q(t1 ) and q(t2 ) at the times t1 and t2 . L is called the Lagrangian of the

system.

This principle (see the proof below) leads to the Lagrange equations

d ‚L ‚L

’ (i = 1, · · · , n).

=0 (15.2)

dt ‚ qi

™ ‚qi

The Lagrangian is not uniquely determined by this requirement because

any (total) time derivative of some function of q and t will have an action

independent of the path and can therefore be added to L.

Proof We prove (15.2). The variation of the action (15.1), when the path

between q(t1 ) and q(t2 ) is varied (but the end points are kept constant),

must vanish if S is a minimum:

t2

‚L ‚L

δS = δq + δq

™ dt = 0.

‚q ‚q

™

t1

Partial integration of the second term, with δ q = dδq/dt and realizing that

™

δq = 0 at both integration limits t1 and t2 because there q is kept constant,

converts this second term to

t2

d ‚L

’ δq dt.

dt ‚q

™

t1

Now

t2

‚L d ‚L

’

δS = δq dt = 0.

‚q dt ‚q

™

t1

Since the variation must be zero for any choice of δq, (15.2) follows.

For a free particle with position r and velocity v the Lagrangian L can only

be a function of v 2 if we assume isotropy of space-time, i.e., that mechanical

laws do not depend on the position of the space and time origins and on

the orientation in space. In fact, from the requirement that the particle

15.3 Hamiltonian mechanics 399

behavior is the same in a coordinate system moving with constant velocity,

it follows1 that L must be proportional to v 2 .

For a system of particles interacting through a position-dependent poten-

tial V (r 1 , . . . , r N ), the following Lagrangian:

N

L(r, v) = ’ V,

1 2

2 mi v i (15.3)

i=1

yields Newtons equations of motion

‚V

mv i = ’

™ , (15.4)

‚r i

as the reader can easily verify by applying the Lagrange equations of motion

(15.2).

15.3 Hamiltonian mechanics

In many cases a more appropriate description of the equations of motions in

generalized coordinates is obtained with the Hamilton formalism. We ¬rst

de¬ne a generalized momentum pk , conjugate to the coordinate qk from the

Lagrangian as

def ‚L

pk = . (15.5)

‚ qk

™

Then we de¬ne a Hamiltonian H in such a way that dH is a total di¬erential

in dp and dq:

n

def

H= pk qk ’ L.

™ (15.6)

k=1

¿From this de¬nition it follows that

n

‚L ‚L

pk dqk + qk dpk ’ dqk ’

dH = ™ ™ d qk .

™ (15.7)

‚qk ‚ qk

™

k=1

The ¬rst and the last terms cancel, so that a total di¬erential in dp and dq

is obtained, with the following derivatives:

‚H

= qk ,

™ (15.8)

‚pk

‚H

= ’pk .

™ (15.9)

‚qk

These are Hamilton™s equations of motion.

1 See Landau and Lifshitz, (1982), Chapter 1.

400 Lagrangian and Hamiltonian mechanics

The reader may check that these also lead to Newton™s equations of mo-

tion for a system of particles interacting through a coordinate-dependent

potential V , where

n

p2

1

H(p, q) = k

+ V (q). (15.10)

2 2mk

k=1

In this case H is the total energy of the system of particles, composed of the

kinetic energy2 K and potential energy V .

If H does not depend explicitly on time, it is a constant of the motion,

since

dH ‚H ‚H

= pk +

™ qk

™

dt ‚pk ‚qk

k

(qk pk ’ pk qk ) = 0.

= ™™ ™™ (15.11)

k

So Hamiltonian mechanics conserves the value of H, or “ in the case of

an interaction potential that depends on position only “ the total energy.

Therefore, in the latter case such a system is also called conservative.

15.4 Cyclic coordinates

A coordinate qk is called cyclic if the Lagrangian does not depend on qk :

‚L(q, q, t)

™

= 0 for cyclic qk . (15.12)

‚qk

For the momentum pk = ‚L/‚ qk , conjugate to a cyclic coordinate, the time