# PHY6200 W07

## Review exam

Everyone will get a chance to work a problem at the board.

## Chapter 7: Lagrange's Equations

### Reading:

In Taylor, read sections 7.10 and 10.9 for today, 10.9 to 10.10 for Wednesday.

### Recall

S = ∫_{1}^{2} L(q_{i}, q_{i}dot, t) dt.

∂L/∂q_{i} - (d/dt)(∂L/∂q_{i}dot) = 0

L = T - U

H = ∑(p_{i}q_{i}dot) - L = T + U = E.

### Lagrange's Equations for Magnetic Forces*

Interesting, but does involve the use of the vector potential, something beyond the present level.
I recommend that students refer back to this when doing E&M at the appropriate level.
This is an example of how topics in physics tie together; our divisions into standalone subjects is somewhat arbitrary.

### Lagrange Multipliers and Constraint Forces*

## Rotational Motion of Rigid Bodies

### Recall

We derived a set of equations for treating the motion of a rigid body in a rotating frame where the principal axes of the body are fixed.
These are the Euler equations:

**L**dot + **ω**×**L** = **Γ**

or in component form

λ_{1}ω_{1}dot - (λ_{2} - λ_{3})ω_{2}ω_{3} = Γ_{1}

λ_{2}ω_{2}dot - (λ_{3} - λ_{1})ω_{3}ω_{1} = Γ_{2}

λ_{3}ω_{3}dot - (λ_{1} - λ_{2})ω_{1}ω_{2} = Γ_{3}

This is a set of coupled, nonlinear differential equations.
We solved these for certain special cases only.
Now that we've learned the Lagrangian formulation of mechanics, we can learn how to treat them in the Lagrangian formalism.
Since the Lagrangian is simply kinetic minus potential energy in an inertial system, we need a way to express the kinetic energy of a rotating body in an inertial system.
This is normally accomplished with by defining 3 angular coordinates that specify the orientation of the body relative to fixed spatial coordinates.
The 3 angular coordinates are called Euler angles.

### Euler Angles

### Motion of a Spinning Top

© 2007 Robert Harr