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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