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