Geodesic Equation

Geodesic Equation & Hamiltonian Dynamics

Hang Shao

Introduction

• Recall Lagrangian Mechanics

• Geodesic Equations

• Hamiltonian Mechanics (Dynamics)

• Applications

Lagrangian Mechanics

• Lagrangian : short for

where

• kinetic minus the potential energy

L(qi, qi, t) L(q1, · · · , qn, q1, · · · , qn, t)

qi = dqi/dt


• Variation of Lagrangian , which is

similar with total differential of

• Then integrate the variation with respect to time and make it to 0,

which can be summarized by Hamilton’s principle

L(qi, qi, t)

�L =NX

i=1

(@L

@qi�qi +

@L

@qi�qi)

Z b

a�L = 0


• Integrating by parts and boundary condition

• Euler–Lagrange equation

d

dt

@L

@qi� @L

@qi= 0

Z b

a�L

Deriving Geodesic Equation via Action

• Euler-Lagrange equation :

• Lagrangian :

d

dt

@L

@qi� @L

@qi= 0

@L

@qj=

1

2

X

i

X

k

@gik@qj

qiqk

@L

@qj=

X

k

gjk qk

L =1

2

X

i

X

j

gij(q)qiqj


d

dt

@L

@qj=

d

dt(X

k

gjk qk)

=X

k

(X

m

@gjk@qm

˙qmqk + gjk qk)

=1

2

X

k

X

m

@gjk@qm

˙qmqk +1

2

X

k

X

m

@gjk@qm

˙qmqk +X

k

gjk qk


• Combine two parts of Euler-Lagrange equation together :

• Time for both sides and do summation respect index of j for above equation :

1

2

X

k

X

m

@gjk@qm

˙qmqk +1

2

X

k

X

m

@gjk@qm

˙qmqk +X

k

gjk qk � 1

2

X

k

X

m

@gkm@qj

˙qmqk = 0

gij

1

2

X

j

gijX

k

X

m

@gjk@qm

qmqk +1

2

X

j

gijX

k

X

m

@gjk@qm

qmqk +X

j

X

k

gjk qk � 1

2

X

j

gijX

k

X

m

@gkm@qj

qmqk = 0


• Exchange indices of m and j for the whole equation :

• Exchange indices of j and k for the first term of equation :

1

2

X

j

X

k

X

m

gim@gmk

@qjqj qk +

1

2

X

j

X

k

X

m

gim@gmk

@qjqj qk � 1

2

X

j

X

k

X

m

gim@gjk@qm

qj qk + qi = 0

1

2

X

j

X

k

X

m

gim@gmk

@qjqj qk ! 1

2

X

k

X

j

X

m

gim@gmj

@qkqk qj


• Rearrange all terms in the equation :

qi +1

2

X

k

X

m

X

j

gim(@gjm@qk

qj qk +@gkm@qj

qj qk � @gjk@qm

qj qk) = 0


• Geodesic equation :

where �ijk =

1

2

X

m

gim(@gjm@qk

+@gkm@qj

� @gjk@qm

)

qi +X

k

X

j

�ijk qj qk = 0

Geodesic Equation

Geodesic Equation

(cos ✓ cos�, sin ✓ cos�, sin�)

e✓ = (� sin ✓ cos�, cos ✓ cos�, 0)

e� = (� cos ✓ sin�,� sin ✓ sin�, cos�)

g =

✓he✓, e✓i he✓, e�ihe�, e✓i he�, e�i

◆

=

✓(cos�)2 0

0 1

◆

• SO(3) ( ) coordinate chart : S2

• Unit tangent vectors:

Geodesic Equation

q(t) = (✓(t),�(t))

✓(t) = t

�(t) = 0

q(t) = 0

• Differential of metric :

• Quaternions :

@g✓✓@�

= �2 sin�

Hamiltonian Mechanics

• Conjugate momenta :

• Total differential of Lagrangian :

(qi, qi) 7! (qi, pi)pi =@L

@qi

dL =NX

i=1

@L

@qidqi +

NX

i=1

@L

@qidqi

=NX

i=1

pidqi +NX

i=1

pidqi


• Set velocities to momenta via a Legendre transformation

• Define “Hamiltonian” as

dH = �NX

i=1

pidqi +NX

i=1

qidpi

dL =NX

i=1

pidqi + d(p · q)�NX

i=1

qidpi

d(p · q � L) = �NX

i=1

pidqi +NX

i=1

qidpi

H = p · q � L


• Hamiltonian equations :

dqidt

=@H

@pidpidt

= �@H

@qi


• Assumed kinetic energy is T =< q, q >=NX

i,j=1

cij qiqj

@T

@qj= 2

NX

i=1

cij qi

H =NX

i=1

pi@L

@qi� L

=NX

i=1

pi@T

@qi� L

= 2T � L

= T + U

Geodesic Equations

• Lagrangian and Hamiltonian

• Metrics and inverse of it

• Hamiltonian Equation

gijgjk = �ik

dqidt

= gijpi

dpidt

= �1

2

@gjk

@qipjpk

L =1

2

X

i

X

j

gij(q)qiqj

H =1

2

X

i

X

j

gij(q)pipj

Applications of Hamiltonian

• Geodesic Flow

• Image Registration

• Hamiltonian Monte Carlo Sampling( HMC Sampling)

Vialard et al. 2011

I1

I2

I3

I4 IN

I �N

�1v1

vN

E(I, vk) =NX

k=1

1

2�2kI � (�k)�1 � Ikk2 + (Lvk, vk)

L = �↵�+ ✏Laplacian operator:: regularization parameter

: noise variance

Single Atlas Estimation

Geodesic Regression

HMC efficiently explores the distribution space with high acceptance rates.

Acceptance-rejection method:

Hamiltonian Dynamics:

- sample , - random auxiliary variable , i.i.d. Gaussian

HMC Sampling

Geodesic Equation

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