bradley j. nartowt Monday, September 09, 2013, 15:23:38
PHYS 6246 – classical mechanics Dr. Whiting
a particle of mass m moves in one dimension such that it has the Lagrangian, L
1
2
m x
12
4
fix typos?
mx V (x ) V 2 (x ) 2
1
2
m x
12
4
mx V (x ) V (x ) 2
2
[I.1]
still ambiguous, even in light of errata, so use
Here, V(x) V(x) is some differentiab differentiable le function function of x. find the equation equation of motion motion for x(t).
L mx 2V x 2V Vx x
L d 4 2 3 m x 2mxV 0 3124 m 2 x 2 x 2m ( xV xV x x ) x dt 12
d dt
We can now write down the equation of motion the net generalized force per unit mass
(m
2
x
2mV ) x mx mx 2V
2
2
x
L x
[I.2]
dtd ( Lx ) ( mx 2 2V )V x m 2 x 2 x 2m (xV V xx 2 ) ; computing
,
2mx
V
2
x
mx
2V mx 2 mx
2
2V
Vx
mx
2
mx
2
2V Vx 2V
Vx
mx mx
Vx [I.3]
Describe the physical physical nature of the system system on the the basis of of this equation equation of motion. motion. ????
If the author intended L
1 12
2
m x
4
mx 2V ( x ) V (x ) 2 , then [I.3] becomes
mx
V x V x , and it is extremely
tempting to identify V(x), which we know only to be a differentiable function of x, as the potential energy. Ordinary mechanics mechanics follows. ????
If the author intended L
1 12
2
m x
4
mx 2V (x ) V (x ) 2 , then [I.3] becomes
mx
V x
V x
. You cannot identify V(x),
then, as the potential, because it would imply (one dimensional) mechanics where particles accelerated “uphill” of potential-gradients. potential-gradients. For For instance, instance, for V ( x ) kx , we would have mx kx , indicating acceleration is in the +x direction. However, the potential slopes potential slopes upward upward in in the +x direction. direction. Even worse, V ( x ) 12 kx would have position 2
x(t ) Ae
t k / m
V ( x)
1 2
kx
2
Bet
k /m
Ae t Be
t
, an exponential acceleration away from the origin, meaning the parabola
we are accustomed to facilitating simple harmonic oscillation would instead act as a repulsing agent. repulsing agent. Only
the inverted-parabola inverted-parabola V ( x ) 12 kx would give a position x (t ) Ae 2
it
Be
i t
C cos(t ) D sin( t ) we are
accustomed to obtaining from hooke’s law. law . This lends support for the case to identify V(x) as the negative of the good ol’ potential energy we know and love which which implies mechanics. mechanics.
