We shall use the following notation given at the first part [1] .

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Notation: Along this paper, we shall consider the variables t and. The derivatives respect to the variable t will be denoted by the symbol “・” while the derivatives respect the variable will be denoted by the symbol apostrophe “'”. In addition, if we will denote:

・

・

・

・

・

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In the first part of this paper, we study the problem of n bodies interacting in a certain medium, which we will assume that it is the vacuum in this part (since the general case is analogous), whose equation of motion (in the vacuum case) is given by

where and are the mass and the component j of the position of the i-body respectively while is the force applied to the i-body.

We saw that if is such that with

then Equation (1) (without the initial conditions) is equivalent to

where

At the same time, we saw that Equation (3) and the initial conditions (and hence Equation (1)) are equivalent to

where h can be any function, i, j, l and m are arbitrary indexes satisfying, and is given by

We also proved that if is a parametrization of C, where C is the image of, then it satisfies conditions (7), (8) and (9). Due to this fact, we call trajectory equation to condition (7), since C represents the trajectory of the system and conditions (8), (9) are just extra conditions for particular cases. We finally saw that if satisfies conditions (7), (8) and (9), then given by also satisfies them, with such that and. In addition

On the other hand, we call temporal equation to Equation (2), since it determines the relationship between t and. We also call temporal equation to Equation (10). We proved that if the force comes from a potential V, then we can write Equation (10) (for) using the mechanical energy of the system as follows

where E is the energy.

Based on these results, we develop the following algorithm for solving Equation (1):

1) Find a solution of the trajectory equation and check that it satisfies conditions (8) and (9).

2) Choose conveniently a function with and in order to build another solution given by.

3) Find the function u given in Equation (10) (or (13)).

4) Solve the temporal equation.

Finally, is the solution of Equation (1).

We saw that if we want to solve the equation of motion, it is more convenient to follow this algorithm.

In this occasion, we will change the last step of this algorithm, i.e., we will find a new way to solve the temporal equation, by curving the horizontal axis (the time axis). In this way we will be able to solve some problems that cannot be solved in the traditional way (without using elliptic integrals) like the pendulum problem. We will also be able to see the period of some periodic systems as the length of a curve. This fact can be compared with the Hamilton-Lagrange’s formalism, where we can see the period of some systems as the area of a surface given by the phase space [2] .

First, we will use the arc length function and its inverse in order to introduce a new way to graph functions by curving the axes. Then, we will use these results in order to complete the algorithm and to graph each component of the solution of Equation (1) in this way. Finally, we will solve the harmonic oscillator, the pendulum, the particle under the action of two elastic springs [3] and Kepler’s problems using all the formalism.

In this section we will use the arc length function and its inverse in order to introduce a new way to graph functions by curving the axes. Before doing this, we will introduce the notation that will be used and we will give some definitions.

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Notation and preliminaries: let C be a curve in the plane given by a parametrization with.

・ We will denote by and, i.e., is the image of and is the image of.

・ We will assume that and we will denote by.

・ If and are the lengths of the curve from to and respectively, we write.

Definition 1: let C be a curve in the plane describing the graph of a function depending of x and given by a parametrization. We define the function as follows: if and only if s is either or, where l is the length of the curve C measured from to and the sign depends of the orientation of the parametrization as shown in Figure 1 and Figure 2.

In a similar way, if C describes the graph of a function depending of y, we define the function as shown in Figure 3 and Figure 4.

Definition 2: let C be a curve in the plane given by a parametrization. We define the function as follows: if and only if s is either or, where l is the length of the curve C measured from to and the sign depends of the orientation of the parametrization as shown in Figure 5 and Figure 6.

In a similar way, we define the function as shown in Figure 7 and Figure 8.

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Before proceeding to the new way to graph functions, we will discuss some points about these functions.

On the one hand, note that all these functions are uniquely determined by the curve C described implicitly by, the initial point and the orientation. In addition, there are infinite parametrizations that define the same function. Then, in the examples that we will see later, we will find F, and the orientation in order to determine these functions, i.e., we will not find the parametrization.

On the other hand, if C describes the graph of a function depending of x, by Definition 1 it is

easily proved that is given by

where the sign + corresponds to the orientation given in Figure 1 and the sign − to the orientation given in Figure 2.

In addition, in this case we can see in Definition 2 that is the inverse function of and then it follows from the inverse function theorem and Equation (14) that is given by

where the sign + corresponds to the orientation given in Figure 5 and the sign-to the orientation given in Figure 6.

If C describes the graph of a function depending of y the process is analogous by changing and by and.

Next, we will use these functions in order to graph functions by curving the axes.

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Curvilinear vertical axis: assume we have a differential equation whose solution is given by the function where the curve C is given by, it is oriented as in Figure 3 and (with). Then, we can graph this function as shown in Figure 9.

This way of making the graph of the function is equivalent to make the graph of the function by curving the X-axis with the function x. Note that the origin of the horizontal axis is (i.e. it does not change) while the origin of the vertical axis is. Note also that the original X-axis is not used to make the graph of the function, it is only used to perform the new y-axis and to make the graph of the function. Finally, we can graph X versus t in a more simple way as shown in Figure 10.

If the curve C is oriented as in Figure 4 the direction of the X-axis changes.

Curvilinear horizontal axes: assume now that we have a differential equation whose solution is given by the function where the curve C is given by, it is oriented as in Figure 5 and (with). Then, we can graph this function as shown in Figure 11.

This way of making the graph of the function is equivalent to make the graph of the function by curving the t-axis. Note that the origin of the vertical axis is (i.e. it does not change) while the origin of the horizontal axis is. Note also that the original t-axis is not used to make the graph of the function, it is only used to perform the new t-axis and to make the graph of the function. Like the previous case, we can graph

X versus t in a more simple way as shown in Figure 12.

If the curve C is oriented as in Figure 6 the direction of the X-axis changes.

If C is not the graph of a function the idea is the same. For example if C is a closed curve and it is positively oriented, the graph of X versus t is shown in Figure 13. In this case we need to say how many times the parametrization of the curve “turns”. If it turns indefinitely, then and we can see in Figure 13 that X is a periodic function whose period is given by the length of C.

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Next, we will complete the algorithm for solving the equation of motion by using these results. We will only

use the functions, since we will just curve the t-axis.

In the first part of this paper we find a more convenient algorithm for solving the equation of motion of a system of n bodies. In this section we will use the functions in order to change the step four of that algorithm, by solving the temporal equation in a different way.

Before doing that, we will need the following mathematical notion.

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Definition 3: let and. We define the following set:

Note 1: in order an element x belongs to it is necessary (but not sufficient) that

in order the term which appears under the radical be no negative.

Note 2: if we do not choose the element appropriately (for instance if) then the set can be empty.

Note 3: although, it may be possible that x belongs to, since the improper integral could converge. Analogously, if, it may be possible that the set is not empty for the same reason.

Proposition 1: let and be such that

Let also be such that

where and the sign is arbitrary.

Then, if u does not change the sign in the interval, is the solution of the differential equation where C is described by, , and the curve is oriented as follows

・ If u is positive, then C is oriented as in Figure 5.

・ If u is negative, then C is oriented as in Figure 6.

Proof: by Equation (17) we have

It follows that

where the sign is + provided and the sign is − provided.

Then, according to Equation (15) we have that is the solution of Equation (16) where and C is described by with its respective orientation.

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By Proposition 1, we will complete the algorithm for solving the equation of motion.

We will assume that the initial time, which implies that Equation (2) is written as Equation (16) by taking. In addition, we can see in Equation (10) (or (13)) that u does not change its sign, and hence we can use Proposition 1. Then, we will change the algorithm as follows

1) Find a solution of the trajectory equation and check that it satisfies conditions (8) and (9).

2) Choose conveniently a function with and in order to build another solution given by.

3) Find the function u given in Equation (10) (or (13)).

4) Find a function y that satisfies Equation (17).

Finally, is the solution of Equation (1) where C is described by, and the curve is oriented as follows

・ If is positive, then C is oriented as in Figure 5.

・ If is negative, then C is oriented as in Figure 6.

where is given by Equation (11).

In this way, we can graph each component of as in Figure 12 (or 13) by curving the horizontal axis, i.e., the time axis.

Note that we have only changed the step four. However, there is also a big difference in the step two.

In the step two of the old algorithm, the statement “choose conveniently a function” refers to choose so that the temporal equation can be solved easily. However, according to Equation (12) we have

This implies that

Hence we arrive to

Then, we can see that choosing so that the temporal equation can be solved easily is equivalent to make a

simple change of variables in the temporal equation. This implies that if the solution of is a non-elemental function, then the solution of will also be a non-elemental function. Hence, we can say that in the old algorithm, the step two is not very important.

In the step two of the new algorithm, the statement “choose conveniently a function” refers to

choose so that, which implies that according to Equation (12), and then

in most cases, and that the integral given in Equation (17) can be solved easily. We will see in the following examples that in this case, the step two is key since if we choose properly, then we will be able to find y analytically and if we do not choose properly, we will not be able to find it.

Finally, suppose that there is just one body and it moves in only one direction and we find a function that satisfies and. Then, the step one is satisfied. In addition, in this case the step two is unnecessary. Hence, the algorithm in this case becomes

1) Choose conveniently a function that satisfies and.

2) Find the function u given in Equation (10) (or (13)).

3) Find a function y that satisfies Equation (17).

As before, the statement “choose conveniently a function “ refers to choose so that and that the integral given in Equation (17) can be solved easily.

Finally, suppose that the universe is only composed of the n bodies. Then, it would be impossible to tell if the bodies move according to and the time t is “an horizontal line” or if the bodies move according to and the time curves according to Equation (17).

Hence, we can say that the master equation is a generalization of Newton’s second law which includes the possibility that the time curves. We can also say that we do not know if Newton’s second law is all right since we do not know if the time curves or does not.

Next we will see four examples. The first three examples will be one dimensional problems and then we will use this last algorithm.

In this section, we will solve the harmonic oscillator, the pendulum, the particle under the action of two elastic springs [2] and Kepler’s problems. Note that all these problems have periodic solutions (at least for some values of the energy). We will use this formalism in order to approximate the periods. In order to do that we will need the following proposition.

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Proposition 2: let C be a curve with elliptical shape with semi-axes a and b and let and be two curves as shown in Figure 14. Then we have

where, and are the length of C, and respectively.

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Suppose that is the solution of one of these problems with C a closed curve with elliptical shape and semi-axes a and b and. If turns indefinitely the solution holds for all t and according to Figure 13 it is periodic. The period is given by the length of C and then, according to this proposition, we have

where is the period.

In other words, where

Hence, if we know a and b, we can approximate the period of the solution using Equation (18) and the error of the approximation is given by Equation (19). If we call to the ratio between the semi-minor axis and the semi-major axis, according to Equations (18) and (19), the relative error is given by

Since, then it is easily proved that

where.

In addition, if is the equilibrium point, we can also calculate the amplitude of the motion by doing

Next we will see some examples and we will use these results in order to approximate the period of the solution. In all the examples we will consider just one body and we will denote by and. In the one dimensional examples we will use the second algorithm.

We used the arc length function and its inverse in order to introduce a new way to graph functions by curving the axes as shown in Figure 10 and Figure 12. Using this, we found a more conveniently method to solve the temporal equation and then we changed the fourth step of the algorithm given in the first part in order to graph each component of the solution of Equation (1) as given in the mentioned figures. Following this algorithm, we solved the harmonic oscillator, the pendulum, the particle under the action of two elastic springs and Kepler’s problems. We were able to solve the pendulum and the particle under the action of two elastic springs problems without using elliptical integrals and to see the periods of both problems as the length of a curve given in Equations (43) and (58) respectively. Then, using this fact, we approximate the periods of both problems in Equations (46) and (61) with a relative error less than 0.17. Finally, in Kepler’s problem, we solved the trajectory equation and we proved that the solution describes an ellipse with focus in the origin (for some values of the energy). We could obtain the semi-major and semi-minor axes, the center of the ellipse and the orbital period by using this formalism.

Federico Petrovich, (2016) A New Formulation of Classical Mechanics—Part 2. Journal of Applied Mathematics and Physics,04,939-966. doi: 10.4236/jamp.2016.45103