EML5215
Analytical Dynamics
Analytical Dynamics –
EML5215
Norman Fitz-Coy
MAE-A 206
352-392-1029
nfc@ufl.edu
© Fitz-Coy
University of Florida
EML5215
Analytical Dynamics
Stability of Hamiltonian Canonical Form
Let’s assume there exists a nominal motion described by

where

q 0   p q 0 , p0   p
p 
0

p


p 0  q q 0 , p 0   q
q 
0

q
Recall the motion is governed by
T
q   p
T
p  q
Now, let the motion be perturbed from a nominal motion by some small quantities p and q;
i.e.,
p  p0   p  p   p
q  q   q  q   q
0
Thus,
 q   pq q   pp p
  q    pq
  
 
  p   
  
qq
 p  qq q  qp p
 pp    q 
 
 
qp    p 

© Fitz-Coy
University of Florida
1
EML5215
Analytical Dynamics
The stability is determined from the eigenvalues of the linearized system

  In  

pq
pp

 






1
 A  det 
 det      0
T 


I


n

qq
pq 

 







det  I2 n


Thus, the Hamiltonian in terms of the generalized momenta and speeds is:


det   In  
pq

I     I     I   
T
n
n
pq
pq
n
qq
1
pq

 0
pp

which results in a polynomial in  with only even coefficients. Thus, the roots of the polynomial
will be symmetric about the origin (from Routh’s array). Now, let’s examine the pendulum
problem
eˆ y
2
1
1 p2
 p  1 p
 mgL cos  
 mgL cos 
   pi q i    p  2  
2
i 1
2
2
mL
mL
mL2


g
eˆ x

L
Thus,
p 

p

p mL2
 

 mgL sin

© Fitz-Coy
University of Florida
EML5215
Analytical Dynamics
Hence,
pp 
 p 
p
p

  p 
1


p  mL2  mL2



 mgL sin   0
p
p


0  det   In  
pq

p 
 
p


  p 

0
  mL2 



 mgL sin   mgL cos


I     I     I   
T
n
pq
n
pq
qq

n
 det    pq    pq      pq  qq    pq  pp

1

2
  2  pq
 qq pp   2  02 
pq

1

 
pp

1
1
mgL cos    2 
mgL cos 
mL2
mL2
1
mgL  stable 
mL2
1
mgL  unstable 
    2 
mL2
  0  2  
© Fitz-Coy
University of Florida
2