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theories:classical_mechanics:newtonian [2018/04/12 16:14]
bogumilvidovic ↷ Page name changed from theories:classical_mechanics to theories:newtonian_mechanics
theories:classical_mechanics:newtonian [2022/09/07 00:08] (current)
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-====== ​Classical ​Mechanics ======+====== ​Newtonian ​Mechanics ======
    
 <tabbox Intuitive> ​ <tabbox Intuitive> ​
  
-Classical ​mechanics is the theory that allows us to describe every day object like an apple that falls to the ground.+Newtonian ​mechanics is the standard ​theory that allows us to describe every day object like an apple that falls to the ground.
  
 We do this by solving, so called equation of motion. The solutions of these equations are particle paths which describe how a particle will move as time passes on.  We do this by solving, so called equation of motion. The solutions of these equations are particle paths which describe how a particle will move as time passes on. 
 +
 +For example, the path of a point-like object could look as follows:
 +
 +{{ :​particletraj2.png?​nolink&​600 |}}
 +
 +
 +
 +The basis of Newtonian Mechanics is summarized by three laws, commonly called "​Newton'​s laws of motion":​
 +
 +  * **First law:** No force is needed to keep an object moving. If an object is at rest, it will remain at rest unless a force acts on it. Similarly, if an object moves with some constant velocity, it will keep moving unless a force acts on it. 
 +  * **[[equations:​newtons_second_law|Second law]]:** The way the movement of an object changes depends only on two things: its mass and the total force acting on it.
 +  * **Third law:** Whenever an object exerts a force on another object, inevitably this second object will also exert a force of equal magnitude on the first object. ​
  
  
 <tabbox Concrete> ​ <tabbox Concrete> ​
  
 +The equations and the theory of Newtonian mechanics were deduced historically from experiments. This worked pretty good but is highly unsatisfactory from a theoretical point of view. Newton proposed his [[equations:​newtons_second_law|second law]]
  
-  * For a nice quick introduction,​ see https://​minireference.com/​static/​tutorials/​mech_in_7_pages.pdf +\begin{equation} \label{newtonssecond} ​ \tag{1} F = m \frac{d²}{dt²}q =m \ddot q,  ​\end{equation}
-  * 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 +
-  * Symmetry in Mechanics: "A GentleModern 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. +
  
 +where $m$ is the mass, $\ddot q$ the acceleration and $F$ the force that acts on the object in question.
  
 +To describe some object we simply have to deduce equations for the forces $F$ that act on the object from experiments and put them on the left-hand side of the equation. This yields a differential equation, which we must solve for $q=q(t)$.
  
 +The solution is called the trajectory of the object and describes the position of the object for every moment in time. This is one framework for classical mechanics and it‘s useful for many, many things.
  
-The standard textbook is "​Classical Mechanics"​ by Herbert Goldstein, Charles Poole, and John Safko +Every student of physics has to solve Newton'​s second law for many different situations.
-<tabbox Abstract>​  +
-Classical mechanics can be formulated geometrically using [[advanced_tools:​fiber_bundles|fibre bundles]].+
  
-The Lagrangian function is defined on the tangent bundle $T(C)$ of the configuration space $C$.+----
  
-The Hamiltonian function is defined on the cotangent bundle $T^\star(C)$,​ which is also called phase space. ​+**First law:**
  
-The map from $T^\star(C) ​\leftrightarrow T(C)$ is called [[advanced_tools:​legendre_transformation|Legendre transformation]].+If the forces acting on an object are balanced, i.e. the total force is zero $\vec F=0$, the velocity of the object will remain constant: $\vec v=\text{const}$. So when the velocity ​is zero, it will remain zero. If the velocity has some other value it will keep it
  
-The phase space is endowed with symplectic structurecalled Poisson BracketThe Poisson Bracket is an operation that eats two scalar fields ​$\Phi$$\Psion the manifold and spits out another scalar field $\theta $:+<​diagram>​ 
 +||||| AA|| |AA=Forces are balanced $ \vec F=0$ 
 +||||||!@4||||||| 
 +|||||BB||||||BB=$ \vec a=0$ 
 +||||,@4| -|^|- |.@4 | | | | 
 +||| AA||BB |AA=Object at rest: $\vec v=0$|BB=Object in motion ​$ \vec v\neq 0$ 
 +||||!@4||||!@4||| 
 +||| AA||BB |AA=Object stays at rest: $\vec v=0$|BB=Object remains in motion $ \vec v \neq 0$; same $ \vec v$. 
 +</​diagram>​
  
-$$  \theta = \{ \Phi,\Psi \}= \frac{\partial \Phi}{\partial p_a}\frac{\partial \Psi}{\partial q^a}-\frac{\partial \Phi}{\partial q_a}\frac{\partial \Psi}{\partial p^a}.$$+**Second law:**
  
-If we leave the $\Psi$ slot blank, we can use the Poisson bracket to define a differential operator ​$\{\Phi,\ \}$. This is 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.+$$ \vec F = m \vec a$$
  
-In this sensethe dynamical evolution of given system is completely described by the Hamiltonian (= a scalar function).+<​diagram>​ 
 +||||| AA|| |AA=Forces are unbalanced $ \vec F\neq 0$ 
 +||||||!@4||||||| 
 +|||||BB||||||BB=$ \vec a \neq 0$ 
 +||||,@4| -|^|- |.@4 | | | | 
 +||| AA||BB |AA=acceleration $\vec a$ depends directly on the net-force $\vec F$ that acts on the object|BB=acceleration $\vec a$ depends inversly on the mass $m$ of the object 
 +</​diagram>​
  
-  ​ 
  
 +-----
  
-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. ​+**Recommended Resources**
  
-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/​ 
  
-----+  * 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. 
 +  * 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>​  
 +[[equations:​newtons_second_law|Newton'​s second law]] is 
 +\begin{equation} 
 + ​F=ma 
 +\end{equation} 
 +which describes a particle moving in $\mathbb{R}^n$. ​
  
-  * 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. +Its positionwhich we call $q$depends on the time $t\in\mathbb{R}$. 
-  * 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>​ 
  
-<​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?> ​ <tabbox Why is it interesting?> ​
Line 82: Line 132:
 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.
  
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 ---- ----
  
 +<​blockquote>​
 +Nature and Nature’s laws lay hid in night:
  
 +God said, "Let Newton be!" and all was light. <​cite>​Alexander Pope</​cite></​blockquote>​
  
  
theories/classical_mechanics/newtonian.1523542490.txt.gz · Last modified: 2018/04/12 14:14 (external edit)