This paper has two parts, in this occasion we will present the first one. Until today, there are two formulations of classical mechanics. The first one is based on the Newton’s laws and the second one is based on the principle of least action. In this paper, we will find a third formulation that is totally different and has some advantages in comparison with the other two formulations.
Until today, there are two formulations of classical mechanics. The first one is based on Newton’s laws and the second one is based on the principle of least action [
The objective of this paper is to introduce a new formulation that has some advantages (and disadvantages) compared to the above formalisms.
Suppose that there are n bodies interacting in a medium where the i-body is subjected to a force
where
where
If we assume that the transverse motion surface of all bodies is constant along the time (this happens if the bodies are spheres or if they move only in one direction without rotation), then the equation of motion of the system, according to classical Newton’s second law, is given by
where
On the one hand, in the vacuum case (
where
On the other hand, if there are just one body (
where
In order to introduce our formalism, first we will find an equation equivalent to Equation (3) (we will call it the master equation of Equation (3)). We will say that our formalism is based on that equation. Then, from the master equation, we will try to generalize the constants of motion given in Equations (4) and (5) for the general case, i.e., for any medium and for the three dimensional case. Finally, we will see another advantage of the master equation. We will define the trajectory and the temporal equations and we will develop a more convenient algorithm for solving the equation of motion.
Notation: along this paper, we shall consider the variables t and
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We propose as a solution of Equation (3)
and will consider a time interval
Then we have
Let
In the first case, using Equation (10) we obtain
Hence, the component j of Equation (3) becomes
In the second case, scaling by
We will call
and we will develop the two members of Equation (12).
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Left member: using Equation (10) we have
Hence, according to Equations (13) and (14) we arrive to
Right member: using Equation (10) we have
In addition, since
By Equations (13), (14), (16) and (17) and using that
Let
Using Equations (2), (19) and (20), Equation (18) turns
Finally, using Equation (21), the fact that
we arrive to
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Using Equations (15) and (23) we infer that Equation (12) becomes
On the other hand, using Equations (10), (13) and (14) we have
where
Then, if we use the fact that
Finally by Equations (24) and (26) we obtain the following set
This equation can be viewed as a differential equation of first order where
Using Equations (11), (14) and (28) we finally obtain that Equation (3) is equivalent to
where
We will say that this is the master equation of Equation (3). It is worthwhile to point out that this equation is as important as Newton’s second law and that our formalism is based on this equation.
Note 1: using that the component j of Equation (3) implies Equation (12) we have that Equation (3) (and the master equation) implies
By taking
where
Note 2: suppose that there is just one body and it moves only in one direction
On the one hand, by condition (9),
On the other hand, by Equation (20),
In addition, since
Making a change of variable we finally obtain
where
Note 3: in the vacuum case, i.e., when
In this section we will try to generalize the constants of motion given in Equations (4) and (5).
In Note 1 of the previous section, we saw that Equation (3) implies Equation (31). It follows that
Using the notation given at the beginning and that according to Equation (33)
where
Since
Hence, we obtain that the following quantity is a constant of motion
If we want to generalize the constants of motion of Equations (4) and (5) we need to express
On the one hand, suppose that we have the following approximation
Hence, using Equations (25), (37) and the notation of the beginning we arrive to
On the other hand, using Equation (32) and the notation of the begining it is easily proved that
and then according to Equation (35) we have
If we call
we arrive to
If we want to write
In that case we have
Hence, taking into account that
However, in order to satisfy Equation (40) we need that
Next we will prove that this equation is equivalent to
In addition, we will prove that if
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Proof: we denote by
According to Equation (39) we have
Since according to Equation (37)
where
Analogously we have
Hence, Equation (43) is equivalent to
Using the definition of Kronecker's delta it is easily proved that this equation is equivalent to Equation (44). In addition, if
and hence Equation (44) turns out to be equivalent to Equation (45).
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Finally, if Equations (37) and (44) (or (45)) are satisfied, then the quantity given in Equation (42) is a constant of motion and it depends on
However, if
Next, we will consider these two cases and we will obtain Equations (4) and (5) from Equation (42). In addition, we will obtain another constant of motion in the three dimensional case (with
In the case
In the case where there is just one body Equation (45) becomes
where we omit the sub-index i, since
In the one dimensional case,
where
Scaling by
In the three dimensional case, we will propose as a solution of Equation (46) the following
where V is the potential of
We have
Then we can see that Equation (46) is satisfied.
In order to satisfy Equation (37), we shall approximate the function
Hence
According to Equation (32) and to the notation of the beginning we have
Then, if we differentiate Equation (37) we arrive to
Assuming that
Hence, Equation (47) becomes
According to Equation (48) its error is given by
where
Then, Equation (49) holds only if
We can see in this equation that there is a problem when
We use Equations (39) and (40) and the fact that
Hence, according to Equation (49) we arrive to
where
Using Equations (49), (51) and (52) we finally obtain that Equation (42) becomes
Taking into account that
where we are considering a time interval where Equation (50) holds.
In this section, we will see other advantages of the formalism. First, we will see an interesting application of the master equation. By means of this equation, we will introduce two equations which are called the trajectory and the temporal equation respectively. Finally, we will develop a more convenient algorithm for solving the equation of motion.
Until now, we have considered
Equations (30) and (31) turns out to be
In addition, according to Equation (3), the equation of motion of the system is given by
Remember also that in this case it is not necessary to ask condition (9).
Next, we will see an application of the master equation.
Let
where
Since
1. Let
The problem is to find which condition is satisfied by
2. Suppose we have an arbitrary parametrization
The problem is to find a way to find the original parametrization
We solve these problems by using the master equation:
1. Let
On the one hand, we know that this equation implies (55) and hence
Then we have
where
Using Equation (10) and the identity
Then we obtain that
On the other hand, using Equation (54) we also have that
Finally, if
2. Suppose we find
By condition (58) we have
where i, j, l and m are indexes satisfying
In addition, there exists λ such that
By condition 60 we have
where i, j, l and m are indexes satisfying
Hence, we can define a function
where h can be any function, λ is given by Equation (61) and i, j, l and m are indexes satisfying
We will prove that this function u satisfies Equation (54) and
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Proof: we will assume that
and
Let
Case
Case
In the first case, by Equation (62) for
In the second case, there exist l and m such that
Since
In addition, since
Then we arrive to
Since this limit should exist and
Then, by l'Hopital's rule we have
Since
Hence we have
Then we also obtain Equation (54) for
Case
In all cases we obtain Equation (54). Then, we proved that the function u given in Equation (62) satisfies the master equation.
In order to prove Equation (63), let
In the first case, by Equation (61),
In the second case,
Hence, by Equation (61) we also obtain
Since i and j were arbitrary, we finally proved that u satisfies Equation (63).
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Let
On the one hand, we saw that u satisfies the master equation and hence
On the other hand, by condition (58), Equation (10) and the fact that
Then,
According to the two solutions of the problems considered above, we can also conclude that the master equation is equivalent to conditions (58), (59) and (60) and to Equation (62).
Next, we will discuss the results obtained, we will give a name to Equations (8) and (58) and we will write them in a better way.
Note: suppose that
According to the previous section, we have that
The first one is that
The second one is that if
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Proof: on the one hand
On the other hand
where we made the change of variable
Since
then there exists
This implies that
Hence, we arrive to
Using the results obtained before, this equation becomes
By calling
we finally have
Then, we proved that
In addition, since
Hence we arrive to
We finally proved that
In order to prove conditions (59) and (60) we have
From these equations we can see that
Hence, using that
Then,
Finally, Equation (64) is easily proved by using that
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Next, we will write the trajectory equation in a better way. We will prove that it is equivalent to the following two equations
where we used the notation given at the beginning and in Equation (65), the sign ± has to be the same for all i.
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Proof: on the one hand, since
is equivalent to
In addition,
Suppose that Equations (65) and (66) hold. From these equations we have
Evaluating this equation in
Then, there exists λ such that
This implies that
On the other hand, since
where the sign of
Hence we arrive to
There are two cases to consider,
In the first case, using this equation we have
In the second case we arrive to
Since the sign of
In both cases we obtain that
which implies
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On the one hand, note that in the one dimensional case, Equation (65) is already solved (except the condition
We will also baptize to Equation (8). Taking into account that it determines the relationship between the “real time” t and
We will prove that if the force comes from a potential V, then we can write this Equation (for
where e is the energy.
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Proof: on the one hand, we saw in the second answer of the previous section that the function u given in Equation (62) satisfies the master equation. On the other hand, we saw in Section 2 that the master equation implies Equation (55). Using the notation given at the beginning it follows that
In addition, if the force comes from a potential V we have
Then we obtain
where we used that
From this equation, we can easily obtain Equation (67).
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Next, we will construct a more convenient algorithm for solving the equation of motion.
Using the results obtained before, we can construct the following algorithm in order to solve Equation (57):
1. Find a solution
2. Choose conveniently a function
3. Find the function u given in Equation (62) (or (67)).
4. Solve the temporal equation.
Finally,
Note that the difficult step of this algorithm is the first one.
However, to find a solution of the trajectory equation is easier than to find the solution of Equation (57) for the following two reasons:
1. There are infinite solutions of the trajectory equation while there are just one solution of Equation (57). In addition, the solution of Equation (57) is also a solution of the trajectory equation.
2. According to the appendix, the trajectory equation is a system of
Due to these facts, if we want to find the motion of the system, it is more convenient to follow this algori- thm.
In the second part of this paper, we will find a more convenient way of solving the temporal equation and then we will change the fourth step of this algorithm. We will also solve some examples using this formalism.
Note 1: in the second step, the phrase “choose conveniently a function
Note 2: according to the appendix, the number of equations of the internal trajectory equation of the i-body is 2. In addition, the number of equations of the external trajectory equation is
Note 3: the set
is a system of
and then solve the system of equations
We obtained an equation equivalent to Equation (3) (Equation (29)) and we called it the master equation. From this equation, we could deduce all the formalism.
We saw that if Equations (37) and (44) were satisfied, then we could generalize the constants of motion given in Equations (4) and (5) in Equation (42). If the force comes from a potential, Equation (44) turns out to be Equation (45) and it can be satisfied only in the vacuum case or in the case where there is just one body. In these cases, we obtain the constants of motion of Equations (4) and (5) from Equation (42) and we can generalize them in Equation (53), provide that Equation (50) is satisfied.
Then, we see another advantage of the master equation. We define the trajectory and the temporal Equations (Equations (8) and (58)) and we develop a more convenient algorithm for solving the equation of motion.
Finally, we can say that we develop a new formalism of classical mechanics based on Equation (29). We can conclude that the main advantages and disadvantages of our formalism, compared to the two formalisms mentioned in the introduction are the following:
• If the force does not come from a potential but it depends on the position, the formalism works well. This is an advantage compared to the Hamilton-Lagrange’s formalism.
• It includes the friction with the medium, considering a drag force proportional to the square of the velocity. This is also an advantage compared to the Hamilton-Lagrange’s formalism which in this case works only in the one dimensional case [
• It has a more convenient algorithm for solving the equation of motion. This is an advantage compared to the other two formalisms.
• It does not work when there are constraint forces or even if the forces depend explicitly on the time or on the velocities (with the exception of the drag force). This is a disadvantage compared to the Hamilton-Lagrange’s formalism.
FedericoPetrovich, (2016) A New Formulation of Classical Mechanics—Part 1. Journal of Applied Mathematics and Physics,04,412-431. doi: 10.4236/jamp.2016.42048
Definition: Let A,
Note 1: if
Note 2:
Note 3:
Remark: Let
Let also
Then, the following conditions are equivalent:
1.
2.
3.
Note 1: we can see in this remark that condition
Note 2: it is easy to find a base of
Proposition: Let A,
1.
2.
Proof:
1
On the one hand, this implies that
On the other hand,
Then, this also implies
2
This implies
On the other hand, since
Let
In the first case, since
In the second case we have
If we use again that
Since i was arbitrary, then
Therefore
Note 1: if
Note 2: if we change