Discrete Geometric Mechanics for

Variational Time Integrators

Ari Stern

Mathieu 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 thetwo-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

Results

http://tinyurl.com/n5sn3xq