最終更新日：2025/12/05
In dynamical systems theory, a theorem stating that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also known n first integrals of motion that are independent and in involution, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the canonical transform is explicitly known.
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Liouville-Arnold theorem
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元となった辞書の項目
Liouville-Arnold theorem
固有名詞
In
dynamical
systems
theory,
a
theorem
stating
that
if,
in
a
Hamiltonian
dynamical
system
with
n
degrees
of
freedom,
there
are
also
known
n
first
integrals
of
motion
that
are
independent
and
in
involution,
then
there
exists
a
canonical
transformation
to
action-angle
coordinates
in
which
the
transformed
Hamiltonian
is
dependent
only
upon
the
action
coordinates
and
the
angle
coordinates
evolve
linearly
in
time.
Thus
the
equations
of
motion
for
the
system
can
be
solved
in
quadratures
if
the
canonical
transform
is
explicitly
known.
日本語の意味
リウヴィル・アーノルドの定理とは、力学系理論において、n自由度のハミルトン系で、独立かつ可換なn個の第一積分が存在する場合に、正準変換を用いて作用角座標系に変換できることを示す定理です。変換後のハミルトニアンが作用座標のみに依存し、角座標が時間に対して線形に進むため、運動方程式が積分可能になるという主張がなされています。
意味（1）
In
dynamical
systems
theory,
a
theorem
stating
that
if,
in
a
Hamiltonian
dynamical
system
with
n
degrees
of
freedom,
there
are
also
known
n
first
integrals
of
motion
that
are
independent
and
in
involution,
then
there
exists
a
canonical
transformation
to
action-angle
coordinates
in
which
the
transformed
Hamiltonian
is
dependent
only
upon
the
action
coordinates
and
the
angle
coordinates
evolve
linearly
in
time.
Thus
the
equations
of
motion
for
the
system
can
be
solved
in
quadratures
if
the
canonical
transform
is
explicitly
known.
（ canonical ）
