## 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

_{j}

### 9. Linear systems

**Eigenvalues:**

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 |

2 imaginary | center | stable |

1 real, positive | improper node | unstable |

1 real, negative | improper node | stable |

### 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:

- equilibrium point;
- cycle;
- 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.