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Subsections

2 Lagrangian mechanics

2.1 The Lagrangian

In Lagrangian mechanics we start by writing down the Lagrangian of the system

L = T - U (1)

where T is the kinetic energy and U is the potential energy. Both are expressed in terms of coordinates (q,$\dot{q}$) where q $\in$ $\reals^{n}_{}$ is the position vector and $\dot{q}$ $\in$ $\reals^{n}_{}$ is the velocity vector.

2.2 The Lagrangian of the pendulum

An example is the physical pendulum (see Figure 1).
 
Figure 1:  The configuration space of the pendulum
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The natural configuration space of the pendulum is the circle. The natural coordinate on the configuration space is the angle $\theta$ . If the mass of the ball is m and the length of the rod is l then we have
T = $\displaystyle{\textstyle\frac{1}{2}}$m(l$\displaystyle\dot{\theta}$)2 (2)
U = - mglcos $\displaystyle\theta$ (3)
Thus, the Lagrangian in coordinates ($\theta$,$\dot{\theta}$) is

L = $\displaystyle{\textstyle\frac{1}{2}}$m(l$\displaystyle\dot{\theta}$)2 + mglcos $\displaystyle\theta$. (4)

2.3 Equations of motion

In Lagrangian mechanics the equations of motion are written in the following universal form:

 
$\displaystyle{\frac{d}{dt}}$$\displaystyle\left(\frac{\partial L}{\partial\dot q}\right)=$$\displaystyle{\frac{\partial L}{\partial q}}$. (5)

2.4 Pendulum--Equations of motion

For example, for the pendulum we have:
$\displaystyle{\frac{\partial L}{\partial\dot\theta}}$ = ml 2$\displaystyle\dot{\theta}$, (6)
$\displaystyle{\frac{\partial L}{\partial\theta}}$ = - mglsin $\displaystyle\theta$. (7)
Thus, the equations of motion are written as

$\displaystyle{\frac{d}{dt}}$(ml 2$\displaystyle\dot{\theta}$) = - mglsin $\displaystyle\theta$. (8)

This equation can be written as second order equation

ml 2$\displaystyle\ddot{\theta}$ = - mglsin $\displaystyle\theta$ (9)

or in the traditional way

$\displaystyle\ddot{\theta}$ = - $\displaystyle{\frac{g}{l}}$sin $\displaystyle\theta$. (10)

2.5 The meaning of dot

We should emphasize that $\dot{\theta}$ has dual meaning. It is both a coordinate and the derivative of the position. This traditional abuse of notation should be resolved in favor of one of these interpretations in every particular situation.

2.6 Lagrangian vs. Newtonian mechanics

In Newtonian mechanics we represent the equations of motion in the form of the second Newton's law:

 
m$\displaystyle\ddot{q}$ = f (q,t) (11)

where f (q,t) is the force applied to the particle.

This equation is identical to the equation obtained from Lagrangian representation if f (q,t) is a conservative field, i.e. it has a potential. A potential is a function U(q,t) such that

f (q,t) = - $\displaystyle{\frac{\partial U}{\partial q}}$. (12)

Indeed, the Lagrangian can be written as

L = $\displaystyle{\textstyle\frac{1}{2}}$m($\displaystyle\dot{q}$)2 - U(q,t). (13)

According to 5 the equations of motion reduce to 11.
next up previous
Next: 3 The variational principle Up: Lagrangian and Hamiltonian mechanics Previous: 1 What does this
Marek Rychlik
9/2/1997