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In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations. The method consists of first rewriting the equations as a system of differential equations that are first-order in time, by introducing additional variables. The original and the new variables form a vector in the phase space. The solution then becomes a curve in the phase space, parametrized by time. The curve is usually called a trajectory or an orbit. The (vector) differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization. Finally, a solution in the phase space is tra

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rdfs:label	位相空間法 (ja) Phase space method (en)
rdfs:comment	応用数学の分野における位相空間法（いそうくうかんほう、英: phase space method）とは、力学系の解を構成し解析するための、すなわち、時間依存の微分方程式を解くためのある手法のことを言う。この手法では、まずはじめに、新たな変数を導入することによって、方程式を時間について一階の連立微分方程式へと書き換える。すると、元の変数と新たな変数は、位相空間におけるあるベクトルを形成する。このとき解は、時間によってパラメータ付けられる、位相空間内の曲線となる。この曲線は通常、軌跡（trajectory）や軌道（orbit）と呼ばれる。微分方程式は、その曲線の幾何的表現として再び定式化される。すなわち、元の時間のパラメータ表現を必要とせず、その位相空間の変数のみについての微分方程式として、再び定式化される。最後に、その位相空間で得られた解が、再び元の設定へと変換される。位相空間法は物理学の分野で幅広く用いられている。例えば、反応拡散系の進行波解を見つける時に、用いられる。 (ja) In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations. The method consists of first rewriting the equations as a system of differential equations that are first-order in time, by introducing additional variables. The original and the new variables form a vector in the phase space. The solution then becomes a curve in the phase space, parametrized by time. The curve is usually called a trajectory or an orbit. The (vector) differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization. Finally, a solution in the phase space is tra (en)
dcterms:subject	Dynamical systems Partial differential equations
Wikipage page ID	10416232 (xsd:integer)
Wikipage revision ID	1034888400 (xsd:integer)
Link from a Wikipage to another Wikipage	Applied mathematics Curve Dynamical system Orbit (dynamics) Phase space Physics Dynamical systems Partial differential equations Trajectory Differential equations Reaction–diffusion system Fisher's equation Traveling wave
sameAs	Phase space method Phase space method Phase space method Phase space method Phase space method
has abstract	In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations. The method consists of first rewriting the equations as a system of differential equations that are first-order in time, by introducing additional variables. The original and the new variables form a vector in the phase space. The solution then becomes a curve in the phase space, parametrized by time. The curve is usually called a trajectory or an orbit. The (vector) differential equation is reformulated as a geometrical description of the curve, that is, as a differential equation in terms of the phase space variables only, without the original time parametrization. Finally, a solution in the phase space is transformed back into the original setting. The phase space method is used widely in physics. It can be applied, for example, to find traveling wave solutions of reaction–diffusion systems. (en) 応用数学の分野における位相空間法（いそうくうかんほう、英: phase space method）とは、力学系の解を構成し解析するための、すなわち、時間依存の微分方程式を解くためのある手法のことを言う。この手法では、まずはじめに、新たな変数を導入することによって、方程式を時間について一階の連立微分方程式へと書き換える。すると、元の変数と新たな変数は、位相空間におけるあるベクトルを形成する。このとき解は、時間によってパラメータ付けられる、位相空間内の曲線となる。この曲線は通常、軌跡（trajectory）や軌道（orbit）と呼ばれる。微分方程式は、その曲線の幾何的表現として再び定式化される。すなわち、元の時間のパラメータ表現を必要とせず、その位相空間の変数のみについての微分方程式として、再び定式化される。最後に、その位相空間で得られた解が、再び元の設定へと変換される。位相空間法は物理学の分野で幅広く用いられている。例えば、反応拡散系の進行波解を見つける時に、用いられる。 (ja)
gold:hypernym	Technique
prov:wasDerivedFrom	wikipedia-en:Phase_space_method?oldid=1034888400&ns=0
page length (characters) of wiki page	1887 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Phase_space_method
is Link from a Wikipage to another Wikipage of	Orbit (dynamics) Phase space Reaction–diffusion system Fisher's equation Phase-space method
is Wikipage redirect of	Phase-space method
is foaf:primaryTopic of	wikipedia-en:Phase_space_method

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