Page 204 - Intro to Tensor Calculus

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where τ is a parameter used to describe the curve C. The principle of stationary action states that of all
curves through the points P 1 and P 2 the one which makes the action an extremum is the curve specified by
Newton’s second law. The extremum is usually a minimum. To show this let

1/2
j
√ dx dx k
φ = 2m (h − V )g jk
dτ dτ
k
in equation (2.2.36). Using the notation ˙x = dx k we find that
dτ
∂φ 2m k
= (h − V )g ik ˙x
∂ ˙x i φ
∂φ 2m ∂g jk j k 2m ∂V j k
= (h − V ) ˙ x ˙x − g jk ˙x ˙x .
∂x i 2φ ∂x i 2φ ∂x i
The equation (2.2.36) which describe the extremum trajectories are found to be

d 2m k 2m ∂g jk j k 2m ∂V j k
(h − V )g ik ˙x − (h − V ) ˙ x ˙x + g jk ˙x ˙x =0.
dt φ 2φ ∂x i φ ∂x i
√ mφ
By changing variables from τ to t where dt = √ we find that the trajectory for an extremum must
dτ 2(h−V )
satisfy the equation
k j k
d dx m ∂g jk dx dx ∂V
m g ik − + =0
dt dt 2 ∂x i dt dt ∂x i
which are the same equations as (2.2.24). (i.e. See also the equations (2.2.22).)
Dynamics of Rigid Body Motion

Let us derive the equations of motion of a rigid body which is rotating due to external forces acting
upon it. We neglect any translational motion of the body since this type of motion can be discerned using
our knowledge of particle dynamics. The derivation of the equations of motion is restricted to Cartesian
tensors and rotational motion.
Consider a system of N particles rotating with angular velocity ω i ,i =1, 2, 3, about a line L through
~ (α)
the center of mass of the system. Let V denote the velocity of the αth particle which has mass m (α) and
(α)
position x , i =1, 2, 3 with respect to an origin on the line L. Without loss of generality we can assume
i
that the origin of the coordinate system is also at the center of mass of the system of particles, as this choice
of an origin simplifies the derivation. The velocity components for each particle is obtained by taking cross
products and we can write

~ (α)
r
V = ~ω × ~ (α) or V (α) = e ijk ω j x (α) . (2.2.37)
i k
The kinetic energy of the system of particles is written as the sum of the kinetic energies of each
individual particle and is
N N
1 X (α) (α) 1 X (α) (α)
T = m (α) V i V i = m (α) e ijk ω j x k e imn ω mx n . (2.2.38)
2 2
α=1 α=1