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--- theories:classical_mechanics:newtonian [2018/04/12 16:23]
bogumilvidovic [Why is it interesting?]
+++ theories:classical_mechanics:newtonian [2022/09/07 00:08] (current)
laserblue
@@ Line 64: / Line 64: @@
 -----
+**Recommended Resources**
   * For a nice quick introduction, see https://minireference.com/static/tutorials/mech_in_7_pages.pdf
   * A great introduction is: http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf
   * and a great textbook is Morin: Introduction to Classical Mechanics and also
-  * Classical Mechanics by John Taylor
+  * Classical Mechanics by John Taylor.
+  * Another great textbook is the [[http://www.feynmanlectures.caltech.edu/I_toc.html|first volume of the Feynman lectures]].
   * Symmetry in Mechanics: "A Gentle, Modern Introduction" by Singer
   * A great resource to understand many of the most important mechanics systems is https://www.myphysicslab.com/. It's a collection of physics simulations, where you can modify the model parameters etc.
   * The standard textbook is "Classical Mechanics" by Herbert Goldstein, Charles Poole, and John Safko
+  * [[https://link.springer.com/book/10.1007/0-306-47122-1| New Foundations for Classical Mechanics]] by David Hestenes
+  * A classic is [[https://openlibrary.org/books/OL5797696M/The_science_of_mechanics| The Science of Mechanics]] by Ernst Mach
 <tabbox Abstract>
-Classical mechanics can be formulated geometrically using [[advanced_tools:fiber_bundles|fibre bundles]].
+[[equations:newtons_second_law|Newton's second law]] is
+\begin{equation}
+ F=ma
+\end{equation}
+which describes a particle moving in $\mathbb{R}^n$.
-The Lagrangian function is defined on the tangent bundle $T(C)$ of the configuration space $C$.
+Its position, which we call $q$, depends on the time $t\in\mathbb{R}$.
+Therefore, the position defines a function,
+\[
+ q \colon \mathbb{R}\longrightarrow\mathbb{R}^n.
+\]
+Using this function $q$ we can define the corresponding __velocity__,
+\[
+ v=\dot{q} \colon \mathbb{R}\longrightarrow\mathbb{R}^n
+\]
+where $\dot{q}=\frac{dq}{dt}$. Analogously, we can define the __acceleration__
+\[
+ a=\ddot{q} \colon \mathbb{R}\longrightarrow\mathbb{R}^n.
+\]
+We call $m>0$ be the __mass__ of the particle. The last puzzle piece in the equation $F$ is a
+vector field on $\mathbb{R}^n$ which we call the __force__.  Newton second law is  2nd-order differential equation
+for $q\colon\mathbb{R}\rightarrow\mathbb{R}^n$. It has a unique
+solution given some $q(t_0)$ and $\dot{q}(t_0)$, provided the vector
+field $F$ is smooth and bounded
+(i.e., $|F(x)|<B$ for some $B>0$, for all $x\in\mathbb{R}^n$).
-The Hamiltonian function is defined on the cotangent bundle $T^\star(C)$, which is also called phase space.
+----
-The map from $T^\star(C) \leftrightarrow T(C)$ is called [[advanced_tools:legendre_transformation|Legendre transformation]].
-The phase space is endowed with a symplectic structure, called Poisson Bracket. The Poisson Bracket is an operation that eats two scalar fields $\Phi$, $\Psi$ on the manifold and spits out another scalar field $\theta $:
-$$  \theta = \{ \Phi,\Psi \}= \frac{\partial \Phi}{\partial p_a}\frac{\partial \Psi}{\partial q^a}-\frac{\partial \Phi}{\partial q_a}\frac{\partial \Psi}{\partial p^a}.$$
-If we leave the $\Psi$ slot blank, we can use the Poisson bracket to define a differential operator $\{\Phi,\ \}$. This is a vector field and when in acts on $\Psi$, we get $\{\Phi, \Psi \}$. If we use instead of $\Phi$, the Hamiltonian $H$, we get an differential operator $\{H,\ \}$ that 'points along' the trajectories on in phase space $T^\star(C)$ and describes exactly the evolution that we get from Hamilton's equations.
-In this sense, the dynamical evolution of a given system is completely described by the Hamiltonian (= a scalar function).
-See page 471 in Road to Reality by R. Penrose, page 167 in Geometric Methods of Mathematical Physics by B. Schutz and http://philsci-archive.pitt.edu/2362/1/Part1ButterfForBub.pdf.
-For some more backinfo why there is a symplectic structure in classical mechanics, have a look at https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/
-----
   * A great introduction to high-level concepts in classical mechanics are the lecture notes by David Tong: http://www.damtp.cam.ac.uk/user/tong/dynamics.html Especially part 4 is amazing and explains, for example, nicely how similar classical and quantum mechanics are, if formulated in the same language. (E.g. in classical mechanics, the generators of translations are also given by the momentum, acting via the Poisson bracket.)
   * A great book that describes the high-level perspective on classical mechanics is "Introduction to Mechanics and Symmetry" by Jerrold E. Marsden and Tudor S. Ratiu and see also the other books by Marsden, many of which are freely available online [[http://www.cds.caltech.edu/~marsden/books/|here]], for example, his "[[http://authors.library.caltech.edu/25029/1/FoM2.pdf|Foundations of Mechanics]]".
-----
-<blockquote>
-a ‘Hamiltonian’ $$H : T^* Q \to \mathbb{R}$$ or a ‘Lagrangian’ $$L : T Q \to \mathbb{R}$$
-Instead, we started with Hamilton’s principal function $$S : Q \to \mathbb{R}$$ where $Q$ is not the usual configuration space describing possible positions for a particle, but the ‘extended’ configuration space, which also includes time. Only this way do Hamilton’s equations, like the [[equations:maxwell_relations|Maxwell relations]], become a trivial consequence of the fact that partial derivatives commute.
-<cite>https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/</cite>
-</blockquote>
-<blockquote>
-Hamilton's principal function is basically just the action
-<cite>https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/</cite>
-</blockquote>
 <tabbox Why is it interesting?>
@@ Line 132: / Line 132: @@
 It makes sense to study these various approaches instead of just one because:
-   - In more modern theories like [[theories:quantum_mechanics|quantum mechanics]] or [[theories:quantum_field_theory|quantum field theory]], we are in exactly the same situation: There are several descriptions that all do the same thing. But for specific situations one approach is stronger than the others. To get a full understanding of the theory it is helpful to know all the various approaches. It is thus sensible to get familiar with this "various approaches" idea.
+   - In more modern theories like [[theories:quantum_mechanics:canonical|quantum mechanics]] or [[theories:quantum_field_theory:canonical|quantum field theory]], we are in exactly the same situation: There are several descriptions that all do the same thing. But for specific situations one approach is stronger than the others. To get a full understanding of the theory it is helpful to know all the various approaches. It is thus sensible to get familiar with this "various approaches" idea.
    - The same three approaches are also applicable in quantum field theory and quantum mechanics, which are the best theories of nature that we currently have. In classical mechanics we deal with objects that we are familiar with in everyday life, whereas in quantum field theory and quantum mechanics, things are more abstract. Thus it makes sense to get familiar with the approaches using everyday objects.

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