Differences

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--- book:chap6:lagrange [2022/01/31 15:40] – abril
+++ book:chap6:lagrange [2022/02/14 14:05] (current) – abril
@@ Line 1: / Line 1: @@
 ====== 6. Integrable Dynamics ======
-<WRAP group>
+<WRAP right #fig_girl-on-swing 115pt>
-<WRAP third column #fig_EXP-dropletExplosion>
+{{girl-on-swing.png}}
-{{Laser_impact_on_a_drop.jpg}}
+[[https://commons.wikimedia.org/wiki/File:Sketch_by_Marguerite_Martyn_of_a_girl_standing_on_a_swing_in_a_bathing_suit_getting_dry_from_the_breeze,_1914.png}{wikimedia/public domain | Marguerite Martyn, 1914]]\\
+Figure 6.1: The point-particle idealization of a girl on a swing is the mathematical pendulum of Figures 1.2 and 1.3.
-Figure 5.1: Impact of a laser pulse on a microdrop of opaque liquid that is thus blown up;
-[[http://link.aps.org/doi/10.1103/PhysRevApplied.3.044018 |Klein, et al, Phys. Rev. Appl. 3 (2015) 044018]]
-</WRAP>
-<WRAP third column #fig_EXP-clackers>
-{{Mensen_met_een_klik_klak_Amsterdam_Bestanddeelnr_924-8383.jpg}}
-Figure 5.2: Girl playing with clackers.
-[[https://commons.wikimedia.org/wiki/File:Mensen_met_een_klik_klak_Amsterdam,_Bestanddeelnr_924-8383.jpg |Punt/Anefo, Amsterdam 1971]], [[https://creativecommons.org/licenses/ |CC0]]
-</WRAP>
-<WRAP third column #EXP-reachTennisBall>
-{{Gluten_free_speed_-_Flickr_-_chascow.jpg}}
-Figure 5.3: Man running to return a tennis ball.
-[[https://commons.wikimedia.org/wiki/File:Gluten_free_speed_-_Flickr_-_chascow.jpg |Charlie Cowins from Belmont, NC, USA]], [[https://creativecommons.org/licenses/by/2.0 |CC by 2.0]]
-</WRAP>
 </WRAP>
+In [[book:chap5:spatial-extension|Chapter 5]] we considered objects that consist of a mass points
+with fixed relative positions, like a flying and spinning ping-pong
+ball. Rather than providing a description of each individual mass
+element, we established equations of motion for their center of mass
+and the orientation of the body in space. From the perspective of
+theoretical mechanics the fixing of relative positions is a constraint
+to their motion, just as the ropes of a swing enforces a motion on
+a one-dimensional circular track, rather than in two dimensions.
+The deflection angle θ of the pendulum, and the center of mass and
+orientation of the ball are examples of generalized coordinates that
+automatically take into account the constraints.\\
-In [[book:chap4:eom-ode|Chapter 4]] we discussed the motion of point particles.
+In this chapter we discuss how to set up generalized coordinates
-However, in our environment the spatial extension of particles in crucial.
+and how to find the associated equations of motion. The discussion
-Physical objects always keep a minimum distance due to their spatial extension.
+will be driven by examples. The examples will be derived from the
-When they had zero extension,
+realm of integrable dynamics. These are systems where conservation laws can be used to break down the dynamics into separate problems that can be interpreted as motion with a single degree of freedom.\\
-one could neither blow up water droplets by impact with a laser ([[#fig_EXP-dropletExplosion |Figure 5.1]]),
-nor work clackers ([[#fig_EXP-clackers |Figure 5.2]]) or hit a ball with a tennis racket ([[#fig_EXP-reachTennisBall |Figure 5.3]]).
-Even giving spin to a ball only works due to the distance between the surface of the racket
-and the center of the ball.
-At the end of this chapter we will be able to discuss the evolution of balls with spin,
+At the end of the chapter you know why coins run away rolling
-and their reflections from flat surfaces.
+on their edge, and how the speed of a steam engine was controlled
-Why is spin of so much importance in table tennis?
+by a mechanical device. Systems where the dynamics is not integrable will subsequently be addressed in Chapter 7.
-How can a wingman score a goal in Handball,
-even when the goal keeper is fully blocking the direct path to the goal?((See [[https://en.wikipedia.org/wiki/Handball#Formations|Wikipedia's description of Handball formations]] and this [[https://de.wikipedia.org/wiki/Kreisl%C3%A4ufer_(Handball)#/media/Datei:Bertrand_Gille_02.jpg|picture of an attack by Bertrand Gille]], the IHF World Player of the Year 2002.))
 ----
-  * [[ 5.1 Motivation and outline | 5.1 Motivation and outline: How do particles collide?]]
+  * [[ 6.1 Motivation and Outline |6.1 Motivation and Outline: How to deal with constraint motion?]]
-  * [[ 5.2 Collisions of particles | 5.2 Collisions of hard-ball particles]]
+  * [[ 6.2 Lagrange formalism |6.2 Lagrange formalism]]
-  * [[ 5.3 Volume integrals | 5.3 Volume integrals — A professor falling through Earth]]
+  * [[ 6.3 Dynamics with one degree of freedom |6.3 Dynamics with one degree of freedom]]
-  * [[ 5.4 Center of mass and spin | 5.4 Center of mass and spin of extended objects]]
+  * [[ 6.4 Several DOF and conservation laws |6.4 Several degrees of freedom and conservation laws]]
-  * [[ 5.5 Bodies with internal degrees of freedom | 5.5 Bodies with internal degrees of freedom: Revisiting mobiles]]
+  * [[ 6.5 Dynamics of 2-particle systems |6.5 Dynamics of 2-particle systems]]
-  * [[ 5.6 Worked example | 5.6 Worked example: Reflection of balls]]
+  * [[ 6.6 Worked problems |6.6 Worked problems: spinning top and running wheel]]
-  * [[ 5.7 Problems | 5.7 Problems]]
+  * [[ 6.7 Problems |6.7 Problems]]