Page 196 - Intro to Tensor Calculus

191

Work and Potential Energy

Define M as the constant mass of the particle as it moves along the curve defined by equation (2.2.1).
r
Also let Q denote the components of a force vector (in appropriate units of measurements) which acts upon
the particle. Newton’s second law of motion can then be expressed in the form

r
Q = Mf r or Q r = Mf r . (2.2.18)

r
r
The work done W in moving a particle from a point P 0 to a point P 1 along a curve x = x (t),r =1, 2, 3,
with parameter t, is represented by a summation of the tangential components of the forces acting along the
path and is defined as the line integral

Z P 1 dx r Z P 1 Z t 1 dx r Z t 1
r
r
W = Q r ds = Q r dx = Q r dt = Q r v dt (2.2.19)
ds dt
P 0 P 0 t 0 t 0
s
where Q r = g rsQ is the covariant form of the force vector, t is the time parameter and s is arc length along
the curve.
Conservative Systems
If the force vector is conservative it means that the force is derivable from a scalar potential function

∂V
2
1
N
V = V (x ,x ,...,x ) such that Q r = −V ,r = − , r =1,... ,N. (2.2.20)
∂x r
In this case the equation (2.2.19) can be integrated and we find that to within an additive constant we will
have V = −W. The potential function V is called the potential energy of the particle and the work done
becomes the change in potential energy between the starting and end points and is independent of the path
connecting the points.

Lagrange’s Equations of Motion

The kinetic energy T of the particle is defined as one half the mass times the velocity squared and can
be expressed in any of the forms
2
1 ds 1 2 1 1
m n
m n
T = M = Mv = Mg mnv v = Mg mn ˙x ˙x , (2.2.21)
2 dt 2 2 2
where the dot notation denotes differentiation with respect to time. It is an easy exercise to calculate the
derivatives
∂T m
= Mg rm ˙x
∂ ˙x r

d ∂T m ∂g rm n m
= M g rm ¨x + ˙ x ˙x (2.2.22)
dt ∂ ˙x r ∂x n
∂T 1 ∂g mn m n
= M ˙ x ˙x ,
∂x r 2 ∂x r
and thereby verify the relation

d ∂T ∂T
− = Mf r = Q r , r =1,... ,N. (2.2.23)
dt ∂ ˙x r ∂x r