Page 192 - Intro to Tensor Calculus

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§2.2 DYNAMICS

Dynamics is concerned with studying the motion of particles and rigid bodies. By studying the motion
of a single hypothetical particle, one can discern the motion of a system of particles. This in turn leads to
the study of the motion of individual points in a continuous deformable medium.

Particle Movement

The trajectory of a particle in a generalized coordinate system is described by the parametric equations

i
i
x = x (t), i =1,...,N (2.2.1)
where t is a time parameter. If the coordinates are changed to a barred system by introducing a coordinate
transformation
i
2
N
1
i
x = x (x ,x ,... ,x ), i =1,... ,N
then the trajectory of the particle in the barred system of coordinates is
i
N
2
1
i
x = x (x (t),x (t),...,x (t)), i =1,... ,N. (2.2.2)
The generalized velocity of the particle in the unbarred system is defined by
dx i
i
v = , i =1,...,N. (2.2.3)
dt
By the chain rule differentiation of the transformation equations (2.2.2) one can verify that the velocity in
the barred system is
r
dx r ∂x dx j ∂x r j
r
v = = = v , r =1,... ,N. (2.2.4)
dt ∂x dt ∂x j
j
i
Consequently, the generalized velocity v is a first order contravariant tensor. The speed of the particle is
obtained from the magnitude of the velocity and is
2
i j
v = g ij v v .
i
The generalized acceleration f of the particle is defined as the intrinsic derivative of the generalized velocity.
The generalized acceleration has the form

m
2 i
δv i i dx n dv i i m n d x i dx dx n
i
f = = v ,n = + v v = + (2.2.5)
δt dt dt mn dt 2 mn dt dt
and the magnitude of the acceleration is
2
i j
f = g ij f f .