(q,ξ) – continuous-time Lagrangian
• h – time-step
• qk+1/2 := 1

2

• qk+1 + qk

– discrete analogue of position q (midpoint rule)

• ξk,k+1 – discrete analogue of velocity ξ

Discrete Hamel’s Equations

Discrete Hamilton’s Principle (Ball and Zenkov [2014])

The sequence (qk+1/2,ξk,k+1) satisfies the discrete Hamel equations if and only if

δ

N−1

• k=0

ld(qk+1/2,ξk,k+1) = 0,

where

δqk+1/2 = 1

2ua(qk+1/2)

• ζa

k+1 +ζa k

• ,

ζ0 = ζN = 0,

and

δξa

k,k+1 = 1 h

• ζa

k+1 −ζa k

• + 1

2ca i j (qk+1/2)ξi k,k+1

• ζj

k+1 +ζj k

• ,

Discrete Hamel’s Equations

• The principal step:

δqk+1/2 = 1

2ua(qk+1/2)

• ζa

k+1 +ζa k

• ,

δξa

k,k+1 = 1 h

• ζa

k+1 −ζa k

• + 1

2ca i j (qk+1/2)ξi k,k+1

• ζj

k+1 +ζj k

• In the continuous-time case,

δq = ua(q)ζa, δξa = ˙ ζa +ca

i j (q)ξi ζj ;

coming from

d dt

• ζi ui
• = δ
• ξi ui
• Less straightforward in the discrete case: A discrete analogue of

time-differentiation needed

• Certain flexibility in discretizing the kinematic equation ˙

q = ξiui(q)

Discrete Nonholonomic Systems

• Discrete Hamel’s formalism with velocity constraints:

1 Fields ui(q) such that continuous-time constraints read

ξm+1 = ξm+2 = ··· = ξn = 0

2 Constrained variations: ζm+1 k

= ··· = ζn

k = 0 3 Discrete constraints: ξm+1 k,k+1 = ξm+2 k,k+1 = ··· = ξn k,k+1 = 0,

imposed after taking the variations

• New version of the discrete Lagrange–d’Alembert principle

1 Ideal discrete constraints 2 Preservation of manifolds/stability of relative equilibria

The Spherical Pendulum (with A. Bloch and M. Leok)

• A point mass moving on a sphere in the presence of gravity. The use of

spherical coordinates is not the best idea

• Pendulum as a degenerate rigid body:

˙ µ = τ, ˙ γ = γ×ξ

• ξ – angular velocity, µ = Jξ – angular momentum, γ – unit vertical vector,

τ – torque due to gravity

• Non-invertible inertia tensor: J = diag
• mr 2,mr 2,0
• Lagrangian 1

2〈Jξ,ξ〉−mgrγ3 is not hyperregular

• m is the mass and r is the length
• Multiple angular velocities for a given µ =
• µ1,µ2,0
• The third component of angular velocity does not affect pendulum’s motion

The Spherical Pendulum (with A. Bloch and M. Leok)

• Energy- and momentum-preserving integrator
• Preservation of the length of γ

2000 4000 6000 8000 10000

• 5. 1015
• 5. 1015
• Conservation of energy

2000 4000 6000 8000 10000

• 1. 1015
• 1. 1014

Concluding Remarks

• Discrete Hamel’s formalism
• Preservation of manifolds of relative equilibria and their stability

References

• A. M. Bloch, J. E. Marsden, and D. V. Zenkov [2009], Quasivelocities and

Symmetries in Nonholonomic Systems, Dynamical Systems 24, 187–222

• D. V. Zenkov, M. Leok, and A. M. Bloch [2012], Hamel’s Formalism and

Variational Integrators on a Sphere, Proc. CDC 51, 7504–7510

• K. Ball and D. V. Zenkov [2014], Hamel’s Formalism and Variational

Integrators, to appear