Mathematical Methods for Physicists: A concise introduction - Site Map

HAMILTON'S PRINCIPLE
Then the function H is
Z
H ˆ
C
‰ 1 2 …xy p
0 y†‡
1 ‡ y 02 Šdx
and the Euler±Lagrange equation gives
!
d 1
dx 2 x ‡ y 0
p
‡ 1
1 ‡ y 02 2 ˆ 0
or
Solving for y 0 , we get
y 0
p
ˆx ‡ c
1 ‡ y 02
1 :
y 0 ˆ dy
dx ˆ x c
q 1
;
2 …x c 1 † 2
which on integrating gives
q
y c 2 ˆ 2 …x c 1 † 2
or
…x c 1 † 2 ‡…y c 2 † 2 ˆ 2 ;
a circle:
Hamilton's principle and Lagrange's equation of motion
One of the most important applications of the calculus of variations is in classical
mechanics. In this case, the functional f in Eq. (8.1) is taken to be the Lagrangian
L of a dynamical system. For a conservative system, the Lagrangian L is de®ned
as the di€erence of kinetic and potential energies of the system:
L ˆ T V;
where time t is the independent variable and the generalized coordinates q i …t† are
the dependent variables. What do we mean by generalized coordinates? Any
convenient set of parameters or quantities that can be used to specify the con®guration
(or state) of the system can be assumed to be generalized coordinates;
therefore they need not be geometrical quantities, such as distances or angles. In
suitable circumstances, for example, they could be electric currents.
Eq. (8.1) now takes the form that is known as the action (or the action integral)
I ˆ
Z t2
t 1
Lq … i …t†; _q i …t†; t†dt; _q ˆ dq=dt …8:17†
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