Classical mechanics

Kinematics

How do you define speed exactly? Cue zeno's paradox to tell you Usain Bolt could never catch up to a turtle or a speeding woman epistemically questioning the cop who stopped her about what a speed limit even means.

Again with divine inspiration, Newton saw that a new concept was needed, specifically that of the infinitesimal (very small) changes that can, if needed, be extrapolated to a hypothetical scenario where things continued as they were in that instant.
In other words, (differential) calculus
- so speed $v = lim_{Δ t \to 0} \frac{Δ x}{Δ t} = \frac{d x}{d t}$ .
- and acceleration $a = \frac{d v}{d t} = \frac{d^{2} x}{d t^{2}}$ .
We reverse this process to add up infinitesimal quantities to get the aggregate result, i.e, integration $x = \int v d t$ and $v = \int a d t$ .

Dynamics

Newton comes across as a force of nature. Once he was done, the world made sense. Motion was computable, both here and in the heavens above. He wrote down three laws:

First law: Inertia, originally discovered by Galileo. Basically, if an object is left alone, and not disturbed, it continues to move with a constant velocity in a straight line if it was originally moving, or it continues to stand still if it was just standing still.
Second law: A rule for finding how an object's motion changes if something is acting on it.
- Newton named this something, a force, which by definition, is an influence that changes an object's motion. It is, roughly speaking, a push or pull.
- The law states that the rate-of-change of momentum (inertia times the velocity vector) of an object is equal to the force acting on it $\vec{F} = \frac{d}{d t} (m \vec{v})$
- Concerns not only changing speeds, but also changing directions of velocities, so an object moving in a circle probably has some perpetual force acting perpendicular to its motion keeping it moving in the circle. and you can find and characterize what that force is once you go looking for it.
  - According to Newton's law, the force needs to be $F = m v^{2} / R$ .
  - See Orbital speed calculation to figure out the effective acceleration of an object moving along a curve.
- Already allows numerical calculation of all sorts of movements (even multi-body systems).
- Another consequence is Galilean Relativity - all laws of physics will look the same as long as we are in an inertial frame (non-accelerating).
Third law: Action equals reaction, i.e, internal forces in a closed system sum to zero, so

\implies \vec{p}_1+\vec{p}_2 = \text{const}$$ giving us the law of conservation of linear momentum.

Lagrangian formulation

It is often more productive to work with generalized coordinates $q_{j}$ instead of natural Cartesian coordinates ${{\vec{r}}_{1}, {\vec{r}}_{2}, \dots {\vec{r}}_{n}}$ of many particles in general constrained systems. It can be shown that Newton's second law is equivalent to Lagrange's equations of motion

\frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{j}}) - \frac{\partial L}{\partial {\dot{q}}_{j}} = 0 $ $ w h e r e $ T $ i s t h e k i n e t i c e n e r g y, $ V $ i s t h e p o t e n t i a l o f t h e s y s t e m a n d $ L = T - V $ i s c a l l e d t h e [[L a g r a n g i a n ∥ L a g r a n g i a n]] . T h e c o o l t h i n g i s t h a t t h e s e e q u a t i o n s a r e v a l i d i n * a n y * c o o r d i n a t e s y s t e m (w i t h i n d e p e n d e n t c o m p o n e n t s), u n l i k e N e w t o n^{'} s l a w s w h i c h o f t e n r e q u i r e t e d i o u s t r a n s f o r m a t i o n s s t a r t i n g f r o m t h e C a r t e s i a n c o o r d i n a t e s w h e r e t h e y a r e v a l i d a s s t a t e d . L a g r a n g e^{'} s e q u a t i o n s c a n a l t e r n a t i v e l y b e f o r m u l a t e d a s t h e [[N o t e s / L e a s t a c t i o n p r i n c i p l e ∥ l e a s t a c t i o n p r i n c i p l e]], i . e ., (f o r a s y s t e m w i t h h o l o n o m i c c o n s t r a i n t s a n d p o t e n t i a l - d e r i v e d f o r c e s) m o t i o n i s d e s c r i b e d b y t h e s t a t i o n a r y p a t h s o f t h e a c t i o n : $ $ S [L] = \int_{t_{1}}^{t_{2}} L (\vec{q}, \dot{\vec{q}}, t) d t = \int_{t_{1}}^{t_{2}} (T - V) d t

It is important to note that many other Lagrangians can be constructed for a given system resulting in the same equations of motion.

A few useful definitions in Lagrangian mechanics:

\begin{aligned} Generalized force & Q_{j} = \sum_{i} {\vec{F}}_{i} \cdot \frac{\partial {\vec{r}}_{i}}{\partial q_{j}} \\ Generalized momentum & p_{j} = \frac{\partial L}{\partial {\dot{q}}_{j}} \\ Energy function & h = \sum_{j} {\dot{q}}_{j} \frac{\partial L}{\partial {\dot{q}}_{j}} - L \end{aligned}

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