Advanced mechanics by Eric Poisson
Author:Eric Poisson
Language: eng
Format: epub
Tags: Advanced Mechanics,
(2.5.3)
and a statement of conservation follows immediately:
Whenever L does not depend explicitly on time, so that dL/dt = 0, we have that
h(q a , q a ) = ^Paia - L (2.5.4)
a
is a constant of the motion, dh/dt = 0.
Surely the function h(q a , q a ) must have something to do with the system's total mechanical energy. Let us first figure out the relationship in the context of a simple example. We go back to the Lagrangian of a particle expressed in cylindrical coordinates,
L= l -m{p 2 + p 2 4> 2 + z 2 )-V{p,cj > ,z),
2.5 Generalized momenta and conservation statements
83
but this time we place no constraints on the potential energy. The generalized momenta are p p = mp, p<p = mp 2 <j), and p z — mz. We then have
h = P P P + P<t>4> + Pz* ~ L
= mp 2 + mp 2 <\> 2 + mz 2 - ^m(p 2 + p 2 (j) 2 + z 2 ) + V(p, (f>, z)
= \m{p 2 +p 2 4> 2 + z 2 ) + V{p,^,z).
This is indeed the total mechanical energy, the sum of kinetic and potential energies.
To verify that h(q ai q a ) is always equal to the total mechanical energy we use the fact that the kinetic energy is usually a quadratic function of the generalized velocities,
T = ^^2A ab q a q b .
a, b
The coefficients A ab may in general depend on the coordinates q a , and without loss of generality we may assume that A ba = A ab . The Lagrangian is then
L = \ ^A ab q a q b - V{q a )
2
a.b
The generalized momentum p a is obtained by differentiating L with respect to q a . To see what this amounts to let us consider a special case in which the mechanical system possesses three degrees of freedom. In this case we have, explicitly,
L = \ All< il + ^129192 + -4139143 + 7,A 22 q 2 + ^23^293 + ^3393 ~ V {<lU 92, 93)-
It follows that
FIT,
= A n qi + A 12 q 2 + A 13 q 3 , = Ai 2 qi + A 22 q 2 + A 23 q 3 ,
= ^1391 + -42392 + A 33 q 3 are the generalized momenta. These relations are summarized by
Pa = ^A ab q b ,
b
and the same expression is always obtained, regardless of the number of degrees of freedom. The function h is then
h = ^2Pa4a-L
a
= ^2(^2A ab q b jq a - ^2,A ab q a q b + V{q a )
\ U / „ h
a.b
= ]^^A ab q a q b + V(q a ),
a, b
and we conclude that
Hq a , q a ) = T(q a , q a ) + V{q a ) = total mechanical energy. (2.5.5)
In all generality, therefore, the function h is the system's total energy, and this is conserved whenever L does not depend explicitly on time.
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