Substitution of variables
Encyclopedia
In mathematics, substitution of variables (also called variable substitution or coordinate transformation) refers to the substitution of certain variable
s with other variables.
Though the study of how variable substitutions affect a certain problem can be interesting in itself, they are often used when solving mathematical or physical
problems, as the correct substitution may greatly simplify a problem which is hard to solve in the original variables. Under certain conditions the solution to the original problem can be recovered by back-substitution (inverting the substitution).
, 
be smooth manifolds and let 
be a 
-diffeomorphism
between them, that is:
is a 
times continuously differentiable, bijective map from 
to 
with 
times continuously differentiable inverse from 
to 
. Here 
may be any natural number (or zero), 
(smooth
) or
(analytic
).
The map
is called a regular coordinate transformation or regular variable substitution, where 
refers to the 
-ness of 
. Usually one will write 
to indicate the replacement of the variable 
by the variable 
by substituting the value of 
in 
for every occurrence of 
.
This may be a potential energy function for some physical problem. If one does not immediately see a solution, one might try the substitution
given by 
.
Note that if
runs outside a 
-length interval, for example, 
, the map 
is no longer bijective. Therefore 
should be limited to, for example 
. Notice how 
is excluded, for 
is not bijective in the origin (
can take any value, the point will be mapped to (0, 0, z)). Then, replacing all occurrences of the original variables by the new expressions prescribed by 
and using the identity 
, we get
.
Now the solutions can be readily found:
, so 
or 
. Applying the inverse of 
shows that this is equivalent to 
while 
. Indeed we see that for 
the function vanishes, except for the origin.
Note that, had we allowed
, the origin would also have been a solution, though it is not a solution to the original problem. Here the bijectivity of 
is crucial.
for a given function
.
The mass can be eliminated by the (trivial) substitution
.
Clearly this is a bijective map from
to 
. Under the substitution 
the system becomes
, Newton
's equations of motion are
.
Lagrange examined how these equations of motion change under an arbitrary substitution of variables
, 
.
He found that the equations
are equivalent to Newton's equations for the function
,
where T is the kinetic, and V the potential energy.
In fact, when the substitution is chosen well (exploiting for example symmetries and constraints of the system) these equations are much easier to solve than Newton's equations in Cartesian coordinates.
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
s with other variables.
Though the study of how variable substitutions affect a certain problem can be interesting in itself, they are often used when solving mathematical or physical
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
problems, as the correct substitution may greatly simplify a problem which is hard to solve in the original variables. Under certain conditions the solution to the original problem can be recovered by back-substitution (inverting the substitution).
Formal introduction
Let, be smooth manifolds and let be a -diffeomorphismDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
between them, that is:
is a times continuously differentiable, bijective map from to with times continuously differentiable inverse from to . Here may be any natural number (or zero), (smoothSmooth
Smooth means having a texture that lacks friction. Not rough.Smooth may also refer to:-In mathematics:* Smooth function, a function that is infinitely differentiable; used in calculus and topology...
) or
(analyticAnalytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
).
The map
is called a regular coordinate transformation or regular variable substitution, where refers to the -ness of . Usually one will write to indicate the replacement of the variable by the variable by substituting the value of in for every occurrence of .Cylindrical coordinates
Some systems can be more easily solved when switching to cylindrical coordinates. Consider for example the equationThis may be a potential energy function for some physical problem. If one does not immediately see a solution, one might try the substitution
given by .Note that if
runs outside a -length interval, for example, , the map is no longer bijective. Therefore should be limited to, for example . Notice how is excluded, for is not bijective in the origin ( can take any value, the point will be mapped to (0, 0, z)). Then, replacing all occurrences of the original variables by the new expressions prescribed by and using the identity , we get.Now the solutions can be readily found:
, so or . Applying the inverse of shows that this is equivalent to while . Indeed we see that for the function vanishes, except for the origin.Note that, had we allowed
, the origin would also have been a solution, though it is not a solution to the original problem. Here the bijectivity of is crucial.Integration
Under the proper variable substitution, calculating an integral may become considerably easier. Consult the main article for an example.Momentum vs. velocity
Consider a system of equationsfor a given function
.The mass can be eliminated by the (trivial) substitution
.Clearly this is a bijective map from
to . Under the substitution the system becomesLagrangian mechanics
Given a force field, NewtonIsaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
's equations of motion are
.Lagrange examined how these equations of motion change under an arbitrary substitution of variables
, .He found that the equations
are equivalent to Newton's equations for the function
,where T is the kinetic, and V the potential energy.
In fact, when the substitution is chosen well (exploiting for example symmetries and constraints of the system) these equations are much easier to solve than Newton's equations in Cartesian coordinates.
See also
- substitution property of equality
- instantiation of universals