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Subsections
It will be convenient to first consider arbitrary linear changes of
variables. Let
L(q,,t) be a Lagrangian and let us perform the
change of variables
where
B(t) : any linear transformation (matrix).
Let us first consider what happens to Newton's equation
m = f (q) under the change of variables
28. We have
It will be convenient to introduce the matrix
= B^{  1} ,
so that the following firstorder matrix differential equation is
satisfied:

= B.
 (30)

Using this equation, we may write
Differentiating the second time we obtain:

=

B( + + Q) + ( + Q)
 

=

B( + + Q) + B( + Q)
 

=

B( + 2 + Q + Q).
 (32) 
We multiply this equation by m and obtain

f (q) = B(m + 2m + mQ + mQ).
 (33)

We introduce new ``force''

F(Q) = B^{  1}f (BQ).
 (34)

(This formula means to a physicist that force is realy a vector,
i.e. it transforms under coordinate changes as above.) We
multiply equation 33 by B^{  1} and obtain:

F(Q) = m + 2m + mQ + mQ.
 (35)

Summarizing the calculations of the previous section,
Newton equations of motion
m = f (q) after a linear
timedependent change of coordinates q = B(t)Q
become (see 35):

m = F(Q)  2m  mQ  mQ
 (36)

where
= B^{  1} .
This is a special case when n = 3 and the matrix B is orthogonal of
determinant 1. In this case the matrix is skewsymmetric.
There is a vector which represents the angular velocity
of the rotating coordinate system, such that for all
Q we
have

Q = x Q
 (37)

where x denotes the usual cross product of vectors. Indeed,
if
= (,,) then the matrix of the linear
transformation
Q x Q is . On the other
hand
  (38)

Thus, we have

=
 (39)

According to the previous section for n = 3 the equations
of motion are:

m = F(Q)  2m x  m x Q  m x ( x Q).
 (40)

The interpretation of these equations is that the fact that
in a rotating coordinate system there are additional forces acting
upon the body, represented by the three additional terms in the
righthand side of this equation. They have their names:
 1.

 2m x is called the Coriolis force;
 2.

 m x ( x Q) is simply the centripetal
force;
 3.

 m x Q is the force of inertia; this
force is 0 if the rotation is uniform, i.e. is constant.
Let
L(q,,t) be a Lagrangian.
According to 20 the
change of variables q = B(t)Q leads to motions described by the new
Lagrangian

K(Q,,t) = L(Q,B + Q,t) = L(Q,B( + Q),t)
 (41)

This formula illustrates the benefits of the Lagrange formalism
when dealing with coordinate changes.
It is interesting to see how the Coriolis, centripetal and inertia
forces can be derived from the Lagrangian formalism. Let L = T  U
where
T = (1/2)m()^{2}. If B is orthogonal then
Bx,By = x,y where
, is the
dot product (so
()^{2} = , ).
Hence, the new Lagrangian is:
K(Q,,t) = m()^{2} + m(Q,) + m(Q,Q)  U(BQ,t).


 (42) 
The extra two terms translate into the additional terms in the
equations of motion which were called Coriolis, centripetal and
inertia forces.
We find the generalized momentum:
and
where
(Q,t) = U(BQ,t) is the new potential.
After identifying

F(Q) = 
 (45)

(we use B^{  1} = B^{ T} here!) we obtain exactly the same
result as in 40.
Having calculated the generalized momentum in rotating coordinates
we may find the Hamiltonian in rotating coordinates:
H(P,Q)

=

P  K(Q,)
 

=

P(P/m  Q)  (P/m  Q)^{2} + (Q,t)
 

=

+ P,Q + (Q).
 (47) 
The Hamiltonian equations of motion are
In three dimensions ( n = 3 ) these equations could also be written
as

=

P/m  x Q,
 

=

  x P.
 (49) 
It is easy to make the sign mistake in determining the direction of
. A simple rule eliminates this mistake: is opposite to the
angular velocity of a point at rest in the original coordinate system
expressed in the rotating coordinate system. For example, if we use
a coordinate system rigidly attached to the earth then a point on the
surface of the earth resting in nonrotating coordinates appears to be
moving west for an observer in the rotating coordinate system. Thus
is pointing towards the north pole, with length equal to

 (50)

Next: 6 Restricted circular threebody
Up: Lagrangian and Hamiltonian mechanics
Previous: 4 Hamiltonian mechanics
Marek Rychlik
9/2/1997