2. Vector kinematics
3. Curvilinear motion
Rigid body's plane rotation
4. Vector dynamics
Sphere in a fluid
5. Rigid bodies dynamics
6. Work and energy
7. Dynamical 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
9. Linear systems
|Eigenvalues λ||Type of point||Stability|
|2 real with opposite signs||saddle point||unstable|
|2 real and positive||repulsive node||unstable|
|2 real and negative||attractive node||stable|
|2 complex with positive real part||repulsive focus||unstable|
|2 complex with negative real part||attractive focus||stable|
|1 real, positive||improper node||unstable|
|1 real, negative||improper node||stable|
10. Nonlinear systems
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.
12. Chaotic systems
Positive limit set: = where curve goes at →∞
Negative limit set: = where curve goes at →−∞
Poincaré-Bendixson theorem. In a system with only two state variables, if or exist, they must be one of the following three cases:
- equilibrium point;
- 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.