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theories:classical_mechanics:newtonian [2018/04/12 16:27]
bogumilvidovic [Abstract]
theories:classical_mechanics:newtonian [2022/09/07 00:08] (current)
laserblue
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 ----- -----
 +
 +**Recommended Resources**
 +
  
   * For a nice quick introduction,​ see https://​minireference.com/​static/​tutorials/​mech_in_7_pages.pdf   * 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   * 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   * 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   * 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.    * 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   * 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> ​ <tabbox Abstract> ​
 +[[equations:​newtons_second_law|Newton'​s second law]] is
 +\begin{equation}
 + F=ma
 +\end{equation}
 +which describes a particle moving in $\mathbb{R}^n$. ​
  
- +Its positionwhich we call $q$depends on the time $t\in\mathbb{R}$. 
-  * 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 explainsfor examplenicely 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 momentumacting via the Poisson bracket. +  
-  * A great book that describes ​the high-level perspective ​on classical mechanics ​is "​Introduction to Mechanics and Symmetry"​ by Jerrold EMarsden and Tudor S. Ratiu and see also the other books by Marsdenmany of which are freely available online [[http://​www.cds.caltech.edu/​~marsden/​books/​|here]], ​for examplehis "​[[http://​authors.library.caltech.edu/​25029/​1/​FoM2.pdf|Foundations of Mechanics]]"​+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 $x\in\mathbb{R}^n$).
  
 ---- ----
  
-<​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 particlebut the ‘extended’ configuration spacewhich also includes time. Only this way do Hamilton’s equationslike the [[equations:maxwell_relations|Maxwell relations]], become a trivial consequence ​of the fact that partial derivatives commute.+  * 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 examplenicely how similar classical and quantum mechanics are, if formulated in the same language. (E.g. in classical mechanicsthe generators of translations are also given by the momentumacting 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]]"​ 
  
-<​cite>​https://​johncarlosbaez.wordpress.com/​2012/​01/​23/​classical-mechanics-versus-thermodynamics-part-2/</​cite>​ 
-</​blockquote>​ 
  
  
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 It makes sense to study these various approaches instead of just one because:  ​ 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.    - 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.
  
theories/classical_mechanics/newtonian.1523543267.txt.gz · Last modified: 2018/04/12 14:27 (external edit)