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1/5-the-lagrangian-for-a-system-of-particles.md

@@ -4,6 +4,8 @@ title: 5. The Lagrangian for a system of particles
 
 Let us now consider a system of particles which interact with one another but with no other bodies. This is called a closed system. It is found that the interaction between the particles can be described by adding to the Lagrangian `LL1/4.2` for non-interacting particles a certain function of the co-ordinates, which depends on the nature of the interaction[^1]. Denoting this function by $-U$, we have
 
+[^1]: This statement is valid in classical mechanics. Relativistic mechanics is not considered in this book. (See `LL2`)
+
 ```load
 LL1/5.1
 ```
@@ -84,4 +86,11 @@ LL1/5.8
 
 To conclude this section, we may make the following remarks concerning the application of Lagrange's equations to various problems. It is often necessary to deal with mechanical systems in which the interaction between different bodies (or particles) takes the form of constraints, i.e. restrictions on their relative position. In practice, such constraints are effected by means of rods, strings, hinges and so on. This introduces a new factor into the problem, in that the motion of the bodies results in friction at their points of contact, and the problem in general ceases to be one of pure mechanics `LL1/25`. In many cases, however, the friction in the system is so slight that its effect on the motion is entirely negligible. If the masses of the constraining elements of the system are also negligible, the effect of the constraints is simply to reduce the number of degrees of freedom $S$ of the system to a value less than $3N$. To determine the motion of the system, the Lagrangian `LL1/5.5` can again be used, with a set of independent generalised co-ordinates equal in number to the actual degrees of freedom.
 
-[^1]: This statement is valid in classical mechanics. Relativistic mechanics is not considered in this book. (See `LL2`)
+## Problems
+
+Find the Lagrangian for each of the following systems when placed in a uniform gravitational field (acceleration $g$).
+
+#### PROBLEM 1. A coplanar double pendulum
+
+`LL1/fig1`
+