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

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:

\( x(t) = A \cos \omega_0 t + B \sin \omega_0 t \)

      with       \( \omega_0^2 = \frac{k}{m} \)

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: $$ \frac{d\vec{l}}{dt} = \frac{d(\vec{r} \times \vec{p})}{dt} = \frac{d\vec{r}}{dt} \times \vec{p} + \vec{r} \times \frac{d\vec{p}}{dt} = \sum_i \vec{r} \times \vec{F}_i $$

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

$$ x' = x - Vt \\ y' = y \\ z' = z \\ t' = t $$

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.

\( \text{K3: } \frac{T^2}{R_a^3} = const \)

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:

\( \vec{v} \equiv \frac{d\vec{r}}{dt} = \frac{d(r\hat{r})}{dt} = \frac{dr}{dt}\hat{r} + r\frac{d\hat{r}}{dt} = v_r \hat{r} + r \frac{d\phi}{dt} \hat{\phi} \\ \quad = v_r \hat{r} + v_\phi \hat{\phi} \)

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 swpet 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: \( \vec{F}_{cf} = - m \vec{\omega} \times \vec{\omega} \times \vec{r}' \)