130_notes.dvi

where theqiare the coordinates of the particle. This equation is derivable from theprinciple of
least action.

δ

∫t^2

t 1

L(qi,q ̇i)dt= 0

Similarly, we can define theHamiltonian

H(qi,pi) =

∑

i

piq ̇i−L

wherepiare the momenta conjugate to the coordinatesqi.

pi=

∂L

∂q ̇i

For a continuous system, like astring, the Lagrangian is an integral of a Lagrangian density function.

L=

∫

Ldx

For example, for a string,

L=

1

2

[

μη ̇^2 −Y

(

∂η

∂x

) 2 ]

whereY is Young’s modulus for the material of the string andμis the mass density. TheEuler-
Lagrange Equationfor a continuous system is also derivable from the principle of least action
states above. For the string, this would be.

∂

∂x

(

∂L

∂(∂η/∂x)

)

+

∂

∂t

(

∂L

∂(∂η/∂t)

)

−

∂L

∂η

= 0

Recall that the Lagrangian is a function ofηand its space and time derivatives.

TheHamiltonian densitycan be computed from the Lagrangian density and is a function of the
coordinateηand its conjugate momentum.

H= ̇η

∂L

∂η ̇

−L

In this example of a string,η(x,t) is asimple scalar field. The string has a displacement at each
point along it which varies as a function of time.

If we apply theEuler-Lagrange equation, we get a differential equationthat the string’s
displacement will satisfy.

L =

1

2

[

μη ̇^2 −Y

(

∂η

∂x

) 2 ]

∂

∂x

(

∂L

∂(∂η/∂x)

)

+

∂

∂t

(

∂L

∂(∂η/∂t)

)

−

∂L

∂η

= 0

∂L

∂(∂η/∂x)

= −Y

∂η

∂x