Classical Mechanics

Newton's Laws

Newton's laws read as:

N1: An object is in a state of uniform motion if no net force acts on it.

N2: $ m \vec{a} = \sum_i \vec{F}_i \text { or } \frac{d\vec{p}}{dt} = \sum_i \vec{F}_i $

N3: $ \vec{F}_{1 \rightarrow 2} = - \vec{F}_{2 \rightarrow 1} \text { or } action = - reaction $

Conservation of Momentum

Two point particles interact via a force $ \vec{F} $, with, by N3, $ \vec{F}_{1 \rightarrow 2} = - \vec{F}_{2 \rightarrow 1} $.
$$ \frac{d\vec{p}_1}{dt} = \vec{F}_{2 \rightarrow 1} $$ $$ \frac{d\vec{p}_2}{dt} = \vec{F}_{1 \rightarrow 2} = - \vec{F}_{2 \rightarrow 1} $$ Adding these shows that whatever the mutual interaction: the total momentum is conserved. $$ \frac{d\vec{p}_1}{dt} + \frac{d\vec{p}_2}{dt} = 0 \rightarrow \vec{p}_1 + \vec{p}_2 = const $$

Work & Kinetic Energy

$ \text{Work: } W_{1 \rightarrow 2} = \int_1^2 \vec{F} \cdot d\vec{s} $

$ \text{Kinetic Energy: } E_{kin} = \frac{1}{2}mv^2 $

Connection:
$$ \begin{align*} W_{1 \rightarrow 2} &= \int_1^2 \vec{F} \cdot d\vec{s} = \int_1^2 \vec{F} \cdot \vec{v} dt \\ &= \int_1^2 m \frac{d\vec{v}}{dt} \cdot \vec{v} dt \\ &= \int_1^2 m \vec{v} \cdot d\vec{v} = \frac{1}{2}mv_2^2 - \frac{1}{2}mv_1^2 \end{align*} $$

Conservative Force

A force is conservative if $ \int_1^2 \vec{F} \cdot d\vec{s} $ is independent of the path from 1 to 2 for all positions 1 and 2.

F is conservative is one of the following holds:

(1) $ \oint \vec{F} \cdot d\vec{s} = 0 \text{ } \forall \text{ closed curve} $

(2) $ \nabla \times \vec{F} = 0 $

(3) $ \vec{F} = -\nabla V $

Potential Energy

If a force is conservative, then a potential or potential energy exists: $$ V(\vec{r}) = - \int_{\vec{r}_{ref}}^{\vec{r}} \vec{F} \cdot d\vec{s} $$ Consequently, energy conservation can be written as: $$ E_{kin} + V = const$$

Harmonic Oscillator

Archetype: mass-spring:

N2: $ m\frac{d^2x}{dt^2} = -kx $

or in terms of energy:

$ \frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2 = E_0 $

General Solution:

$ x(t) = A \cos \omega_0 t + B \sin \omega_0 t \quad \textrm{with} \quad \omega_0^2 = \frac{k}{m} $

Angular Momentum

Angular momentum of a point particle is defined as:
$\vec{l} = \vec{r} \times \vec{p}$
N2 for angular momentum reads as:
$\frac{d\vec{l}}{dt} = \sum_i \vec{r}_i \times \vec{F}_i$
Proof: $$ \begin{align*} \frac{d\vec{l}}{dt} &= \frac{d(\vec{r} \times \vec{p})}{dt}\\ &= \frac{d\vec{r}}{dt} \times \vec{p} + \vec{r} \times \frac{d\vec{p}}{dt} = \sum_i \vec{r} \times \vec{F}_i \end{align*} $$

Central Force

A force is called a central force if $$\vec{F} = F(r) \hat{r}$$ Consequently, the angular momentum is not changed by such a force: $$ \frac{d\vec{l}}{dt} = \vec{r} \times \vec{F} = 0 $$ since $ \vec{r} // \vec{F} $

Galilei
Transformation

$$ \begin{align*} x' &= x - Vt \\ y' &= y \\ z' &= z \\ t' &= t \end{align*} $$

Kepler's Laws

K1: The orbit of a planet is an ellipse with the sun at one of the focal points.

K2: A line from the sun to a planet weeps equal areas in equal times.

K3: $\frac{T^2}{R_a^3} = const $

Cylindrical
Coordinates

coordinate transformation
$ x = r \cos \phi \\ y = r \sin{\phi} $

unit vectors
$ \hat{x} = \cos \phi \text{ } \hat{r} - \sin \phi \text{ } \hat{\phi} $
$ \hat{y} = \sin \phi \text{ } \hat{r} + \cos \phi \text{ } \hat{\phi} $
velocity: $$ \begin{align*} \vec{v} \equiv \frac{d\vec{r}}{dt} &= \frac{d(r\hat{r})}{dt} = \frac{dr}{dt}\hat{r} + r\frac{d\hat{r}}{dt} = v_r \hat{r} + r \frac{d\phi}{dt} \hat{\phi} \\ \quad &= v_r \hat{r} + v_\phi \hat{\phi} \end{align*} $$

Kepler's Law of Area

As the sun's gravity on the earth is a central force: $$ \frac{d\vec{l}_{earth}}{dt} = 0 \rightarrow \vec{l}_{earth} = mr^2\frac{d\phi}{dt} \hat{z} = const $$ Area swept by earth: $ dA = \frac{1}{2} r \cdot rd\phi $

Combining: $$ \frac{dA}{dt} = \frac{1}{2}r^2\frac{d\phi}{dt} = \frac{l_{earth}}{2m} = const $$

Center of Mass

definition center of mass: $$ \vec{R} \equiv \frac{\sum m_i \vec{x}_i}{\sum_i m_i} \text{ and } M\equiv \sum m_i$$ momentum: $$ \vec{P} \equiv M \dot{R} \leftrightarrow \frac{d\vec{P}}{dt} = \sum \vec{F}_{ext, i}$$ angular momentum: $$ \vec{L} \equiv \sum_i \vec{x}_i \times \vec{p}_i \leftrightarrow \frac{\vec{L}}{dt} = \sum_i \vec{x}_i \times \vec{F}_{ext, i} $$

Solid Bodies

Moment of Inertia: $ I = \sum_i m_i r_i^2 = \int r^2 dm $

Angular Momentum for rotation around a fixed axis: $$ \frac{d\vec{L}}{dt} = I \frac{d\vec{\omega}}{dt}= \sum \vec{x}_i \times \vec{F}_i$$ Angular momentum: $ \vec{L} = \vec{R} \times \vec{P} + \vec{L}_{cg} $

with cg denoting 'with respect to center of gravity'.

Solid Bodies II

Kinetic Energy: $ I = \frac{1}{2}MV^2 + \frac{1}{2}I\omega^2 $

Parallel axis theorem
(Steiner):

$ I = Md^2 + I_{cg} $

Rotating Frame
of Reference

N2: $ m\vec{a}' = m\vec{a} - 2m \vec{\omega} \times \vec{v}' - m \vec{\omega} \times \vec{\omega} \times \vec{r}' $

Coriolis Force: $ \vec{F}_{cor} = - 2m \vec{\omega} \times \vec{v}' $

Centrifugal Force: $ \vec{F}_{cf} = - m \vec{\omega} \times \vec{\omega} \times \vec{r}' $