Thesis defence

Introduction
Geometric theory of Lie systems
The theory of Quasi-Lie schemes
Conclusions

Lie systems and applications to Quantum
Mechanics

Javier de Lucas Araujo

November 29, 2009

Javier de Lucas Araujo Lie systems and applications to Quantum Mechanics

Introduction
History of Lie systems
Conclusions

Lie’s work

About one century ago, Lie investigated those linear homogeneous
systems of ordinary diﬀerential equations
n
dx j
= ajk (t)x k , j = 1, . . . , n,
dt
k=1

admitting their general solution to be written as
n
j
x j (t) = λk x(k) (t), j = 1, . . . , n,
k=1

with {x(1) (t), . . . , x(n) (t)} being a family of linear independent
particular solutions and {λ1 , . . . , λn } a set of real constants.

Introduction
Conclusions

Lie noticed that any non-linear change of variables x → y
transforms the previous linear system into a non-linear one of the
form
dy j
= X j (t, y ), j = 1, . . . , n, (1)
dt
whose solution could be expressed non-linearly as

y j (t) = F j (y(1) (t), . . . , y(n) (t), λ1 , . . . , λn ), (2)

with {y(1) (t), . . . , y(n) (t)} being a family of certain particular
solutions for (1). He called the expressions of the above kind
superposition rules.


Introduction
Conclusions

Lie characterized those non-autonomous systems of ﬁrst-order
diﬀerential equations admitting the general solution to be
written in terms of certain families of particular solutions and
a set of constants.
In his honour, these systems are called nowadays Lie systems
and the expressions of the form (2) are still called
superposition rules.
Many work have been done since then and many applications
and developments for this theory have been done by many
authors.


Fundamentals
Introduction Superpositions and connections
Geometric theory of Lie systems Lie systems and Lie groups
The theory of Quasi-Lie schemes Integrability of Riccati equations
Conclusions SODEs
Quantum Mechanics

Non-autonomous systems and vector ﬁelds

In the modern geometrical lenguage, we describe any
non-autonomous system on Rn , written in local coordinates

dx j
= X j (t, x), j = 1, . . . , n, (3)
dt
by means of the t-dependent vector ﬁeld
n
∂
X (t, x) = X j (t, x) , (4)
∂x j
j=1

whose integral curves are those given by the above system.


Fundamentals
Conclusions SODEs
Quantum Mechanics

Definition
The system (3) is a Lie system if and only if the t-dependent
vector field (4) can be written
r
X (t) = bα (t)Xα , (5)
α=1

where the vector fields Xα close on a finite-dimensional Lie algebra
of vector fields V , i.e. there exist r 3 real constants cαβγ such that

[Xα , Xβ ] = cαβγ Xγ , α, β = 1, . . . , r .

The Lie algebra V is called a Vessiot-Guldberg Lie algebra of
vector fields of the Lie system (5).


Examples of Lie systems
Linear homogeneous and inhomogeneous systems of
differential equations.
Riccati equations and Matrix Riccati equations.
Other non-autonomous systems related to famous
higher-order differential equations:
1 Time-dependent harmonic oscillators.
2 Milne-Pinney equations.
3 Ermakov systems.
4 Time-dependent Lienard equations.

The study of the previous systems related to higher-order
differential equations is developed in this work by the first time
and, furthermore, many applications to Physics have been found.

Fundamentals
Conclusions SODEs
Quantum Mechanics

Deﬁnition
The system (3) is said to admit a superposition rule, if there
exists a map Φ : U ⊂ Rn(m+1) → Rn such that given a generic
family of particular solutions {x(1) (t), . . . , x(m) (t)} and a set of
constants {k1 , . . . , kn } , its general solution x(t) can be written as

x(t) = Φ(x(1) (t), . . . , x(m) (t), k1 , . . . , kn ).

Lie’s theorem
Lie proved that any Lie system (5) on Rn admits a superposition
rule in terms of m generic particular solutions with r ≤ mn.


Fundamentals
Conclusions SODEs
Quantum Mechanics

Superposition Rule for Riccati equations
We asserted that the Riccati equation is a Lie system and therefore
it admits a superposition rule. This is given by

x1 (x3 − x2 ) − kx2 (x3 − x1 )
x= .
(x3 − x2 ) − k(x3 − x1 )

Taken three diﬀerent solutions of the Ricatti equation x(1) (t),
x(2) (t), x(3) (t) and a constant k ∈ R ∪ {∞} this expression allows
us to get the general solution.


Given a Lie system (5), it admits a superposition rule

x = Φ(x(1) , . . . , x(m) , k1 , . . . , kn ).

where the map Φ can be inverted on the last n variables to give
rise to a new map

(k1 , . . . , kn ) = Ψ(x, x(1) , . . . , x(m) ),

Differentiating, we get
m n
j ∂Ψj
X (t, x )Ψ ≡
˜ X i (t, x(β) (t)) i
= 0, j = 1, . . . , n.
β=0 i=1
∂x(β)

Hence the functions {Ψj | j = 1, . . . , n} are first-integrals for the
distribution D spanned by the vector fields X (t, x ). Note that for
˜
the sake of simplicity we call x(0) the variable x.

Fundamentals
Conclusions SODEs
Quantum Mechanics

Definition
n i
Given a t-dependent vector field X (t, x(0) ) = i=1 X (t, x(0) )∂x(0) ,
i

on Rn , we call prolongation X of X to the manifold Rn(m+1) to the
vector field
m n
∂Ψj
X (t, x ) ≡
˜ X i (t, x(β) (t)) i
.
β=0 i=1
∂x(β)

Under certain conditions, n first-integrals for the prolongated
t-dependent vector fields allow us to get the superposition rule.


Fundamentals
Conclusions SODEs
Quantum Mechanics

Lie systems and equations in Lie groups

Lemma
Given any Lie algebra V of complete vector fields on Rn and a
connected Lie group G with Te G V , there exists an effective, up
to a discrete set of points, action ΦV ,G : G × Rn → Rn such that
their fundamental vector fields are those in V .
This facts implies important consequences in the theory of Lie
systems. As we next show, it provides a way to reduce the study of
Lie systems to a particular family of them.


Fundamentals
Conclusions SODEs
Quantum Mechanics

Proposition
Given the Lie system X (t) = n bα (t)Xα related to a Lie
α=1
Vessiot Gulberg Lie algebra V , we can associate it with the
equation
n
R
g =−
˙ bα (t)Xα (g ), g (0) = e,
α=1

in a connected Lie group G with Te G V and with the Xα R

right-invariant vector ﬁelds clossing on the same commutation
relations as the Xα . Moreover, the corresponding action
ΦV ,G : G × Rn → Rn determines that the general solution for X is

x(t) = ΦV ,G (g (t), x0 ).


Fundamentals
Conclusions SODEs
Quantum Mechanics

Integrability of Riccati equations

Fixing the basis {∂x , x∂x , x 2 ∂x } of vector fields on R for a
Vessiot–Gulberg Lie algebra V for Riccati equations, each Riccati
equation can be considered as a curve (b0 (t), b1 (t), b2 (t)) in R3 .
¯
A curve A of the group G of smooth curves in G = SL(2, R)
transforms every curve x(t) in R into a new curve x (t) in R
¯
given by x (t) = ΦV ,G (A(t), x(t)).
Correspondingly, the t-dependent change of variables
¯
x (t) = ΦV ,G (A(t), x(t)) transforms a Riccati equation with
coefficients b0 , b1 , b2 into a new Riccati equation with new
t-dependent coefficients, b0 , b1 , b2 .


Fundamentals
Conclusions SODEs
Quantum Mechanics

Theorem
The group G of curves in a Lie group G associated with a Lie
system, here SL(2, R), acts on the set of these Lie systems, here
Riccati equations.

The group G also acts on the left on the set of curves in
¯
SL(2, R) by left translations, i.e. a curve A(t) transforms the
¯
curve A(t) into a new one A (t) = A(t)A(t).
If A(t) is a solution of the equation in SL(2, R) related to a
Riccati equation, then the new curve A (t) satisﬁes a new
equation in SL(2, R) but with a diﬀerent right hand side a (t)
associated with a new Riccati equation.


Fundamentals
Conclusions SODEs
Quantum Mechanics

Lie group point of view
The relation between both curves in sl(2, R) is

¯ ¯ ˙
a (t) = A(t)a(t)A−1 (t) + A(t)A−1 (t).
¯ ¯

If the right hand is in a solvable subalgebra of sl(2, R) the system
can be integrated.

J.F. Cari˜ena, J. de Lucas and A. Ramos, A geometric approach to
n
integrability conditions for Riccati equations, Electr. J. Diﬀ. Equ.
122, 1–14 (2007).
˙
¯
If not, putting apart A(t), the previous equation becomes a Lie
system related to a Vessiot–Gulberg Lie algebra isomorphic to
sl(2, R) ⊕ sl(2, R).

Fundamentals
Conclusions SODEs
Quantum Mechanics

Some restrictions on the solutions of this system make it to
become a solvable Lie system.
On one hand, this restriction make the system not having
always solution and some integrability conditions arise.
On the other hand, this system allows to characterize many
integrable families of Riccati equations.

J.F. Cari˜ena and J. de Lucas, Lie systems and integrability
n
conditions of diﬀerential equations and some of its applications,
Proceedings of the 10th international conference on diﬀerential
geometry and its applications, World Science Publishing, Prague,
(2008).


Fundamentals
Conclusions SODEs
Quantum Mechanics

SODE Lie systems
Deﬁnition
Any system x i = F i (t, x, x), with i = 1, . . . , n, can be studied
¨ ˙
through the system corresponding for the t-dependent vector field
n
X (t) = (v i ∂x i + F i (t, x, v )∂v i ).
i=1

We call SODE Lie systems those SODE for which X is a Lie
system.

J.F Cari˜ena, J. de Lucas and M.F. Ra˜ada. Recent Applications
n n
of the Theory of Lie Systems in Ermakov Systems, SIGMA 4, 031
(2008).

Fundamentals
Conclusions SODEs
Quantum Mechanics

Generalization of the theory of SODEs
Even if here we are dealing with second-order differential
equations, this procedure can be applied to any higher-order
differential equation.
The study of superposition rules for second-order differential
equations is new, only it is cited once by Winternitz.
There exist a big number of systems of higher “Lie systems”.

Example of SODEs
Time-dependent frequency harmonic oscillators.
Milne–Pinney equations.
Ermakov systems.


Fundamentals
Conclusions SODEs
Quantum Mechanics

Harmonic Oscillators and Milne–Pinney equation

The Milne–Pinney equation with t-dependent frequency is
x = −ω 2 (t)x + kx −3 can be associated with the following system
¨
of first-order differential equations

 x = v,
˙
 v = −ω 2 (t)x + k .
˙
x3
This system describes the integral curves for the t-dependent
vector field
∂ k ∂
X (t) = v + −ω 2 (t)x + 3 .
∂x x ∂v


Fundamentals
Conclusions SODEs
Quantum Mechanics

This is a Lie system because X can be written as
X (t) = L2 − ω 2 (t)L1 , where the vector fields L1 and L2 are given
by

∂ ∂ k ∂ 1 ∂ ∂
L1 = x , L2 = v + 3 , L3 = x −v ,
∂v ∂x x ∂v 2 ∂x ∂v

which are such that

[L1 , L2 ] = 2L3 , [L3 , L2 ] = −L2 , [L3 , L1 ] = L1

Therefore, the Milne–Pinney equation is a Lie system related to a
Vessiot-Gulberg Lie algebra isomorphic to sl(2, R). If k = 0 we
obtain the same result for t-dependent frequency harmonic
oscillators.


Fundamentals
Conclusions SODEs
Quantum Mechanics

As the Milne–Pinney equation is a Lie system, the theory of Lie
systems can be applied and, for example, a new superposition rule
can be obtained:
1/2
2 2 4 4 2 2
x =− λ1 x1 + λ2 x2 ±2 λ12 (−k(x1 + x2 ) + I3 x1 x2 ) . (6)

J.F. Cari˜ena and J. de Lucas, A nonlinear superposition rule for
n
the Milne–Pinney equation, Phys. Lett. A 372, 5385–5389 (2008).


Fundamentals
Conclusions SODEs
Quantum Mechanics

Also, if we apply the theory of integrability to the Milne–Pinney
equation or the harmonic oscillator, we can many integrable
particular cases, as the Caldirola-Kanai Oscillator. Moreover, many
new results and integrable cases can be obtained.

J.F. Cari˜ena and J. de Lucas, Integrability of Lie systems and
n
some of their applications in Physics, J. Phys. A 41, 304029
(2008).
J.F. Cari˜ena, J. de Lucas and M.F. Ra˜ada, Lie systems and
n n
integrability conditions for t-dependent frequency harmonic
oscillators, to appear in the Int. J. Geom. Methods in Mod. Phys.


Fundamentals
Conclusions SODEs
Quantum Mechanics

Mixed superposition rules

Definition
We say that a mixed superposition rule is an expression allowing
us to obtain the general solution of a system in terms of a family
of particular solutions of other differential equations.

Theorem
Lie systems related to t-dependent vector fields of the form
X (t) = r bα (t)Xα with vector fields Xα closing on the same
α=1
commutation relations, i.e. [Xα , Xβ ] = cαβγ Xγ , cαβγ ∈ R, admit a
mixed superposition rule.


Fundamentals
Conclusions SODEs
Quantum Mechanics


The Milne–Pinney equation and the harmonic oscillator
The Milne–Pinney equation and the harmonic oscillator in
ﬁrst-order

 x = v,
˙ x = v,
˙
k
2
 v = −ω (t)x + ,
˙ v = −ω 2 (t),
˙
x3
can be related to the vector ﬁelds X (t) = L2 − ω 2 (t)L1 for
arbitrary k and k = 0.


Moreover, the chosen basis of fundamental vector ﬁelds closes on
the same commutation relations. Therefore, there exists a mixed
superposition rule relating both of them. It can be seen to be
given by
√
2 1/2
x= I2 y 2 + I1 z 2 ± 4I1 I2 − kW 2 yz .
|W |

Furthermore, a certain Riccati equation satisﬁes the same
conditions, therefore, it can be obtained the new mixed
superposition rule

(C1 (x1 − x2 ) − C2 (x1 − x3 ))2 + k(x2 − x3 )2
x= .
(C2 − C1 )(x2 − x3 )(x2 − x1 )(x1 − x3 )

Fundamentals
Conclusions SODEs
Quantum Mechanics

Quantum Lie systems

Consider the below facts
Any (maybe infinite-dimension) Hilbert space H can be
considered as a real manifold.
As there exists a isomorphism Tφ H H we get that
T H H ⊕ H.
Each operator T on a Hilbert space H can be used to define a
map X A : ψ ∈ H → (ψ, Aψ) ∈ T H
Any t-dependent operator A(t) can be related at any time t
as a vector field X A(t) .
The X A(t) is a t-dependent vector field on a maybe infinite
dimensional manifold H.


Fundamentals
Conclusions SODEs
Quantum Mechanics

Geometric Schrodinger equation

Given a t-dependent hermitian Hamiltonian H(t), the associated
Schr¨dinger equation ∂t Ψ = −iH(t)Ψ describes the integral curves
o
for the t-dependent vector ﬁeld X −iH(t) .

Theorem
Given two skew self-adjoint operators A, B associated with two
vector ﬁelds X A and X B , we have that

[X A , X B ] = −X [A,B] .


Fundamentals
Conclusions SODEs
Quantum Mechanics

Deﬁnition
We say that a t-dependent hermitian Hamiltonian H(t) is a
Quantum Lie system if we can write
r
H(t) = bα (t)Hα ,
α=1

with the Hα hermitian operators such that the iHα satisfy that
there exist r 3 real constants such that
r
[iHα , iHβ ] = cαβγ iHγ , α, β = 1, . . . , r .
γ=1


Fundamentals
Conclusions SODEs
Quantum Mechanics

Note that if H(t) is a Quantum Lie system, we can write
r
X (t) ≡ X −iH(t) = bα (t)Xα ,
α=1

where Xα = X −iHα , and
r
[Xα , Xβ ] = cαβγ Xγ , α, β = 1, . . . , r .
γ=1

Schr¨dinger equations are the equations of the integral curves for a
o
t-dependent vector ﬁeld satisfying the analogous relation to Lie
systems but in a maybe inﬁnte-dimensional manifold H.


Flows of “operational” vector fields
Note that the vector fields X A admit the flows

Fl : (t, ψ) ∈ R × H → Flt (ψ) = exp(tA)(ψ) ∈ H.

Action of a quantum Lie system
In case that H(t) is a quantum Lie system, there exist certain
vector fields {Xα | α = 1, . . . , r } closing on a finite-dimensional Lie
algebra V and we can define an action ΦV ,G : G × H → H, with
Te G V , satisfying that for a basis of Te G given by
{aα | α = 1, . . . , r } closing on the same commutation relations
than the Xα , we get

d
ΦV ,G (exp(−taα ), ψ) = Xα (ψ),
dt t=0

that is, the Xα are the fundamental vector fields related to the aα .

Fundamentals
Conclusions SODEs
Quantum Mechanics

Reduction to equations in Lie groups

In case that H(t) is a Quantum Lie system, the related action
ΦG ,V : G × H → H allows to relate the Schr¨dinger equation
o
related to H(t) with an equation
n
R
g =−
˙ bα (t)Xα (g ), g (0) = e,
α=1

in a connected Lie group G with Te G V and with
R
Xα (g ) = Rg ∗e aα and aα a basis of Te G . Moreover, the general
solution for X is
Ψt = ΦV ,G (g (t), Ψ0 ).


Fundamentals
Conclusions SODEs
Quantum Mechanics

Spin Hamiltonians
Consider the t-dependent Hamiltonian

H(t) = Bx (t)Sx + By (t)Sy + Bz (t)Sz ,

with Sx , Sy and Sz being the spin operators. Note that the
t-dependent Hamiltonian H(t) is a quantum Lie system and its
Schr¨dinger equation is
o
dψ
= −iBx (t)Sx (ψ) − iBy (t)Sy (ψ) − iBz (t)Sz (ψ),
dt


Fundamentals
Conclusions SODEs
Quantum Mechanics

that can be seen as the diﬀerential equation determinating the
integral curves for the t-dependent vector ﬁeld

X (t) = Bx (t)X1 + By (t)X2 + Bz (t)X3 ,

where (X1 )ψ = −iSx (ψ), (X2 )ψ = −iSy (ψ), (X3 )ψ = −iSz (ψ).
Therefore our Schrodinger equation is a Lie system related to a
quantum Vessiot-Guldberg Lie algebra isomorphic to su(2).

Another example of Quantum Lie systems is given by the family of
time dependent Hamiltonians

H(t) = a(t)P 2 + b(t)Q 2 + c(t)(QP + PQ) + d(t)Q + e(t)P + f (t)I .


Fundamentals
Conclusions SODEs
Quantum Mechanics

The theory of Lie systems allow us to investigate all these systems
to obtain exact solutions, see

J.F. Cari˜ena, J. de Lucas and A. Ramos, A geometric approach to
n
time evolution operators of Lie quantum systems. Int. J. Theor.
Phys. 48 1379–1404 (2009).

or analyse integrability conditions in Quantum Mechanics, see

J.F. Cari˜ena and J. de Lucas, Integrability of Quantum Lie
n
systems. accepted for publication in the Int. J. Geom. Methods
Mod. Phys.


Introduction
Properties of Quasi-Lie schemes
Lie families
Dissipative Milne-Pinney equations
Conclusions

Definition of quasi-Lie scheme

Definition
Let W , V be non-null finite-dimensional real vector spaces of
vector fields on a manifold N. We say that they form a quasi-Lie
scheme S(W , V ), if the following conditions hold:
1 W is a vector subspace of V .
2 W is a Lie algebra of vector fields, i.e. [W , W ] ⊂ W .
3 W normalises V , i.e. [W , V ] ⊂ V .


Introduction
Lie families
Conclusions

Quasi-Lie schemes are tools to deal with non-autonomous systems
of first-order differential equations. Let us describe those systems
that can be described by means of a scheme.
Definition
We say that a t-dependent vector field X is in a quasi-Lie scheme
S(W , V ), and write X ∈ S(W , V ), if X belongs to V on its
domain, i.e. Xt ∈ V|NtX .

Lie systems and Quasi-Lie schemes
Given a Lie system related to a Vessiot-Guldber Lie algebra of
vector fields V , then S(V , V ) is a quasi-Lie scheme. Hence, Lie
systems can be studied through quasi-Lie schemes.


Introduction
Lie families
Conclusions

Abel equations

Abel equations
Abel differential equations of the first kind are of the form

x = f3 (t)x 3 + f2 (t)x 2 + f1 (t)x + f0 (t).
˙

Consider the linear space of vector fields V spanned by the basis
∂ ∂ ∂ ∂
X0 = , X1 = x , X2 = x 2 , X3 = x 3 ,
∂x ∂x ∂x ∂x
and define the Lie algebra W ⊂ V as

W = X0 , X1 .


Introduction
Lie families
Conclusions

Moreover, as

[X0 , X2 ] = 2X1 , [X0 , X3 ] = 3X2 ,
[X1 , X2 ] = X2 , [X1 , X3 ] = 2X3 ,

then [W , V ] ⊂ V and S(W , V ) is a scheme. Finally, the Abel
equation can be described through this scheme because
3
X (t, x) = fα (t)Xα (x),
α=0

and thus X ∈ S(W , V ).


Introduction
Lie families
Conclusions

Symmetries of a scheme

Definition
We call symmetry of a scheme S(W , V ) to any t-dependent
change of variables transforming any X ∈ S(W , V ) into a new
X ∈ S(W , V ).

We have charecterized different kinds of infinite-dimensional
groups of transformations of a scheme:
The group of the scheme, G(W ).
The extended group of the scheme, Ext(W ).


Theorem
Given a scheme S(W , V ) and an element g ∈ Ext(W ), then for
any X ∈ S(W , V ) we get that g X ∈ S(W , V ).

Deﬁnition
Given a scheme S(W , V ) and a X ∈ S(W , V ), we say that X is a
quasi-Lie system with respect to this scheme if there exists a
symmetry of the scheme such that g X is a Lie system.

Theorem
Every Quasi-Lie system admits a t-dependent superposition rule.

J.F. Cari˜ena, J. Grabowski and J. de Lucas, Quasi-Lie systems:
n
theory and applications, J. Phys. A 42, 335206 (2009).

Introduction
Lie families
Conclusions

We can characterise those non-autonomous systems of ﬁrst-order
diﬀerential equations admitting a t-dependent superposition rule.
Such systems are called Lie families.
Theorem
A system (1) admits a t-dependent superposition rule if and only if
n
¯
X = ¯
bα (t)Xα ,
α=1

where the Xα satisfy that there exist n3 functions fαβγ (t) such that
n
¯ ¯
[Xα , Xβ ] = ¯
fαβγ (t)Xγ , α, β = 1, . . . , n.
γ=1


Introduction
Lie families
Conclusions

The applications of quasi-Lie systems and Lie families are very
broad, we can just say some of them
Dissipative Milne–Pinney equations
Emden equations
Non-linear oscillators
Mathews–Lakshmanan oscillators
Lotka Volterra systems
Etc.
Moreover, they can be applied to Quantum Mechanics and PDEs
also.


Introduction
Lie families
Conclusions

Dissipative Ermakov systems are
1
x = a(t)x + b(t)x + c(t)
¨ ˙ .
x3
As a first-order differential equation it can be written

x
˙ = v,
v
˙ = a(t)v + b(t)x + c(t) x13 .

Consider the space V spanned by the vector fields:
∂ ∂ 1 ∂ ∂ ∂
X1 = v , X2 = x , X3 = 3 , X4 = v , X5 = x .
∂v ∂v x ∂v ∂x ∂x


Introduction
Lie families
Conclusions

Take the three-dimensional Lie algebra W ⊂ V generated by the
vector ﬁelds
∂ ∂ ∂
Y1 = X1 = v , Y2 = X2 = x , Y3 = X5 = x .
∂v ∂v ∂x
What is more, as

[Y1 , X3 ] = −X3 , [Y1 , X4 ] = X4 ,
[Y2 , X3 ] = 0, [Y2 , X4 ] = X5 − X1 ,
[Y3 , X3 ] = 0, [Y3 , X4 ] = −3X3 .

Therefore [W , V ] ⊂ V and S(W , V ) is a quasi-Lie scheme.


Introduction
Lie families
Conclusions

Moreover, the above system describes the integral curves for the
t-dependent vector ﬁeld

X (t) = a(t)X1 + b(t)X2 + c(t)X3 + X4 ,

Therefore X ∈ S(W , V ). The corresponding set of t-dependent
diﬀeomorphisms of TR related to elements of Ext(W ) reads

x = γ(t)x ,
, α(t) = 0, γ(t) = 0.
v = α(t)v + β(t)x ,


Introduction
Lie families
Conclusions

The transformed system of diﬀerential equations obtained through
our scheme is
 dx
 dt = β(t)x + α(t)v ,


dv b(t) β(t) β 2 (t) ˙
β(t)
dt = α(t) + a(t) α(t) − α(t) − α(t) x
a(t) − β(t) − α(t) v + c(t) 13 .
˙

+


α(t) α(t) x

Note that if β(t) = 0 this system is associated with

d 2x dx 1
= a(t) + b(t) x + c(t) 3 .
dt 2 dt x


Introduction
Lie families
Conclusions

Related Lie systems

Thus, the resulting transformation is determined by α(t) = c(t) k
and β(t) = 0, for a certain constant k. As a consequence, we
obtain a t-dependent superposition rule
√
2 1/2
x(t) = ¯2 ¯2
I2 x1 (t) + I1 x2 (t) ± 4I1 I2 − kW 2 x1 (t)¯2 (t)
¯ x ,
W
with x1 , x2 solutions for the diﬀerential equation
¯ ¯

x = a(t)x + b(t)x.
¨ ˙


Introduction
Conclusions

Advances in Lie systems

Let us schematize some developments obtained during this work:

Usual Lie systems Lie systems in this work

First-order Higher-order
Finite-dimensional Infinite-dimensional
manifolds manifolds (QM)
Ordinary differential Partial differential
equations equations


Introduction
Conclusions

Other advances

1 The theory of integrability of Lie systems have been developed. It has been shown that Matrix Riccati
equations can be used to analyse the integrability of Lie systems.
2 The theory of Quantum Lie systems has been analysed. Many integrable cases have been understood
geometrically and new ones have been provided. These methods allow us to get integrable models to
analyse Quantum Systems.
3 The theory of Lie systems have been applied to analyse higher order diﬀerential equations. The theory of
integrability has been aplied here and many results have been obtained and explained. We have found
many applications to Physics of many Lie systems
4 The theory of Quasi-Lie schemes has been started. The number of applications of this theory is very broad
and the number of systems analysed with it is still very small.


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