I
legendre's dual transformation
Start with given Function
F
F Lu
of variable
we introduce a new we
Vi
Now we define a
⼆
new
G⼆
As
we express
器
U
Un
⼀
function G
三 uivi
ui interms
G G Cu
⼆
un
F
we can express G in terms
of Vi
Vi
un
SG ⼆ 三 器 SVi
uisvitrifuij 4⼆
in other hand SG⼆三
4
⼆三 扎 it Lvi_
Since
we
of
vi
点
⼆
無Hug
get theform
SGIUisv bghhtiut.in
Ui 点
⼆
This result
remarkable dnaliy of Legendre's Fansformation
expresses a
Old System
u
un
GCV
We enlarge
afunction
The
Un
Vi
un l
F 三 FLU
Vi ⼆ 岳
G ⼆ 三 Uivi F
G
New System
以
G Gw
Ui
⼆
1⼆ 三 Viui
Flu
1
our transformation in a
of two sets of variable
ui are independent
無
of ui
The
atg
transformation is
symmetrical
以
G
un
further respect.Assume that Fisatndg
EF
cm
my
in thetransformation
un Uh
and do not participate
of us and
p p
process oftransformation and get
ndepen
We had to the
2
all
ui
the uansl
a new
on
relation
器
器
We
n
and vi the wire
the
the passive
wi
Legendre's transformation applied to the Lagrangian function
L Lcq
qi
qn
qi
we qī and the pass in andqibeaetire in
introduce
pi
a
obtain
意
wwfwetion
H⼆
thus
⼆
H
三 piqi_ L
H Lq
Gip
⼆
pniy
called Hanlan function
Hamilton function
Lagrangian function
pi
legendre's transformation
点
q ⼆哥
1⼤ 三Pīqi_ L
H
L
⼆
三 pi
Hcp.irpniq quit
L Lcq
And the form of passive varih
蚩
是
器
⼆
⼀
年
qiq in
⼀
it
3 Fansformation
of the Lagrangian equations of motion
By the definition of moment
pi
莹⼀
pi ⼆ 点
According to legendre'stransformation
器
pi
器
器
the Lagrangian equations of motion have been replacedby a new see
of differential equations called the canonical time of Hamilton
Thus
qi
These wins
岩
⼆⼀
莹
entirely equivalent to the original Lagrangian wins
mathematically newform Yet the equations are
vastly
are
and are merely a
superior to the originals
4 The
i
canonical integral