The Calculus of
Variations!
A Primer by Chris Wojtan
Variational Calculus:
What’s in Store






What are all these big words?
Simple applications
Euler-Lagrange equation
Hamilton’s Principle
Noether’s theorem
Graphics applications
Variational Calculus:
What’s in Store






What are all these big words?
Simple applications
Euler-Lagrange equation
Hamilton’s Principle
Noether’s theorem
Graphics applications
What is the Calculus of Variations?

Please interrupt me if I go too fast!
• No point in giving this talk
if nobody gets anything out of it
What is the Calculus of Variations?

Calculus uses functions
• Function: maps real numbers to real numbers

Variational Calculus uses functionals
• Functional: maps functions to real numbers
2.0
-213.6
0.99
Optimization Review




Input: scalar or vector
Output: scalar
Extremize that output!
Zero first derivative at local extremum
Cost Function
Input Vector
25.1
-0.5
6.2
-4.0
100
9
-1.1
500
8.2
10.1
2.9
7.4
Variational Calc as Optimization




Input: function
Output: scalar
Extremize that output!
Zero first variation at local extremum
Cost Functional
Input Function
10.1
2.9
7.4
One Dimensional Calculus
Input Value
Input Scalar
Three Dimensional Calculus
Input Values
Input Vector
Ten Dimensional Calculus
Input Values
Input Vector
50 Dimensional Calculus
Input Values
Input Vector
Infinite
Dimensional
Calculus
Variational
Calculus
Input Value
Input Vector
Function
Variational Calculus:
What’s in Store






What are all these big words?
Simple applications
Euler-Lagrange equation
Hamilton’s Principle
Noether’s theorem
Graphics applications
Geodesics

Find the curve that minimizes arclength
Plane
Shortest distance is a straight line
Geodesics

Find the curve that minimizes arclength
Sphere
Shortest distance is a great circle
Catenary


What shape does a hanging rope form?
Resting shape minimizes potential energy
Catenary forms a
Hyperbolic Cosine
Catenoid


What shape does a bubble form?
Resting shape minimizes surface area
Photograph
3d Plot
Variational Calculus:
What’s in Store






What are all these big words?
Simple applications
Euler-Lagrange equation
Hamilton’s Principle
Noether’s theorem
Graphics applications
Gradients and Extrema

v
Cost function  (v )

We want to find

Zero gradient at extremum

Input vector
v that extremizes  (v)
  0
Variations and Extrema


Input function
Cost function
y (x)
x1
J ( y )   f ( x, y, y)dx
x0


We want to find
y that extremizes J ( y )
Euler-Lagrange equation satisfied
at extrememum
d  f  f

 
0
dx  y  y
Euler-Lagrange Equation


Analogous to gradient
To extremize J ( y ), find function
that satisfies this equation
y (x)
d  f  f

 
0
dx  y  y
Example: Geodesics in the Plane


Arclength
Minimize
dx 2  dy 2  1  y2 dx
1
J ( y)  
0

Euler-Lagrange
y
1  y

2
 const
2

1  y dx
d  f  f
d  y

 
0
dx  y  y dx  1  y2

y  c1

0  0


y( x)  c1 x  c2
Shortest distance between 2 points is a line!
Variational Calculus:
What’s in Store






What are all these big words?
Simple applications
Euler-Lagrange equation
Hamilton’s principle
Noether’s theorem
Graphics applications
The Lagrangian

Kinetic minus Potential Energy
L(t , q, q )  T (q, q )  V (t , q)

Principle of Least Action:
• Equations of motion result
from extremizing Lagrangian
d  L  L
  
0
dt  q  q
Example: Ballistic Motion
1
2
2


L(t , qx, qy )  T (m
(
x

y
q, q)  V (t),q)mgy
2
d  L  L d  L  L
d  L  L
 0     0
 0   
 
dt  x  x dt  q  q
dt  y  y
mx  0
x(t )  x0  x0t
my  mg  0
1 2
y (t )  y0  y 0t  gt
2
Variational Calculus:
What’s in Store






What are all these big words?
Simple applications
Euler-Lagrange equation
Hamilton’s Principle
Noether’s theorem
Graphics applications
Variational Symmetry

Study the change in Lagrangian
with respect to perturbations
L
J (q)   L(tq,,qq,q)qdt
)dt  L  const
t0
q
t1

Invariance (symmetry) in the Lagrangian
creates conservation laws
1
2


L(q, q)  mq  U (q)
2
1 L2
1
2
qmq 
U
L (q) mconst
q  U (q)
2 q
2
Noether’s Theorem





Variational symmetry = conservation law
Invariance wrt time conserves energy
Invariance wrt translation
conserves linear momentum
Invariance wrt rotation
conserves angular momentum
All of these properties are implicit
in a single function!
(the Lagrangian)
Variational Calculus:
What’s in Store






What are all these big words?
Simple applications
Euler-Lagrange equation
Hamilton’s Principle
Noether’s theorem
Graphics applications
Variational Calc In Graphics

Two main approaches
• Discretize problem,
reduce to optimization
• Solve problem analytically,
plug variables into resulting equation

Many applications in graphics!
Variational Integrators

Break integral into discrete time steps
J (q)   L(q, q )dt  k 0 Ld (q k , q k 1 )
t1
N 1
t0

Explicitly set gradient to zero
• Principle of Least Action says
Lagrangian extremizes J (q )

J (q)
0
q k
Automatically conserves momentum and energy!
Useful Variational Integrators

Symplectic Euler
vn  vn1  f ( xn1 , vn1 )t
xn  xn1  vn t

Newmark
t 2
 f ( xn1 , vn1 )  f ( xn , vn )
xn  xn1  vn1t 
4
t
vn  vn 1   f ( xn 1 , vn 1 )  f ( xn , vn ) 
2

Others…
Variational Integrators
Variational Integrators
Variational Tetrahedral Meshing

Find the mesh that minimizes energy
Energy 

i 1... N
Vi
|| x  xi ||2 dx
Shape Transformation
Using Variational
Implicit Functions



Interpolate between 3D shapes
Scattered data interpolation
Find shape that minimizes energy
Energy   f (x)  2 f (x)  f (x)

2
xx
2
xy
2
yy
Variational Solid-Fluid Coupling



Pressure minimizes
kinetic energy
Solve Euler-Lagrange equation
to derive Navier-Stokes
Replace density term with
accurate mass matrix
Variational
Eulerian Geometry Processing



Surface operations with Eulerian grids
Mean curvature flow presented
as minimization of surface area
Mass conserved explicitly
The End
Have fun!