Dynamics and Dynamical Systems

Summary

1. Kinematics


2. Vector kinematics


Relative motion


3. Curvilinear motion


Circular motion


Rigid body's plane rotation


4. Vector dynamics


Sphere in a fluid

 ( )
 ( )

5. Rigid bodies dynamics


6. Work and energy


7. Dynamical systems


Conservative systems


Equilibrium points: (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

= constraint forcej

9. Linear systems


Eigenvalues:

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:

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


12. Chaotic systems

Positive limit set: = where curve goes at →∞

Negative limit set: = where curve goes at →−∞

Divergence:

Poincaré-Bendixson theorem. In a system with only two state variables, if or 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.