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{\nopagenumbers
\titlea{On Lie point symmetries in mechanics}{}
\vfill
\centerline{\it G. Cicogna, Dipartimento di Fisica dell'Universita', I-56126
Pisa, Italy}
\medskip
\centerline{\it G. Gaeta\footnote{$ ^*$}{{\petit present address: Mathematical
Institute RUU, P.O.Box 80.010, 3508 TA Utrecht (Netherlands)}}, C.Ph.Th., Ecole
Polytechnique, F-91128 Palaiseau, France}
\vfill
{\bf Summary}:
{\petit We present some remarks on the existence and the properties of Lie
point
symmetries of finite dimensional dynamical systems expressed either in
Newton-Lagrange or in Hamilton form. We show that the only Lie symmetries
admitted by Newton-Lagrange-type problems are essentially linear symmetries,
and
construct the most general problem admitting such a symmetry. In the case of
Hamilton problems, we discuss the differences and the relationships between the
existence of time-independent Lie point symmetries and the invariance
properties
of the Hamiltonian under coordinate transformations.}
\vfill\eject }
\pageno=1
\titleb{Introduction}
In this paper we want to present some considerations on the \sy properties
of the dynamical problems which are expressed as systems either of
second-order ordinary differential \eqs in Newton-Lagrange form, or of
first-order \eqs in Hamilton form.
According to the old idea due to S. Lie, and recently reconsidered by various
authors (see e.g. [1-8] and references therein, see also e.g. [9-11] for more
specific applications to dynamical problems), one may introduce the very
general
notion of {\it Lie point \sy} of a dynamical system, which is any
transformation
of the dynamical variables $t\to t'=t'(q,t),\ q\to q'=q'(q,t)$, where
$t'(q,t),\
q'(q,t)$ are smooth (e.g. $C^\infty$) functions on the whole domain of
definition \big[respectively $t\to t'=t'(q,p,t),\ q\to q'=q'(q,p,t),\ p\to
p'=p'(q,p,t)$ in the case of Hamiltonian formulation\big], with the property of
transforming into itself the set of the solutions of the given problem (i.e.:
if
$q(t)$ is a solution, then $q'(t')$ is too: for a more complete definition of
Lie symmetry see the above mentioned references). This transformation is
assumed
to depend smoothly on a real parameter $\ep$, and its infinitesimal generator
will be denoted by $\g$.
The interest in the notion of Lie point symmetry, apart from its intrinsic
abstract -~algebraic and geometrical - features, is motivated by some
successful
applications in the analysis of several concrete
problems: we refer to the already mentioned references for a list of these
applications, which cover, especially, time-evolution problems (both in the
form
of ordinary and of partial differential equations), bifurcation theory, fluid
dynamics. Other applications deal with Lagrangian formulation of mechanics, and
the connections between Lie point \ss and Noether theorem in its most general
form [3].
In this paper, we shall avoid as far as possible any
abstract approach, and we concentrate our attention on the existence and the
properties of the Lie point \ss admitted by problems in Newton-Lagrange and in
Hamilton mechanics. We are mainly interested in autonomous systems, and in
Lie point \ss which are time-independent (LPTI), i.e. in the "purely
geometrical
"
symmetries. However, to obtain more generality and to allow some useful
comparisons, we shall occasionally consider also nonautonomous systems and/or
possibly time dependent Lie symmetries. We shall carefully state our
assumptions
case by case. In any case, we do not consider transformations which change the
time variable $t$; from now on, when speaking about Lie point symmetries, we
shall always refer to transformations of this type (i.e. leaving $t$ fixed).
In sect.1 we discuss the Lie \ss of Newton-Lagrange-type problems; we shall
show - in particular - that second-order systems of the form $\"q=F(q,t)$
\big(or even, more in general, $\"q = F(\.q,q,t)$, with $F$ depending
linearly on the "velocities" $\.q$\big) either do not admit LPTI \ss (which is
the most common case), or admit merely "linear" symmetries. Some interesting
remarks are suggested by the construction of the most general system having
such
a symmetry.
After some short considerations on LPTI \ss of first-order differential
systems (sect. 2), we shall consider in sect. 3 the dynamical problems
expressed in Hamiltonian form: here, a richer geometrical structure is
obviously present, and more symmetries as well - as a natural consequence of
the fact that transformations of both variables $q$ and $p$ are allowed; these
would correspond, in the Lagrangian formalism, to (first order) contact
transformations [3,5,8,12] rather than purely Lie point symmetries (enlarging
th
e
class of allowed \ss is actually one of the main advantages of Hamiltonian
formulation of mechanics). Various, quite different, notions of symmetry for
Hamiltonian problems can be introduced, related to the invariance of the
Hamiltonian under coordinate transformations, or the existence of constants of
motion. We shall examine these notions and discuss their relationships with
the
notion of LPTI symmetry.
\titleb{1. Newton and Newton-Lagrange problems}
Let us start considering systems of second-order differential \eqs of
the general "Newton form"
$$\"q_i=F_i(q,\.q,t) \qquad \quad (i=1,\ldots,n)\eqno(1.1)$$
for the $n$ (generalized) Lagrangian coordinates $q_i=q_i(t)$, where the
"forces
"
$F_i$ are assumed to be smooth enough (for our purposes, a $C^r$
differentiability with $r\ge 3$ would be sufficient). Let us look for the Lie
point symmetries admitted by this problem: writing the Lie generator $\g$ of
this \sy (possibly depending on time) in the form (sum over repeated indices)
$$\g =\th_i(q,t){\pd\over{\pd q_i}} \eqno(1.2)$$ the determining equations for
the functions $\th_i(q,t)$ are, as well known (see, e.g. [5,6]),
$$ \th_j {\pd F_i\over{\pd q_j}}+\Big({\pd \th_j\over{\pd t}}+\.q_k{\pd
\th_j\over{\pd q_k}}\Big){\pd F_i\over{\pd \.q_j}}-F_j{\pd \th_i\over{\pd
q_j}}-\.q_j\.q_k{\pd^2\th_i\over{\pd q_j\pd q_k}}-2\.q_j{\pd^2\th_i\over{\pd t
\pd q_j}}-{\pd^2\th_i\over{\pd t^2}} = 0 \eqno(1.3)$$
Observing that these are identities on $t, q, \.q$, and
that $\th_i$ do not depend on $\.q$, these \eqs can be splitted into
several conditions, once the explicit form of $F_i$ is given.
Assume now that the forces $F_i$ in (1.1)
depend at most linearly on the "velocities" $\.q_i$, i.e.
$$ {\pd^2F_i\over{\pd \.q_j\pd\.q_k}} = 0\eqno (1.4)$$
We stress that this is also the case if (1.1) are obtained as Euler-Lagrange
\eqs
from a Lagrangian function ${\cal L}={\cal L}(q,\.q,t)$: in this case,
indeed, ${\cal L}$ must be of the form
$${\cal L}={1\over 2}\.q_i\.q_i - {\cal V}(q,t)+{\cal A}_i(q,t)\ \.q_i$$
where ${\cal V}$ and ${\cal A}_i$ are respectively the scalar and the vector
potentials. This form of ${\cal L}$ implies (1.4) and actually covers many
physically relevant problems (e.g. electromagnetism and
conservative mechanics; notice that (1.4) covers linear dissipation as well).
>From (1.3), one gets using (1.4), $${\pd^2\th_i\over{\pd q_j \pd q_k}} = 0
\eqno(1.5)$$ which implies immediately that the only admitted Lie point \ss
are purely "linear" symmetries, or - more precisely - that the functions
$\th_i$
must be of the form
$$\th_i(q,t) = A_{ij}(t)\ q_j + b_i(t) \qquad\quad (i,j=1,\ldots,n)\eqno(1.6)$$
The result that the determining equations acquire a significant structure, or
at least are greatly simplified when the r.h.s. of (1.1) depend linearly on the
higher-order derivatives of the variables, is not peculiar to the case at
hand:
a similar situation occurs in fact also in different and more general
contexts, as already remarked and discussed [5,8].
Next, assume more in particular that the dependence of the $F_i$ on $\.q_i$ has
the form
$$F_i=\l \.q_i+f_i(q,t) \eqno(1.7)$$
where in general $\l=\l (t)$ may be any function of time (possibly $\l=0$ or a
constant, of course): in this case, again from (1.3), one obtains
$$ {\pd^2\th_i\over{\pd t \pd q_j}} = A_{ij}={\rm const} $$
and the determining equations become
$$ (A_{jk}q_k+b_j){\pd f_i\over{\pd q_j}}-A_{ij}f_j = \"b_i - \l \.b_i \ .
\eqno(1.8)$$
Clearly, this shows - first of all - that there are relatively "few" problems
(1.1), with $F_i$ given by (1.7), which can admit Lie point symmetries:
it is in fact a quite exceptional case that, given $F_i$, some
$A,b$ can be found to satisfy (1.8): for instance, the only 1-dimensional
problems ($n=1$), admitting a Lie point symmetry, are (this can be easily
verified differentiating (1.8) with respect to $q$)
$$\"q = \l(t)\ \.q + k(t)\ q + c(t) \eqno(1.9)$$
where $\l(t),\ k(t),\ c(t)$ are arbitrary functions of time.
The symmetries admitted by (1.9) are given by
$$\g^{(1)}=\big(q+b^{(1)}(t)\big){\pd\over{\pd q}}\qquad {\rm
and} \qquad \g^{(0)}=b^{(0)}(t){\pd\over{\pd q}}$$
where $b^{(1)}(t)$ and $b^{(0)}(t)$ satisfy respectively the equations
$$\"b^{(1)}=\l\ \.b^{(1)} + k\ b^{(1)}-c \qquad {\rm and}
\qquad \"b^{(0)}=\l \ \.b^{(0)} + k\ b^{(0)}.$$
The presence of the first \sy asserts that if $q_0(t)$ is a solution
of (1.9), then
$$q'(t)\equiv q(t,\ep)=q_0(t)\ {\rm e}^\ep +b^{(1)}(t)\ ({\rm e}^\ep-1)$$
is again a
solution for any $\ep$, the second simply states the elementary property that
$$q'(t) = q_0(t)+\ep b^{(0)}(t)$$
is a solution for any $b^{(0)}(t)$ solving the homogeneous equation.
Less trivial situations, involving nonlinear $F(q)$ or $F(q,t)$ can be
constructed in dimension $n>1$, as we shall see later on.
Another simple but interesting consequence of (1.8) can be deduced in the
hypothesis that $F_i$ do not depend on time: differentiating in fact (1.8) with
respect to $t$, one easily obtains that either $b_i$ are constants, or $f_i(q)$
depend linearly on the $q_i$.
Therefore, we can state the following result.
\bigskip
\pn
{\bf Proposition 1}. {\sl The Lie point symmetries (1.2) admitted by
Newton-Lagrange problems (1.1), with $F_i$ depending at most linearly on
$\.q_i$, have - if they exist - the form (1.6). If in particular
$$F_i = \l(t)\ \.q_i + f_i(q,t)$$
(including $\l=0$ or constant), the admitted \ss have the form (1.6)
where $A_{ij}$ are constants and $b_i(t)$ satisfy (1.8).
In the further hypothesis that $F_i$ are independent of
time - and excluding the (rather trivial) case
of $f_i$ linearly dependent on $q$ - then also $b_i$ are constants, and the
admitted symmetries - if any - have the form
$$\g=\big( A_{ij}q_j+b_i\big){\pd\over{\pd q_i}} \eqno(1.10)$$
where the constants
$A_{ij}$ and $b_i$ have to satisfy}
$$(A_{jk}q_k+b_j){\pd f_i\over{\pd q_j}}-A_{ij}f_j =0\ . \eqno(1.11)$$
\bigskip
In the remaining of this section, we restrict our analysis to
problems (1.1) where the forces $F_i$ are independent on $\.q$,
and to Lie point \ss which are independent of time.
To examine more carefully the existence of Lie point symmetries of a given
Newton-type system, we shall reverse the problem, and construct the more
general
Newton system admitting a given LPTI symmetry.
In order to do this, and also in view of an application in sect. 3, let us come
back preliminarily to a generic LPTI \sy generator in the form (1.2) with
$\th_i=\th_i(q)$, and introduce the set of the real functions $\Phi(q)$ which
are left invariant by the action of the symmetry $\g$, i.e.
$\Phi(q)=\Phi(q')$.
This property is verified if and only if the condition
$$\g \cdot \Phi \equiv \th_i \ {\pd\Phi\over{\pd q_i}} = 0 \eqno(1.12)$$
holds. From a geometrical point of view, the
equation $\Phi(q)=k={\rm const}$ describes an invariant manifold under $\g$.
Solving the above equation (1.12) by means of its characteristic equations:
$${dq_1\over{\th_1(q)}}=\ldots={dq_n\over{\th_n(q)}} \eqno(1.13)$$
one naturally introduces a set of $(n-1)$ independent functions $h_\a(q), \
\a=1,\ldots,n-1$, such that
$$h_\a(q)=k_\a={\rm const} \qquad \quad (\a=1,\ldots,n-1) \eqno(1.14)$$
where the $k_\a$ are the "constants of integration" of (1.13). Each of these
characteristic integrals $h_\a(q)$ of (1.13), and any function
$\Phi(q)=\Phi\big( h_1(q),\ldots,h_{n-1}(q)\big)$ of them, is an invariant
function under $\g$.
We construct now the most general field of forces $F_i$ which admits a
given LPTI \sy (1.10). Given $\g$, the conditions (1.11) become a system of
partial differential \eqs for the $F_i$; this is a system "with the same
principal part" [13], and its characteristic \eqs are
$${dq_1\over{A_{1i}q_i+b_1}}=\ldots={dq_n\over{A_{ni}q_i+b_n}}=\eqno(1.15a)$$
$$={dF_1\over{A_{1i}F_i}}=\ldots={dF_n\over{A_{ni}F_i}} \eqno(1.15b)$$
To solve these, one first sees from (1.11,15) that the dependence of the $F_i$
o
n
the variables $q_i$ must be of the form
$$F_i=K_{ij}q_j+c_i\eqno(1.16)$$
where the matrix $K$ and the vector $c$ satisfy
$$KA\ q + K\ b=AK\ q+ A\ c\ . \eqno(1.17)$$
Then $K$ is the most general matrix such that
$$K\ A = A\ K \eqno(1.18)$$
whereas the solution of the remaining condition
$$K\ b= A\ c \eqno(1.19)$$
depends on whether $A$ is invertible or not. In the first case,
$c=A^{-1}K\ b=K\
A^{-1}b$, and the most general form of $F$ is
$$F=K\ (q+A^{-1}b)=K \ q+A^{-1}K \ b \eqno(1.20)$$
with $K$ the most general nonzero matrix commuting with $A$. If $A$ is
singular, one has to further distinguish various cases: if $K$ is chosen such
that $K\ b\in R(A)=$ the range of $A$, then $c$ is any vector such that $A\
c=K\ b$\ ; if $b$ is such that for no $K\ne 0$ one has $K \ b\in R(A)$, then
(1.19) can be solved by
$$K=0, \quad {\rm and}\quad c\in N(A)= {\rm \ the\ kernel\ of\ } A
\eqno (1.21)$$
If finally $K b$ is "generic", one may decompose it as a sum of two
terms and obtain a combination of the two above possibilities, as easily
verified (a sort of Fredholm alternative theorem). In conclusion, even if $A$
is singular, one has again (1.16), i.e. $F=K\ q+c$, with the only difference
tha
t
also the choice $K=0$ and $c\ne 0$ is admitted. To obtain finally the complete
general solution of (1.15), one has to take into account the subsystem (1.15a),
and one can conclude [13] that the general solution is obtained assuming that
th
e
entries $K_{ij}$ of $K$ in (1.16) are arbitrary functions of the characteristic
integrals $h_\a(q)$ of (1.15a). These functions $h_\a(q)$ are precisely the
invariant functions under the symmetry $\g$: indeed, they solve the equation
(cf. (1.12-13)) $$\g\cdot h \equiv (A_{ij} q_j+b_i){\pd h\over{\pd q_i}}\ .
\eqno(1.22)$$ Notice that the above result holds for both autonomous and
nonautonomous systems: in fact, time $t$ is left invariant
by $\g$, just as the functions $h_\a(q)$.
Then, summarizing, we have the following conclusion.
\bigskip
\pn
{\bf Proposition 2}. {\sl Given a LPTI symmetry of the form (according to
Prop. 1)
$$\g=(A_{ij}q_j+b_i){\pd \over{\pd q_i}}$$
where $A_{ij},\ b_i$ are constants, the most general Newton problem (autonomous
or not) admitting the \sy generated by $\g$ has the form
$$ \"q_i= K_{ij}\big(h_\a(q),t\big)\ q_j+c_i(t) \eqno(1.23)$$
where $K_{ij}$ are arbitrary functions of the quantities $h_\a(q)$ which are
left invariant under the action of $\g$:
$$\g\cdot h_\a =0 $$
(and possibly of the time), restricted by the commutativity condition
$A_{im}K_{mj}=K_{im}A_{mj}$, and $c_i$ are determined by $A, b, K$. }
\bigskip
We give a simple example to describe this result. Let $n=2,\ q\equiv (q_1, q_2)
\equiv (x, y)$ and
$$A=\pmatrix{1&0\cr
3&-2\cr} \quad ; \quad b=0 $$
i.e. consider the LPTI \sy
$$\g = x{\pd\over{\pd x}}+(3x-2y){\pd\over{\pd y}} \eqno(1.24)$$
A function $h=h(x,y)$ invariant under $\g$ is
$$h=x^3-x^2y$$
and the matrices $K$ are
$$K=\pmatrix{k_1&0\cr
k_1-k_2&k_2\cr}$$
Then, the most general problem admitting the LPTI symmetry $\g$ (1.24) is
$$ \eqalign{ \"x =& k_1(x^3-x^2y,t)\ x \cr
\"y =& \big( k_1(x^3-x^2y,t)-k_2(x^3-x^2y,t)\big)\ x+k_2(x^3-x^2y,t)\ y \cr}
\eqno(1.25)$$
where $k_1, k_2$ are arbitrary functions of $h, t$. The existence of the
symmetry $\g$ means that, if $x_0(t), y_0(t)$ is any solution of (1.25), then
al
so
$$ \eqalign { x'(t)&\equiv x(t,\ep) = x_0(t)\ {\rm e}^\ep \cr
y'(t)&\equiv y(t,\ep) = y_0(t)\ {\rm e}^{-2\ep}+
x_0(t)\ ({\rm e}^\ep -{\rm e}^{-2\ep}) } $$
is a solution of (1.25) for any $\ep$.
\titleb{2. Newton-Lagrange problems in first-order form}
In order to find a richer geometrical structure (and to find other, possibly
nonlinear Lie symmetries) for Newton-Lagrange problems, one could - as well
known [1-8] - look for \ss more general than Lie point ones, of the form
(contact or dynamical symmetries)
$$ \^ \g = \th_i(q,\.q,t){\pd\over{\pd q_i}}+\th_i^{(1)}(q,\.q,t){\pd\over{\pd
\.q_i}}\eqno(2.1)$$
or transform the original system into a first-order system in $2n$ variables,
putting
$$ \eqalign{ \.q_i =& \ p_i \cr
\.p_i =& \ F_i(q,p,t) } \eqno (2.2)$$
In this second way, one falls into the more general problem of finding the Lie
point \ss
$$\g=\phi_j(u,t){\pd \over{\pd u_j}}\eqno(2.3)$$
of a first-order system
$$\.u_j=f_j(u,t)\eqno(2.4)$$
The determining \eqs for this general case (2.3-4) are [1,8,10]
$$ \phi_i{\pd f_j\over{\pd u_i}}-f_i{\pd \phi_j\over{\pd u_i}}=
{\pd\phi_j\over{\pd t}} \eqno(2.5)$$
In the special case of (2.2), putting
$$\g=\th_i(q,p,t){\pd\over{\pd q_i}}+\psi_i(q,p,t){\pd\over{\pd
p_i}}\eqno(2.6)$$
the above conditions (2.5) become explicitly
$$ \eqalign {\psi_j=&\ p_i{\pd \th_j\over{\pd q_i}}+
F_i {\pd\th_j\over{\pd p_i}} \cr
\th_i{\pd F_j\over{\pd q_i}}=&\ p_i{\pd\psi_j\over
{\pd q_i}}+F_i{\pd\psi_j\over{\pd p_i}} } \eqno(2.7)$$
The general problem of finding the Lie point \ss of first-order
systems has been considered in detail especially in [1-2] and also in
[8,10,11],
and we do not pursue here the analysis of this point. Let us add just another
remark along the lines of the previous section. Assume that in (2.2)
$F_i=F_i(q)$, and let us look for those particular LPTI \ss which have the form
$$\g=\th_i(q){\pd\over{\pd q_i}}+\psi_i(q,p){\pd\over{\pd p_i}}\eqno(2.8)$$
where $\th_i$ depend only on the variables $q$ (this restriction
is suggested by the desire of preserving, as far as possible, the physical
meaning of the "coordinates" $q$). Then, it is easy to obtain, using (2.7), the
following result.
\bigskip\pn
{\bf Proposition 3}. {\sl The only LPTI \ss having the special form (2.8),
which
are admitted by the problem (2.2) with $F_i=F_i(q)$, are merely of the
following
type
$$\g=(A_{ij}q_j+b_i){\pd\over{\pd q_i}}+A_{ij}p_j{\pd\over{\pd p_i}}$$
where $A_{ij}, b_i$ are constants}.
\bigskip
Another property of LPTI \ss of first-order systems which should be mentioned
is the following. Let $\g_1, \g_2, \ldots, \g_r$ be a set of LPTI \sy
generators admitted by the system (2.4), and let $\{h_\a=h_\a(u)\}$ be a set of
constants of motion of (2.4), i.e. of functions satisfying
$${{\rm d}h_\a\over{{\rm d}t}} = f_j{\pd h_\a\over{\pd u_j}} = 0 \eqno(2.9)$$
\big(the constants of motions $h_\a$ are precisely the functions satisfying
$\g_f\cdot h_\a = 0$, i.e. the invariant fuctions under the operator
$$\g_f\equiv f_j{\pd\over{\pd u_j}}\eqno(2.10)$$
expressing the time-evolution of the dynamical flow, which is in fact a LPTI
\sy
generator for the problem (2.4)\big); then also
$$\^\g = h_1\g_1 + \ldots+h_r\g_r \eqno(2.11)$$
is a LPTI \sy for (2.4). This means that the set of the LPTI \ss for
first-order systems is not only an ($\infty-$dimensional) algebra, but rather
a finite dimensional {\it module} (cf.[14]) over the set of constants of
motions.
\titleb{3. Hamilton problems}
In this section, we consider exclusively autonomous problems and time
independent symmetries.
We specialize here our discussion to the problems expressed in the Hamiltonian
form
$$\.q_i={\pd H\over{\pd p_i}} \qquad \qquad \.p_i=-{\pd H\over{\pd q_i}}
\qquad\qquad (i=1,\ldots,n)\eqno(3.1)$$
with $H=H(q,p)$. There are various notions related to the
symmetry properties of Hamiltonian problems. In order to clearly show the
differences and discuss the relationships between them, let us recall some
well known definitions, writing down the following list.
\bigskip
\pn
{\it A) (LPTI symmetries)}. The operator
$$\g=\th_i (q,p){\pd \over{\pd q_i}}+\psi_i(q,p){\pd\over{\pd p_i}}\eqno(3.2)$$
is a LPTI symmetry for (3.1) if it generates a transformation
$q\to q', \ p\to p'$
with the property that solutions $q(t),\ p(t)$ of (3.1) are mapped into
solution
s
$q'(t),\ p'(t)$ of (3.1).
\bigskip \pn
{\it B) (invariance of H)}. Given a (continuously differentiable) coordinate
transformation $q\to~\~q$, $p\to \~p$ generated by the infinitesimal operator
$$\~\g =\~\th_i (q,p){\pd \over{\pd q_i}}+\~\psi_i(q,p){\pd\over{\pd
p_i}}\eqno(3.3)$$
the Hamiltonian is invariant under $\~\g$ if $H(\~q,\~p)=H(q,p)$, or
equivalently if
$$\~\g\cdot H = \~\th_i (q,p){\pd H \over{\pd q_i}}+\~\psi_i(q,p)
{\pd H\over{\pd p_i}}=0\eqno(3.4)$$
\bigskip\pn
{\it C) (canonical transformations)}. If an infinitesimal canonical
transformation leaves invariant the Hamiltonian $H(q,p)$, then the generatrix
function $G=G(q,p)$ of this transformation has the property
$${\pd G\over{\pd q_i}}{\pd H\over{\pd p_i}}-{\pd G\over{\pd p_i}}{\pd H\over
{\pd q_i}}={d G\over{d t}}=0\eqno(3.5)$$
and (equivalently) $G(q,p)$ is a constant of motion.
\bigskip
Let us remark, incidentally, that another commonly mentioned symmetry
transformation, i.e. the scaling $H\to \a H$, is not included in the present
scheme, because it requires a scaling of the time variable.
Now, notice first of all that A) and B) are different.
For instance, consider the following problem, with
$n=1$, and $H=q(p^2+1)$\ : $$ \eqalign{ \.q=&2qp \cr
\.p=&-p^2 -1} \eqno(3.6)$$
It is easy to verify that
$$\g=q{\pd\over{\pd q }}$$
generates a LPTI \sy for the problem (3.6), but does not satisfy B);
instead, for
$$\~\g={2qp\over{p^2+1}}\ {\pd\over{\pd q}}-{\pd \over{\pd p}}$$
one has $\~\g\cdot H=0$, i.e. the Hamiltonian is invariant under $\~\g$, but
$\~\g$ does not generates a LPTI symmetry for (3.6). Notice also that neither
$\
g$
nor $\~\g$ above generate canonical transformations.
The determining \eqs (2.5) for the functions $\th_i(q,p),\ \psi_i(q,p)$ in
order
that $\g$ (3.2) be a LPTI \sy admitted by the Hamilton problem (3.1), become
$$ \eqalign { {\pd \th_j\over{\pd p_i}}{\pd H\over{\pd
q_i}}+\th_i{\pd^2H\over{\pd q_i \pd p_j}}-{\pd \th_j\over{\pd q_i}}{\pd
H\over{\pd p_i}}+\psi_i{\pd^2 H\over{\pd p_i\pd p_j}}=0 \cr
- {\pd \psi_j\over{\pd p_i}}{\pd H\over{\pd
q_i}}+\th_i{\pd^2H\over{\pd q_i \pd q_j}}+{\pd \psi_j\over{\pd q_i}}{\pd
H\over{\pd p_i}}+\psi_i{\pd^2 H\over{\pd p_i\pd q_j}}=0 } \eqno(3.7)$$
Differentiating with respect to $q_j$ and to $p_j$ the condition (3.4)
expressin
g
the invariance of $H$ under $\~\g$, one has instead
$$ \eqalign { {\pd \~\th_i\over{\pd q_j}}{\pd H\over{\pd
q_i}}+\~\th_i{\pd^2H\over{\pd q_i \pd q_j}}+{\pd \~\psi_i\over{\pd q_j}}{\pd
H\over{\pd p_i}}+\~\psi_i{\pd^2 H\over{\pd p_i\pd q_j}}=0 \cr
{\pd \~\th_i\over{\pd p_j}}{\pd H\over{\pd
q_i}}+\~\th_i{\pd^2H\over{\pd q_i \pd p_j}}+{\pd \~\psi_i\over{\pd p_j}}
{\pd H\over{\pd p_i}}+\~\psi_i{\pd^2 H\over{\pd p_i\pd p_j}}=0 } \eqno(3.8)$$
Comparing with (3.7), one sees that if $\~\g$ satisfies B) and the following
conditions
$$ \eqalign { {\pd \~\th_i\over{\pd p_j}} = {\pd \~\th_j\over{\pd p_i}} \qquad
& \qquad {\pd \~\psi_i\over{\pd q_j}} = {\pd \~\psi_j\over{\pd q_i}} \cr
{\pd \~\th_i\over{\pd q_j}} =& - {\pd \~\psi_j\over{\pd p_i}} }
\eqno(3.9)$$
then $\~\g$ is also a LPTI symmetry. But (3.9) are just the
"integrability conditions" for a infinitesimal transformation
$$\eqalign {q\to q' +\ep \~\th (q,p) \cr
p\to p' +\ep \~\psi (q,p) }\eqno(3.10)$$
to be canonical [12]: then, in these hypotheses, $\~\g$ determines also a
canonical transformation leaving invariant the Hamiltonian, and there is a
generatrix function $G=G(q,p)$ which is a constant of motion ${\rm d}G/{\rm
d}t=0$. Let us observe that, instead, there may exist LPTI \ss satisfying also
the integrability conditions (3.9), but which do not satisfy the invariance
condition B). For instance, for the above example (3.6), the operator
$$\g'={1\over{p^2+1}}\ {\pd \over{\pd q}}$$
generates a LPTI \sy for (3.6), satisfies (3.9) and then generates a canonical
transformation, but the Hamiltonian of the problem is not invariant under this
$\g'$.
Obviously, the evolution operator (cf. (2.10))
$$\g_f={\pd H\over{\pd p_i}}{\pd\over{\pd q_i}} - {\pd H\over{\pd
q_i}}{\pd\over{\pd p_i}}\eqno(3.11)$$
generating the dynamical flow of the problem (3.1), satisfies in any case all
of
the three above conditions A), B), C).
We can then conclude with the following statement.
\bigskip\pn
{\bf Proposition 4}. {\sl Given the Hamiltonian problem
(3.1), there may be LPTI \ss admitted by (3.1) which are not canonical
transformations, or which are also canonical transformations but do not leave
invariant the Hamiltonian. Similarly, there may be infinitesimal coordinate
transformations leaving invariant the Hamiltonian but which are not LPTI
symmetries. If a coordinate transformation leaving invariant the Hamiltonian is
canonical, then it is also a LPTI \sy admitted by the problem; and each
constant
of motion $G=G(q,p)$ generates not only a canonical transformation which leaves
invariant the Hamiltonian, but also a LPTI \sy which is given by }
$$\g^{(G)}=\th_i^{(G)}(q,p){\pd\over{\pd q_i}}+\psi_i^{(G)}(q,p)
{\pd\over{\pd p_i}} \qquad {\rm where}\qquad \th_i^{(G)}={\pd G\over{\pd p_i}}
,
\quad \psi_i^{(G)}=-{\pd G\over{\pd q_i}} \eqno(3.12)$$
\bigskip
A final remark on the integrability conditions (3.9) can be useful. Clearly, if
$\~\g$ satisfies B), then also $k(q,p)\ \~\g\ $ does, for any function
$k(q,p)$;
if in particular there is an "integrating factor" $\~k=\~k(q,p)$ for the
differential form
$$-\~\psi_i\ {\rm d}q_i + \~\th_i\ {\rm d}p_i \eqno(3.13)$$
then $\~k\ \~\g$ satisfies (3.9) and determines a canonical transformation.
Denoting by $G_{\~k} = G_{\~k}(q,p)$ the generatrix function of this
transformation, one has in particular the invariance property under $\~\g$
$$\~\g\cdot G_{\~k} = 0 \ . \eqno(3.14)$$
The situation is quite different for the LPTI symmetries: in fact, as remarked
in sect. 2, if $\g$ generates a LPTI symmetry, also $h(q,p)\g$ does
if (and only if) $h = h(q,p)$ is a constant of motion (and no more an arbitrary
function). Then, the problem of finding an integrating factor for a LPTI \sy is
restricted to the set of constants of motion. This shows another difference
between the notion of Lie point \sy and the one related to the invariance of
the
Hamiltonian under generic coordinate transformations.
\vfill \eject
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\bye