# Lagrangian Mechanics Made Simple

Part 2

Rotational Motion

**Ron Steinke <rsteinke@w-link.net>**

Last month we discussed how to use Lagrangian mechanics to do simple one dimensional motion. This month, we go on to motion in multiple dimensions, and discuss how to handle rotation.

# Vectors and Multiple Dimensions

Since we're going to be working in multiple dimensions, it's
easiest to write our equations of motion using vectors.
The usual notation uses for the position vector,

(1) |

**boldface**indicates a vector, so is the position vector, and is its magnitude

Lagrangian mechanics in multiple dimensions is very similar to
the one dimensional case. We begin by writing down the kinetic
energy and the potential energy ,

(2) |

In this case, we have chosen and for a particle moving in three dimensions with no external forces. We then write down the Lagrangian,

Next, we find the equations of motion. Since we now have
three variables, for , we get three
Lagrange equations,

showing that the free floating particle is not accelerating, as expected.

# Rotating Coordinate Systems

The equation of motion (5)
is simple enough, since we're working in a fixed
rectangular coordinate system. Of course, many other
choices of coordinates are possible ^{1}.
A particularly interesting choice is a rotating coordinate
system. Imagine you are riding inside a large spinning drum,
like those found in amusement parks. If you stay
at a "fixed" position in the drum, you're really moving,
since the drum is rotating. It is useful to describe
the physics of object inside the drum in terms of
a rotating coordinate , which has a fixed value
at a given point on the drum. For a drum rotating
with frequency about the axis, we can write your real
position in terms of your "drum position"
,

If you aren't tied to the drum, you can move around
inside. Your "drum position" and real position
are both functions of the time . By
taking the time derivative in
(6), we can calculate
your velocity in terms of ,

# Mechanics of Rotating Systems

We are now in a position to use Lagrange's equation
(4) to find the equations of motion
in terms of the coordinate . Since we know the
velocity from (7), we can write down
the Lagrangian,

Notice that the functions and don't appear in the Lagrangian. This is a reflection of the

**time translation invariance**of the equations of motion. It doesn't matter how long the drum has been rotating, the equations of motion will be the same. This is a feature of the coordinate system we have chosen, and would not be true for, say, rotation around an ellipse, where the equations of motion would depend on what part of the ellipse you were on.

Looking at Eq. (8), you can see that
it's the sum of three squares. This suggests that is
actually the square of some vector, with each of the terms
in the sum being the square of one of the vector's components.
We use this fact to rewrite in vector notation,

(9) |

and the right hand side is

Combining (10) and (11) gives the equation of motion

# Pseudo Forces

We have arrived at the result (12), which seems to say that the object at position is accelerating. This may seem mysterious at first, but is actually quite simple. Looking at the two terms on the right hand side of (12), we see that the second, , is the familiar centripetal "force." This term is equal to multiplied by the part of which is perpendicular to the axis of rotation. It points away from the axis, so the object will tend to get further away with time. The inward force you feel on amusement rides is the wall providing a compensating acceleration to keep you in the "same" place.

The term is commonly known as the Coriolis "force." It depends on the velocity instead of the position , so it is not as much a part of common experience as the centripetal "force." It causes objects moving towards or away from the rotation axis to be deflected around the axis, and is an important feature of rotational kinematics.

Current Issue: February 2003, Recent Issues: January 2003, November 2002, October 2002 |