Discrete Geometric Mechanics for

Variational Time Integrators

Ari SternMathieu Desbrun

Geometric, Variational

Integrators for Computer Animation

L. KharevychWeiweiY. Tong

E. KansoJ. E. MarsdenP. SchröderM. Desbrun

Time Integration• Interested in Dynamic Systems

• Analytical solutions usually difficult or impossible

• Need numerical methods to compute time progression

Local vs. Global Accuracy• Local accuracy (in scientific applications)

• In CG, we care more for qualitative behavior

• Global behavior > Local behavior for our purposes

• A geometric approach can guarantee both

Simple Example: Swinging Pendulum

• Equation of motion:

• Rewrite as first-order equations:

𝑞 (𝑡)

𝑙

Discretizing the Problem• Break time into equal steps of length :

• Replace continuous functions and with discrete functions and

• Approximate the differential equation by finding values for

• Various methods to compute

Taylor Approximation• First order approximation using tangent to curve:

v

• As , approximations approach continuous values

(𝑞𝑘 ,𝑣𝑘)

(𝑞𝑘+1 ,𝑣𝑘+1)

Explicit Euler Method• Direct first order approximations:

• Pros:• Fast

• Cons:• Energy “blows up”• Numerically unstable• Bad global accuracy

Implicit Euler Method• Evaluate RHS using next time step:

• Pros:• Numerically stable

• Cons:• Energy dissipation• Needs non-linear solver• Bad global accuracy

Symplectic Euler Method• Evaluate explicitly, then :

• Energy is conserved!• Numerically stable• Fast• Good global accuracy

Symplecticity• Sympletic motions preserve the

two-form:

• For a trajectory of points inphase space:

• Area of 2D-phase-space region is preserved in time

• Liouville’s Theorem

Geometric View: Lagrangian Mechanics

• Lagrangian: • Action Functional:• Least Action Principle:

• Action Functional “Measure of Curvature”• Least Action “Curvature” is extremized

𝑡 0

𝑇

Euler-Lagrange Equation

=

= 0

Lagrangian Example: Falling Mass

The Discrete Lagrangian• Derive discrete equations of motion from a Discrete

Lagrangian to recover symplecticity:

• RHS can be approximated using one-point quadrature:

The Discrete Action Functional• Continuous version:

• Discrete version:

Discrete Euler-Lagrange Equation

Discrete Lagrangian Example: Falling Mass

More General: Hamilton-Pontryagin Principle

• Equations of motion given by critical points of Hamilton-Pontryagin action

• 3 variations now:

• is a Lagrange Multiplier to equate and

• Analog to Euler-Lagrange equation:

Discrete Hamilton-Pontryagin Principle

Faster Update via Minimization• Minimization > Root-Finding

• Variational Integrability Assumption:

• Above satisfied by most current models in computer animation

Minimization: The Lilyan

Resultshttp://tinyurl.com/n5sn3xq