Variational Principle in Physics: From Lagrange to Hamilton, Study notes of Physics

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Lecture 5 Outline - Variational Principle

one final Lagrange example (Section 1.6)...

Hamilton’s Variational Principle (Section 2.1)

Calculus of Variations (Section 2.2)

Another Lagrange Example

Time-dependant rotating (straight) wire

space

Hamilton’s Variational Principle

derivable from scalar potentials (except constraints!)For monogenic systems in which all forces are

which may depend on

q i , dq

i

dt

, t

The motion of the system from time

t 1

to

t 2

is such that

I

t 2

t 1

L

dt

t 2

t 1 ( T − V )

(^) dt

has a stationary value for the actual path of motion:

δI

δ

∫

t 2

t 1 L ( q 1

, q

2 ,... ,

dq

1

dt

dq

2

dt

,... , t

dt

Sometimes

I

is called the action (units of energy/time).

Variational principle’s appeared in different forms

Calculus of variations setup

Assume correct path

y ( x,

with set of paths

y ( x, α

y ( x,

αη

x

)

by defn

η ( x

1 ) =

η ( x

2 ) = 0

(nice functions

y ( x,

η ( x

)

)

J

α

) =

x 2

x 1

f (^) ( y ( x, α

dx dy

x, α

, x

dx

want stationary point when

dαdJ

α

since

α

is independent of

x

, differentiate under the sign.

dα dJ

x 2

x 1

dα df

dx

x 2

x 1

[

∂y ∂f

∂α∂y

∂ ∂f

˙y

∂

˙y

∂α

∂x ∂f

∂α∂x

]

dx

x 2

x 1

[

∂y ∂f

∂α∂y

∂ ∂f

˙y

2 y

∂x∂α

]

dx

now second part we integrate by parts

Calculus of variations technique

integrating by parts:

vdu

uv

udv

with

u

∂α∂y

and

v

∂ ∂f

˙y

so

dxdv

d

dx

∂ ∂f (^) ˙y )

thus

dv

d

dx

∂ ∂f (^) ˙y )

dx

x 2

x 1

∂^ ∂f

˙y

2 y

∂x∂α

dx

[

∂ ∂f

˙y

∂α∂y

]

x 2

x 1 − ∫ x 2

x 1

d

dx

∂ ∂f

y

)

∂α ∂y

dx

but

∂α∂y

x

1 ) =

n

( x

1 ) =

∂α∂y

x

2 ) =

n

( x

2 ) = 0

dα dJ

x 2

x 1

[

∂y ∂f

d

dx

∂ ∂f

˙y )]

∂α ∂y

dx

thus at the stationary value of

α

dα dJ

α

x 2

x 1

[

∂y ∂f

d

dx

∂ ∂f

y

)]

∂α∂y

α

dx

but

∂α∂y

η ( x )

is an arbitrary function!

Simple c of v example

Shortest distance between two points (in a plane)