Modified Hamiltonian formalism for higher

Transkrypt

Outline
Motivation
Alternative formalism
Many degrees of freedom
Remarks and conclusions
Modified Hamiltonian formalism for
higher-derivative theories
Krzysztof Andrzejewski
together with Joanna Gonera, Piotr Kosiński and Paweł Maślanka
XXIX Max Born Symposium. Wrocław 28-30 June 2011
Modified Hamiltonian formalism for higher-derivative theories
Wydział Fizyki i Informatyki Stoswanej. Uniwesytet Łódzki
Outline
Motivation
1
Motivation
2
The case of even derivatives
The case of odd derivatives
Example
3
The case of second derivatives
The case of third derivatives
4
Outline
Motivation
Motivation
Standard framework for dealing with higher-derivative theories is
provided by Ostrogradski formalism.
The main disadvantages of the Ostrogradski approach are:
the Hamiltonian, being linear function of some momenta, is
necessarily unbounded from below (this cannot be cured)
the Ostrogradski Lagrangian leads to incorrect equations of
motion.
there is no straightforward transition from the Ostrogradski
Lagrangian to the Hamiltonian formalism (the Legendre
transformation cannot be performed).
Outline
Motivation
Alternative approach
An alternative approach.
It leads directly to the Lagrangians which, gives correct
equations of motion i.e.
n
X
dk
(−1)
dt k
k=0
k
∂L
∂q (k)
=0
(1)
no additional Lagrange multipliers are necessary
for Lagrangians nonsingular in Ostrogradski sense the
Legendre transformation takes the standard form.
Outline
Motivation
Ostrogradski approach
First, let us recall Ostrogradski formulation. Let L be of the form
..
L = L(q, q̇, q, . . . , q (n) ),
(2)
and we define new variables
q1 = q,
q2 = q̇,
...
, qn = q (n−1) .
(3)
Then L takes the form
L = L(q1 , q2 , . . . , qn , q̇n ),
(4)
and leads to incorrect equations of motion.
Moreover, the canonical momenta provide the following primary
constraints: p1 ≈ 0, p2 ≈ 0, . . . , pn−1 ≈ 0.
Outline
Motivation
One deals with the latter problem by adding Lagrange multipliers
enforcing the proper relation between new coordinates and time
derivatives of the original ones qi = q (i−1)
L −→ L +
n−1
X
λi (q̇i − qi+1 ).
(5)
i=1
The theory becomes constrained (in spite of the fact that the
initial theory may be nonsingular in the Ostrogradski sense),
∂2L
∂2L
=
6= 0.
∂ q̇n2
∂q (n)2
(6)
Outline
Motivation
Then, using Dirac method for constrained systems, we obtain the
Ostrogradski Hamiltonian:
H=
n−1
X
pi qi+1 + pn q̇n − L(q1 , . . . , qn , q̇n ).
(7)
i=1
The canonical equations following from H are equivalent to the
initial Lagrangians ones.
Of course, the above Hamiltonian is not the Legendre
transformation of Ostrogradski Lagrangian.
Outline
Motivation
We have seen that the Ostrogradski approach is based on the idea
that the consecutive time derivatives of the initial coordinate form
new coordinates, qi ∼ q (i−1) .
However, it has been suggested by
H.J. Schmidt, Phys. Rev. D49 (1994), 6345
S. Hawking, T. Hertog, Phys. Rev. D65 (2002), 103515
T.-C. Cheng, P.-M. Ho, M.-C Yeh, Phys. Rev. D65 (2002),
085015
that one can use every second derivative as a new variable,
qi ∼ q (2i−2) .
(8)
We generalize this idea by introducing new coordinates as some
functions of the initial ones and their time derivatives.
Outline
Motivation
Let us start with the Lagrangian with one degree of freedom
depending on time derivatives up to some even order
L = L(q, q̇, . . . , q (2n) ),
and define new variables
qi = q (2i−2) ,
q̇i = q (2i−1) ,
(9)
i = 1, . . . , n + 1,
i = 1, . . . , n.
(10)
Then
L = L(q1 , q̇1 , q2 , q̇2 , . . . , qn , q̇n , qn+1 ).
(11)
Let further F be any function of the following variables
F = F (q1 , q̇1 , q2 , q̇2 , . . . , qn , q̇n , qn+1 , qn+2 , . . . , q2n ).
(12)
obeying
∂F
∂L
+
= 0,
∂qn+1 ∂ q̇n
"
∂2F
det
∂qi ∂ q̇j
#
6= 0,
n 2.
(13)
n+2¬i¬2n
1¬j¬n−1
(for n = 1 only the first condition remains)
Outline
Motivation
Finally, we define a new Lagrangian
L=L+
n X
∂F
k=1
∂qk
2n
X
∂F
∂F
qk+1 +
q̇j .
∂ q̇k
∂qj
j=n+1
q̇k +
(14)
Let us have a look on Lagrange equations. For i = n + 1, . . . , 2n
we have
n
X
∂2F
..
(qk+1 − q k ) = 0,
∂q
∂
q̇
i
k
k=1
i = n + 1, . . . , 2n.
(15)
By assumption about F one gets
..
qk+1 = q k ,
k = 1, . . . , n.
(16)
Outline
Motivation
Furthermore, from Lagrange equations for i = 1, . . . , n we have
∂L
d
−
∂qi
dt
∂L
∂ q̇i
∂F
d2
+
− 2
∂ q̇i−1 dt
∂F
∂ q̇i
= 0,
i = 1, . . . , n. (17)
By combining these equations and the definition of F , one obtains
2n
X
dk
(−1)
dt k
k=0
k
∂L
∂q (k)
= 0.
(18)
So, we arrive at the initial Lagrange equation.
Outline
Motivation
Let us now consider the Hamiltonian formalism.
The Legendre transformation can be immediately performed;
neither additional Lagrange multipliers nor constraints analysis are
necessary. One obtains
pi =
∂F
,
∂qi
i = n + 1, . . . , 2n,
(19)
n
2n
X
∂L X
∂2F
∂F
∂2F
∂2F
pi =
+
q̇k +
qk+1 +
q̇j +
,
∂ q̇i k=1 ∂ q̇i ∂qk
∂ q̇i ∂ q̇k
∂ q̇i ∂qj
∂qi
j=n+1
!
(20)
for i = 1, . . . , n.
By the assumption about F , equations (19) can be solved for
velocities
q̇i = fi (q1 , . . . , q2n , pn+1 , . . . , p2n ),
i = 1, . . . , n.
(21)
Outline
Motivation
Now, eqs. (20) are linear with respect to q̇i for i = n + 1, ..., 2n
and can be easily solved.
Finally, the Hamiltonian is calculated according to the standard
prescription.
H=
2n
X
i=1
pi q̇i − L =
n X
pi −
i=1
∂F
∂qi
q̇i −
n
X
∂F
i=1
∂ q̇i
qi+1 − L,
(22)
where everything is expressed in terms of q’s and p’s.
Outline
Motivation
In order to compare the present formalism with the Ostrogradski
approach let us define new (Ostrogradski) variables
q̃k , p̃k , 1 ¬ k ¬ 2n:
q̃2i−1
q̃2i
p̃2i−1
p̃2i
=
=
=
=
qi ,
fi (q1 , . . . , q2n , pn+1 , . . . , p2n ),
∂F
pi − ∂q
(q1 , f1 (. . .), . . . , qn , fn (. . .), qn+1 , . . . , q2n ),
i
∂F
− ∂fi (q1 , f1 (. . .), . . . , qn , fn (. . .), qn+1 , . . . , q2n ),
(23)
where i = 1, . . . , n and fi = q̇i .
It is easily seen that the above transformation is a canonical one
and the relevant generating function reads
Φ(q1 , . . . , q2n , p̃1 , q̃2 , p̃3 , q̃4 , . . . , p̃2n−1 , q̃2n ) =
n
X
qk p̃2k−1 + F (q1 , q̃2 , q2 , q̃4 , . . . , qn , q̃2n , qn+1 , . . . , q2n ).
(24)
k=1
Outline
Motivation
Let us now consider the case of Lagrangian depending on time
derivatives up to some odd order. Again, we define
qi = q (2i−2) ,
q̇i = q (2i−1) ,
i = 1, . . . , n + 1
i = 1, . . . , n + 1.
(25)
Now, we select a function F ,
F = F (q1 , q̇1 , q2 , q̇2 . . . , qn , q̇n , qn+1 , qn+2 , . . . , q2n+1 ),
(26)
subject to the single condition (let us note that no condition of
the form (13) is here necessary),
"
∂2F
det
∂qi ∂ q̇j
#
6= 0,
(27)
n+2¬i¬2n+1
1¬j¬n
and define the Lagrangian
L=L+
n X
∂F
k=1
2n+1
X ∂F
∂F
q̇k +
qk+1 +
q̇j .
∂qk
∂ q̇k
∂qj
j=n+1
(28)
Outline
Motivation
As before, the above Lagrangian gives the initial Lagrange
equation. Similarly, we can perform the Legendre transformation
obtaining
H=
n+2
X
k=1
pk q̇k − L −
n X
∂F
k=1
∂F
∂F
q̇k +
qk+1 −
q̇n+1 . (29)
∂qk
∂ q̇k
∂qn+1
Also this Hamiltonian is related to the Ostrogradski one by
q̃2i−1 = qi ,
q̃2j = fj (q1 , . . . , q2n+1 , pn+1 , . . . , p2n+1 ),
∂F
(q1 , f1 (. . .), . . . , qn , fn (. . .), qn+1 , . . . , q2n+1 ),
p̃2i−1 = pi − ∂q
i
∂F
p̃2j = − ∂fj (q1 , f1 (. . .), . . . , qn , fn (. . .), qn+1 , . . . , q2n+1 ),
where i = 1, . . . , n + 1, j = 1, . . . , n.The generating function reads
Φ(q1 , . . . , q2n+1 , p̃1 , q̃2 , p̃3 , q̃4 , . . . , p̃2n−1 , q̃2n , p̃2n+1 ) =
=
n+1
X
qk p̃2k−1 + F (q1 , q̃2 , q2 , q̃4 , . . . , qn , q̃2n , qn+1 , . . . , q2n+1 ).(30)
k=1
Outline
Motivation
Example
Let us conclude this part with a simple example. Consider the
Lagrangian
1
ω2 2
.. 2
L = q̇ 2 −
q − gq q
(31)
2
2
and define
..
q1 = q, q2 = q.
(32)
Let F be a function of the form
F (q1 , q̇1 , q2 ) = 2gq1 q̇1 q2 .
(33)
Then new Lagrangian reads
1
ω2 2
q + gq1 q22 + 2g q̇12 q2 + 2gq1 q̇1 q̇2 .
L = q̇12 −
2
2 1
(34)
Outline
Motivation
Example
It is now straightforward to construct the relevant Hamiltonian
H=
p1 p2
(1 + 4gq2 ) 2 ω 2 2
p2 +
−
q − gq1 q22 .
2gq1
2 1
8g 2 q12
(35)
The generating function to the Ostrogradski variables reads
Φ(q1 , q2 , p̃1 , q̃2 ) = q1 p̃1 + 2gq1 q2 q̃2 ,
(36)
and gives
q1 = q̃1
q2 = −
p1 = p̃1 −
p̃2 q̃2
q̃1
p̃2
2g q̃1
p2 = 2g q̃1 q̃2 ,
In terms of new variables we get the Ostrogradski Hamiltonian
H = p̃1 q̃2 −
p̃22
ω2 2
1
− q̃22 +
q̃
4g q̃1 2
2 1
(37)
Outline
Motivation
The case of many degrees of freedom.
First, we will consider Lagrangians depending on first and second
time derivatives. We expect that the counterpart of the condition
(13) should be of the form
∂F
∂L
= 0,
µ +
∂q2
∂ q̇1µ
(38)
but this condition is incorrect because the order in which we take
partial derivatives is irrelevant.
Therefore we must slightly modify our earlier considerations.
Outline
Motivation
Let us start with Lagrangians containing time derivatives up to the
second order
..
L = L(q, q̇, q),
(39)
here q = (q µ ), µ = 1, ..., N.
The nonsingularity condition of Ostrogradski reads
det
!
∂2L
..
..
∂ qµ∂ qν
6= 0.
(40)
We define new coordinates q1 , q̇1 , q2
q µ = q1µ ,
q̇ µ = q̇1µ ,
.. µ
q = χµ (q1 , q̇1 , q2 ),
(41)
where χ are the functions specified below.
Outline
Motivation
First, we select a function, F = F (q1 , q̇1 , q2 ), subject to the single
condition
!
∂2F
6= 0.
(42)
det
∂ q̇1µ ∂q2ν
Now, χµ (q1 , q̇1 , q2 ) are defined as the unique (at least locally due
to (40)) solution to the following set of equations:
∂L(q1 , q̇1 , χ)
∂F (q1 , q̇1 , q2 )
=−
.
µ
∂χ
∂ q̇1µ
(43)
The new Lagrangian is given by
∂F (q1 , q̇1 , q2 ) µ
q̇1
∂q1µ
∂F (q1 , q̇1 , q2 ) µ ∂F (q1 , q̇1 , q2 ) µ
+
q̇2 +
χ (q1 , q̇1 , q2 ). (44)
∂q2µ
∂ q̇1µ
L(q1 , q̇1 , q2 , q̇2 ) = L(q1 , q̇1 , χ(q1 , q̇1 , q2 )) +
Outline
Motivation
The equation of motion for q2µ yields
∂2F
.. ν
(χν − q 1 ) = 0;
∂ q̇1ν ∂q2µ
(45)
.. µ
which, by virtue of (42) implies, q = χµ (q1 , q̇1 , q2 ).
For the remaining variables q1µ one obtains
∂L
d
µ −
∂q1
dt
∂L
∂ q̇1µ
!
d2
+ 2
dt
∂L
∂χµ
= 0,
(46)
which together with the first equation gets the initial Lagrange
equations.
Outline
Motivation
One can also directly pass to the Hamiltonian picture:
p2µ =
p1µ =
∂L
∂F (q1 , q̇1 , q2 )
µ =
∂ q̇2
∂q2µ
(47)
∂2F
∂2F
∂F
∂2F
∂L
ν
ν
q̇
+
χ
+
q̇2ν + µ . (48)
µ
µ +
µ
µ
1
ν
ν
ν
∂ q̇1
∂q1 ∂ q̇1
∂ q̇1 ∂ q̇1
∂ q̇1 ∂q2
∂q1
Using (42) the first set of equations can be solved for q̇1µ and then
the second set is linear in terms of q̇2µ . So, we get q̇2µ .
The Hamiltonian H is computed in standard way:
H = p1µ q̇1µ − L −
∂F µ
∂F µ
χ ,
µ q̇1 −
∂q1
∂ q̇1µ
(49)
where everything is expressed in terms of q1 , q2 , p1 , p2 .
Outline
Motivation
One can check, by direct calculation, that the canonical equations
following from H are equivalent to the initial Lagrangians ones.
There exists canonical transformation which relates our
Hamiltonian to the Ostrogradski one. It reads
q̃1µ = q1µ ,
q̃2µ = f µ (q1 , q2 , p2 ),
∂F
p̃1µ = p1µ − ∂q
µ (q1 , f , q2 ),
(50)
1
∂F
p̃2µ = − ∂f
µ (q1 , f , q2 ),
where tilde refers to Ostrogradski variables and f µ = q̇1µ .
Outline
Motivation
Let us consider a nonsingular Lagrangian of the form
.. ...
L = L(q, q̇, q, q).
(51)
It is slightly surprising that this case (and, in general, the case
when the highest time derivatives are of odd order) is simpler.
We define the new variables q1 , q̇1 , q2 , q̇2
q µ = q1µ ,
q̇ µ = q̇1µ ,
.. µ
q = q2µ ,
...µ
q = q̇2µ .
(52)
Outline
Motivation
Let F be a function such that
F = F (q1 , q̇1 , q2 , q3 ),
det
∂2F
∂ q̇1µ ∂q3ν
!
6= 0.
(53)
The modified Lagrangian reads
L=L+
∂F µ
∂F µ
∂F µ
∂F µ
q ,
µ q̇1 +
µ q̇2 +
µ q̇3 +
∂q1
∂q2
∂q3
∂ q̇1µ 2
(54)
and it gives Lagrange equations for the initial variables for the
original variable
Outline
Motivation
As in the second-order case, the Legendre transformation can be
directly performed and Hamiltonian is of the form
H = p1µ q̇1µ + p2µ q̇2µ − L −
∂F µ
∂F µ
∂F µ
q̇ .
µ q̇1 −
µ q2 −
∂q1
∂ q̇1
∂q2µ 2
(55)
The canonical transformation which relates our formalism to the
Ostrogradski is of the form
q̃1µ = q1µ , q̃3µ = q2µ ,
q̃2µ = f µ (q1 , q2 , q3 , p3 ),
∂F
p̃1µ = p1µ − ∂q
µ (q1 , f (q1 , q2 , q3 , p3 ), q2 , q3 ),
1
p̃3µ = p2µ −
(56)
∂F
(q , f (q1 , q2 , q3 , p3 ), q2 , q3 ),
∂q2µ 1
∂F
p̃2µ = − ∂f
µ (q1 , f (q1 , q2 , q3 , p3 ), q2 , q3 ),
where f µ = q̇1µ .
Outline
Motivation
Presented approach is also applicable to singular Lagrangians
and the results agree with the conclusions for Ostrogradski
formalism
and possess generalization to the case of field theory.
More detailed discussion and references:
”Modified Hamiltonian formalism for higher-derivative theories”,
Physical Review D 82 2010, 045008
For example if we consider formulation of f (R) gravity with the
metric of the form
ds 2 = −N 2 dt 2 + a2 d~x 2 .
(57)
Outline
Motivation
Then the curvature of the metric reads
R=6
ȧ
NA
.
+ 12
ȧ
Na
2
(58)
and the Lagrangian of f (R) gravity takes the form
1
L(a, N) = Na3 f (R).
2
(59)
In our case the curvature R can be one of the basic variables
a1 = a, ȧ1 = ȧ, N1 = N, Ṅ1 = Ṅ, a2 = R.
(60)
Outline
Motivation
Thank you for your attention

Modified Hamiltonian formalism for higher

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