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In the first part of this paper, we found a more convenient algorithm for solving the equation of motion of a system of n bodies. This algorithm consists in solving first the trajectory equation and then the temporal equation. In this occasion, we will introduce a new way to solve the temporal equation by curving the horizontal axis (the time axis). In this way, we will be able to see the period of some periodic systems as the length of a certain curve and this will allow us to approximate the period in a different way. We will also be able to solve some problems like the pendulum one without using elliptic integrals. Finally, we will solve Kepler’s problem using all the formalism.

We shall use the following notation given at the first part [

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Notation: Along this paper, we shall consider the variables t and

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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

We saw that if

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

We also proved that if

On the other hand, we call temporal equation to Equation (2), since it determines the relationship between t and

where E is the energy.

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

1) Find a solution

2) Choose conveniently a function

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

4) Solve the temporal equation.

Finally,

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 [

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 [

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

・ We will denote by

・ We will assume that

・ If

Definition 1: let C be a curve in the plane describing the graph of a function depending of x and given by a parametrization

In a similar way, if C describes the graph of a function depending of y, we define the function

Definition 2: let C be a curve in the plane given by a parametrization

In a similar way, we define the function

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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

On the other hand, if C describes the graph of a function

easily proved that

where the sign + corresponds to the orientation given in

In addition, in this case we can see in Definition 2 that

where the sign + corresponds to the orientation given in

If C describes the graph of a function

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

This way of making the graph of the function is equivalent to make the graph of the function

If the curve C is oriented as in

Curvilinear horizontal axes: assume now that we have a differential equation whose solution is given by the function

This way of making the graph of the function is equivalent to make the graph of the function

X versus t in a more simple way as shown in

If the curve C is oriented as in

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

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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

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

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

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Definition 3: let

Note 1: in order an element x belongs to

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

Note 2: if we do not choose the element

Note 3: although

Proposition 1: let

Let also

where

Then, if u does not change the sign in the interval

・ If u is positive, then C is oriented as in

・ If u is negative, then C is oriented as in

Proof: by Equation (17) we have

It follows that

where the sign is + provided

Then, according to Equation (15) we have that

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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

1) Find a solution

2) Choose conveniently a function

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

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

Finally,

・ If

・ If

where

In this way, we can graph each component of

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

This implies that

Hence we arrive to

Then, we can see that choosing

simple change of variables in the temporal equation. This implies that if the solution of

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

choose

Finally, suppose that there is just one body and it moves in only one direction

1) Choose conveniently a function

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

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

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 [

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Proposition 2: let C be a curve with elliptical shape with semi-axes a and b and let

where

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Suppose that

where

In other words,

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

Since

where

In addition, if

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

The force and the potential in this case are given by

where k is the elastic constant.

Before proceeding, we will call

We propose as a solution the following one

where v is given by

with E the mechanical energy.

We will choose

Then, the solution satisfies

In order to find u, according to Equations (5), (23), (24), (25) and (26) we have

In addition, according to Equations (11) and (25),

Since v is positive, then

Finally, by Equations (13), (28), (29) and (30) the function u is given by

Then, the second step of the algorithm is complete.

In order to find y, according to Equation (31) we have

Then, by making the change of variable

and choosing conveniently the value of

Without being quite formal, it follows that the curve C is given by

Hence, the third step is complete.

Finally,

Since C is a closed curve and we are considering that

in

In addition, according to Equations (22), (24), (25) and (26) the amplitude is given by

We can find the period and the amplitude in the traditional way and we obtain the same results.

We will call

where l is the length of the rope.

Then, the equation of motion can be written as Equation (1) where the force is given by

and g is the gravity.

We will choose conveniently the potential as follows

Before proceeding, using this equation and the fact that the kinetic energy is positive, note that

In addition, we will define conveniently

We propose the solution of Equation (25) and we will take

By condition (36) we can see that v is well defined.

As in the harmonic oscillator example, the solution satisfies

In order to find u, according to Equations (5), (25), (35) and (39) we have

In addition, in a similar way we can prove Equation (32).

Finally, according to Equations (13), (32), (40) and (41) the function u is given by

Then, the second step of the algorithm is complete.

In order to find y, according to Equation (42) we have

Then, by making the change of variable

and choosing conveniently the value of

Without being quite formal, it follows that the curve C is given by

According to Equation (39) we can also write C as follows

Hence, the third step is complete.

Finally,

If

It is easily proved that

Since C is a closed curve and we are considering that

The relative error is given by Equation (20) where in this case

We will study the limit cases, i.e., when

The first case corresponds to small oscillations and according to Equation (38)

Equations (44) and (45) we have that

In the second case we have that

Since

Taking into account that in the previous case

Finally, we can also find the amplitude and according to Equations (22), (25), (39) and (44) it is given by

We can find the amplitude in the traditional way and see that we arrive to the same result.

If

If we graph X versus t as in

However, it never reaches that value.

If

In this case, if we graph X versus t as in

describes a rotating circular motion. The time

the curve from

As in Proposition 2, we can obtain lower bounds and upper bounds for this length and approximate

it is easily proved that

with a and b given by Equations (48) and (49).

The force and the potential in this case, considering small oscillations, are given by [

where k is the elastic constant and

We propose as a solution

where

with

We will choose

Then, the solution satisfies

In order to find u, according to Equations (5), (50), (51), (52) and (53) we have

In addition, according to Equations (11) and (51),

Since A,

Finally, according to Equations (13), (30), (55) and (56) the function u is given by

Then, the second step of the algorithm is complete.

In order to find y, according to Equation (57) we have

Then, by making the change of variable

and choosing conveniently the value of

Without being quite formal, it follows that the curve C is given by

Using well known properties of the trigonometric and the hyperbolic functions we can write this equation as follows

Hence, the third step is complete.

Finally,

It is proved that C is a closed curve with elliptical shape. The semi-axes a and b are given by

In this case

Since C is a closed curve and we are considering that

The relative error, according to Equation (20) and taking into account that

is given by

Note that if

By means of the procedure indicated before, it is easily proved that A is the amplitude.

The force and the potential in this case are given by

where

We will assume that the initial conditions are given by

where

In addition, we will choose

Then, since

where according to Equations (4), (6) and (62)

We propose the following solution

where

We have

Let

then we arrive to

In order to solve the integrals given in Equation (67) and (68), we will ask that this expression is a perfect square. It happens if and only if

where we assume that

In this case we obtain

We will assume that

and hence we finally arrive to

On the other hand, according to Equation (69) we have

Then, by Equation (69) it follows that

We can write these expressions using Equation (70) as follows

Hence, by Equations (73), (75) and (76), we finally obtain that Equations (67) and (68) becomes

Let

where we made the change of variables

We can write Equations (77) and (78) as follows

On the other hand, by Equations (5) and (74) we have

Using Equation (70) we arrive to

Finally, by Equations (81), (82), (83) and (84) we can turn Equation (65) into an algebraic equation as follows

This equation can be written as

In order to satisfy this equality for all

Then, Equation (85) becomes

This equation depends only on

In order to satisfy

By Equation (74), the other initial condition

where

Finally, the trajectory equation is satisfied if and only if Equations (71), (86) and (87) hold, provided that

where l is the angular momentum (which is known that is constant).

However, this solution holds only if

By this condition, we can obtain the minimum value of

If

Next we will choose

In the first case, according to Equations (89) and (90),

Using this result, and taking into account that

In the second case, according to Equations (89), (90) and (91),

Using these results it is proved that condition (72) is satisfied.

Finally, using that

the solution of the trajectory equation is given by

where

It is worthwhile to point out that we choose

If

We can see in Equation (93) that

Hence, conditions (8) and (9) are satisfied and then the first step of the algorithm is complete.

We will just choose

In order to find u, on the one hand, according to Equations (63) and (73) we have

By Equation (89) this expression turns out to be

On the other hand, according to Equations (83) and (84) we obtain

According to Equation (71) we arrive to

Finally, according to Equations (13), (95) and (96) we have

Using Equations (89) and (91) we can write this expression as follows

We will choose

Note that

where

Finally, Equation (97) turns out to be

On the other hand, using Equation (88) and the fact that

In addition, by Equations (94) and (98) it is easily proved that

which implies

Finally, by Equations (100) and (101) and using condition (72) we obtain

However, according to Equation (89)

and then we just can write

Using Equations (70), (89) and (91) this equation can be written as follows

where

and

Note that since

Finally we complete the third step.

In this case we will not find the function y since it is not easy to find a function

It is easily proved that the solution is given by

By Equation (98) we can write the solution depending the case as follows

where

Finally,

Finally, we will obtain Kepler’s third law which says that the square of the orbital period is directly proportional to the cube of the semi-major axis.

In Equation (93) we can see that if

Using Equations (89) and (99) we finally arrive to

and then we proved Kepler’s third law.

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

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