dynamics

Dynamics and Dynamical Systems

Equations sheet

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Formulário
  1. Kinematics
  2. Vector kinematics
  3. Curvilinear motion
  4. Vector dynamics
  5. Rigid bodies dynamics
  6. Work and energy
  7. Dynamical systems
  8. Lagrangian dynamics
  9. Linear systems
  10. Nonlinear systems
  11. Limit cycles and population dynamics
  12. Chaotic systems

1. Kinematics

v = d s d t
a t = d v d t
a t = v d v d s
v x = d x d t
a x = d v x d t
a x = v x d v x d x

2. Vector kinematics

a · b = a b cos θ = a x b x + a y b y + a z b z
| a |= a · a
r = x ˆ ı + y ˆ + z ˆ k
v = d r d t
a = d v d t
r = r 0 + t 0 v d t
v = v 0 + t 0 a d t

Relative motion

r P = r P/Q + r Q
v P = v P/Q + v Q
a P = a P/Q + a Q

3. Curvilinear motion

a × b =− b × a
a × b = a b sin θ ˆ n = ⏐⏐ ⏐ ⏐ ⏐ ⏐ ˆ ı ˆ ˆ k a x a y a z b x b y b z ⏐⏐ ⏐ ⏐ ⏐ ⏐
v = ˙ s ˆ e t
a = ˙ v ˆ e t + v 2 R ˆ e n
a 2 = a 2t + a 2n

Circular motion

s = s 0 + R θ
v = R ω
a t = R α
ω = d θ d t
α = d ω d t
α = ω d ω d θ

Rigid body's plane rotation

ω = ω ˆ e axis
v = ω × r
a = α × r + ω × v

4. Vector dynamics

I = t 2 t 1 F d t = p 2 − p 1
p = m v
F = m a
F g = m g
F s ≤ µ s N
F k = µ k N

Sphere in a fluid

N R = r v ρ η
F r = 6 πη r v  ( N R < 1 )
F r = π 4 ρ r 2 v 2  ( N R > 10 3 )

5. Rigid bodies dynamics

M P = F d
M O = r × F
M z = ⏐⏐ ⏐ ⏐ x y F x F y ⏐⏐ ⏐ ⏐
r cm = 1 m r d m
v cm = 1 m v d m
a cm = 1 m a d m
n i = 1 F i = m a cm
n i = 1 M z , i = I z α
I z = R 2 d m

6. Work and energy

W 12 = s 2 s 1 F t d s
W 12 = E c (2) − E c (1)
E c = 1 2 m v 2cm + 1 2 I cm ω 2
U =− r r 0 F · d r
W 12 = U (1) − U (2)
U g = m g z
U e = 1 2 k s 2
E m = E c + U
s 2 s 1 F nct d s = E m (2) − E m (1)
Ω = k m = 2 π f
s = A sin( Ω t + φ 0 )
E m = 1 2 m v 2 + 1 2 k s 2

7. Dynamical systems

˙ x 1 = f 1 ( x 1 , x 2 )
˙ x 2 = f 2 ( x 1 , x 2 )
u = f 1 ( x 1 , x 2 )ˆ e 1 + f 2 ( x 1 , x 2 )ˆ e 2
¨ x = f ( x , ˙ x )
y = ˙ x
u = y ˆ ı + f ( x , y )ˆ

Conservative systems

∂ f 1 ∂ x 1 + ∂ f 2 ∂ x 2 = 0
f 1 = ∂ H ∂ x 2
f 2 =− ∂ H ∂ x 1

Equilibrium points: u = 0 (stable or unstable).

Cycle: closed curve in phase space.

Homoclinic orbit: starts and ends in the same unstable equilibrium point.

Heteroclinic orbit: links several unstable equilibrium points.

8. Lagrangian dynamics

d d t ∂ E c ∂ ˙ q j − ∂ E c ∂ q j + ∂ U ∂ q j = Q j
Q j = i F i · ∂ r i ∂ q j
d d t ∂ E c ∂ ˙ q j − ∂ E c ∂ q j + ∂ U ∂ q j − λ∂ f ∂ q j = Q j
λ∂ f ∂ q j = constraint forcej

9. Linear systems

d r d t = A r
r = x 1 x 2
A = A 11 A 12 A 21 A 22

Eigenvalues:  λ 2 − tr( A ) λ + det( A ) = 0

Eigenvalues λType of pointStability
2 real with opposite signssaddle pointunstable
2 real and positiverepulsive nodeunstable
2 real and negativeattractive nodestable
2 complex with positive real partrepulsive focusunstable
2 complex with negative real partattractive focusstable
2 imaginarycenterstable
1 real, positiveimproper nodeunstable
1 real, negativeimproper nodestable

10. Nonlinear systems

Jacobian matrix:   J = ⎡⎢ ⎢ ⎣ ∂ f 1 ∂ x 1 ∂ f 1 ∂ x 2 ∂ f 2 ∂ x 1 ∂ f 2 ∂ x 2 ⎤⎥ ⎥ ⎦

At each equilibrium point, it is the matrix of the linear approximation to the system near that point.

11. Limit cycles and population dynamics

Limit cycle: Isolated cycle in phase space.

Two-species systems

˙ x = f ( x , y )
˙ y = g ( x , y )
lim x → 0 f ( x , y ) = 0
lim y → 0 g ( x , y ) = 0

12. Chaotic systems

Positive limit set:  ω ( Γ ) = where curve Γ goes at t →∞

Negative limit set:  α ( Γ ) = where curve Γ goes at t →−∞

Divergence:  ∇· u = ∂ f 1 ∂ x 1 + ∂ f 2 ∂ x 2

Poincaré-Bendixson theorem. In a system with only two state variables, if α ( Γ ) or ω ( Γ ) exist, they must be one of the following three cases:

  1. equilibrium point;
  2. cycle;
  3. homoclinic or heteroclinic orbit.

With 3 or more state variables, a limit set that does not belong to any of those three classes is called a strange attractor.

Bendixson criterium. In a dynamical system with only two state variables, if the divergence is always positive or always negative in a simply-connected region of the phase space, then there are no cycles or orbits in that region.

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