We analyze Noether and
In a fluid medium, drag forces are the major sources of energy loss for moving objects. Fuel consumption may have reduced to minimize the drag work. This can be achieved by the selection of optimum path. The drag force depends on the density of fluid, the drag coefficient, the cross-sectional area, and the velocity. These parameters are the combination of the altitude-dependent parameters which can be expressed as a single arbitrary function. If all parameters are assumed to be constants, then the minimum drag work path would be a linear path. But these parameters change during the motion. And all parameters can be defined as the function of altitude [
The main purpose of the work is to study Noether and
The second type of classification that is called
The other classification that we discuss in our study is how to obtain
The outline of this work is as follows. In the next section, we present the necessary preliminaries. In Section
Let us assume that
For each
Let us consider an
Let (
On the other hand the
Let
If
The differential equation describing the path of the minimum drag work is given in the form
For this case the solution of (
The associated infinitesimal generators turn out to be
Thus, the first integrals by
For the linear case of
The solution of determining equations for the form of
The associated five-parameter symmetry generators take the form
For this case, the infinitesimal functions read
The corresponding Noether symmetry generators are
For this choice of
Noether symmetry classification table of path equation.
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Invariant solutions that satisfy the original path equation can be obtained by first integrals according to the relation
(b) For the same
The relationship between
In terms of Find a first integral The solution of ( Let The solution of
Let us consider an
Thus the invariance criterion of (
However, for the path equation (
For arbitrary
Using (
If we substitute
It is clear that a solution of (
To write (
Taking derivative of (
It is easy to see that the general solution of this equation is
According to (
Then the conserved form satisfies the following equality:
For For
For another case
Thus, we can calculate
Therefore, the characteristic is written as
By using (
A solution of (
By using these equalities (
To find the integration factor one can write above equation in terms of
If we substitute
For this case the eight-parameter symmetry generators are obtained as follows:
Now let us consider
Using these infinitesimals we find the characteristic
By using (
A solution of (
This equation can be written as
By differentiation of (
To define
Therefore, by using the relation (
If we rewrite (
For this case the infinitesimal generators of path equation are
If we consider, for example,
If we apply the operator
We find the
By applying (
By differentiating
To define this equality in terms of variable
Finally one can write the conservation law
If
By using
By considering (
The solution of (
To write (
By taking derivative (
If we substitute
By using (
It is easy to see that the conserved form satisfies the following equality:
Table of
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In this section we analyze
The nonlocal approach [
For this case the divergence of the path equation yields
Another special form we consider here is
For this case of
For this case we find that
Therefore, we find new
The divergence of the path equation yields
If we substitute (
It is clear that we should analyze two specific values for
In summary all new
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In this section we present some invariant solutions based on Jacobi multiplier approach.
The aim of this study is to classify Noether and
In the literature, as a different and a new concept,
In our study, additionally, the Jacobi last multiplier concept is presented as a new and an alternative approach to construct