“This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent powerful Being” – Sir. Issac Newton
When I was seven years old, I had the notion that all planets dance around the sun along a wavy orbit (see Fig. 1).
Many years later, I took on a challenge to show mathematically the orbit of my ‘dancing planet’ . This post is a long overdue report of my journey.
Shown in Fig. 2 is the sun and a planet in a x-y-z coordinate system. The sun is at the origin. The moving planet’s position is being described by .
According to Newton’s theory, the gravitational force sun exerts on the planet is
where is the gravitational constant, the mass of the sun and planet respectively. .
By Newton’s second law of motion,
(0-3) (0-2) yields
it must be true that
where is a constant.
where are constants.
If then by the following well known theorem in Analytic Geometry:
“If A, B, C and D are constants and A, B, and C are not all zero, then the graph of the equation Ax+By+Cz+D=0 is a plane“,
(0-7) represents a plane in the x-y-z coordinate system.
For , we have
where is a constant. Simply put,
Hence, (0-7) still represents a plane in the x-y-z coordinate system (see Fig. 3(a)).
The implication is that the planet moves around the sun on a plane (see Fig. 4).
By rotating the axes so that the orbit of the planet is on the x-y plane where (see Fig. 3), we simplify the equations (0-1)-(0-3) to
It follows that
Integrate with respect to ,
where is a constant.
We can also re-write (0-6) as
where is a constant.
Using polar coordinates
we obtain from (1-2) and (1-3) (see Fig. 5):
If the speed of planet at time is then from Fig. 6,
Suppose at , the planet is at the greatest distance from the sun with and speed . Then the fact that attains maximum at implies . Therefore, by (1-4) and (1-5),
We can now express (1-4) and (1-5) as:
By chain rule,
Notice that .
By doing so, (1-14) can be expressed as
Take the first case,
Integrate it with respect to gives
where is a constant.
By (1-11), it is
Solving (1-15) for yields
Studies in Analytic Geometry show that for an orbit expressed by (1-16), there are four cases to consider depend on the value of :
We can rule out parabolic and hyperbolic orbit immediately for they are not periodic. Given the fact that a circle is a special case of an ellipse, it is fair to say:
The orbit of a planet is an ellipse with the Sun at one of the two foci.
In fact, this is what Kepler stated as his first law of planetary motion.
from which we obtain
This is an ellipse. Namely, the result of rotating (1-16) by hundred eighty degrees or assuming attains its minimum at .
The second case
can be written as
Integrate it with respect to yields
from which we can obtain (1-16) and (1-17) again.
Over the time duration , the area a line joining the sun and a planet sweeps an area (see Fig. 9).
is a constant. Therefore,
A line joining the Sun and a planet sweeps out equal areas during equal intervals of time.
This is Kepler’s second law. It suggests that the speed of the planet increases as it nears the sun and decreases as it recedes from the sun (see Fig. 10).
Furthermore, over the interval , the period of the planet’s revolution around the sun, the line joining the sun and the planet sweeps the enire interior of the planet’s elliptical orbit with semi-major axis and semi-minor axis . Since the area enlosed by such orbit is (see “Evaluate a Definite Integral without FTC“), setting in (2-1) to gives
Research on rocket flight performance has shown that typical single-stage rockets cannot serve as the carrier vehicle for launching satellite into orbit. Instead, multi-stage rockets are used in practice with two-stage rockets being the most common. The jettisoning of stages allows decreasing the mass of the remaining rocket in order for it to accelerate rapidly till reaching its desired velocity and height.
Optimizing flight performance is a non-trivial problem in the field of rocketry. This post examines a two-stage rocket flight performance through rigorous mathematical analysis. A Computer Algebra System (CAS) is employed to carry out the symbolic computations in the process. CAS has been proven to be an efficient tool in carry out laborious mathematical calculations for decades. This post reports on the process and the results of using Omega CAS explorer, a Maxima based CAS to solve this complex problem.
A two-stage rocket consists of a payload propelled by two stages of masses (first stage) and (second stage), both with structure factor . The exhaust speed of the first stage is , and of second stage . The initial total mass, is fixed. The ratio is small.
Based on Tsiolkovsky’s equation, we derived the multi-stage rocket flight equation . For a two-stage rocket, the final velocity can be calculated from the following:
Let , so that (1) becomes
We seek an appropriate value of that maximizes .
Consider as a function of , its derivative is computed (see Fig. 1)
We have .
That is, .
As shown in Fig. 2, can be expressed as
Notice that .
Solving for gives two solutions (see Fig. 3)
We rewrite the expression under the square root in and as a quadratic function of : and compute (see Fig. 4)
If , . It implies that is positive since . When , where is still positive since as a result of , the zero point of function is .
The expression under the square root is positive means both and are real-valued and (see Fig. 5), i.e., .
From (3) where , we deduce the following:
For all , if then
For all , if then
For all , if then
Moreover, from Fig. 6,
Since the expression in the numerator of , namely
It follows that
The implication is that has at least one zero point between and .
However, if both and , the two known zero points of are between and , by () and (), must be positive, which contradicts (4). Therefore, must have only one zero point between and .
We will proceed to show that the only zero lies between and is .
There are two cases to consider.
Case 1 () since and . But this contradicts (). Therefore, must not be positive.
Case 2 () The denominator of is negative since . However, , the terms not under the square root in the numerator of can be expressed as . This is a positive expression since implies that . Therefore, .
The fact that only lies between and , together with () and () proves that is where the global maximum of occurs.
can be simplified to a Taylor series expansion (see Fig. 7)
The result produced by CAS can be written as . However, it is incorrect as would suggest that is a negative quantity when is small.
To obtain a correct Taylor series expansion for , we rewrite as first where
Its first order Taylor series is then computed (see Fig. 8)
The first term of the result can be written as . Bring the value of into the result, we have:
To compute from (6) , we substitute for in (2) and compute its Taylor series expansion about (see Fig. 9 )
Writing its first term as and substituting the value yields:
It is positive when is small.
We have shown the time-saving, error-reduction advantages of using CAS to aid manipulation of complex mathematical expressions. On the other hand, we also caution that just as is the cases with any software system, CAS may contain software bugs that need to be detected and weeded out with a well- trained mathematical mind.
This post is an introduction to deterministic models of infectious diseases and their Computer Algebra-Aided mathematical analysis.
We assume the followings for the simplistic SI model:
(A1-1) The population under consideration remains a constant.
(A1-2) The population is divided into two categories: the infectious and the susceptible. Their percentages are denoted by and respectively. At .
(A1-3) The infectious’ unit time encounters with other individuals is . Upon an encounter with the infectious, the susceptible becomes infected.
When a infectious host have encounters with the population, susceptible become infected. There are infectious in total at time . It means that within any time interval , the infectious will increase by . i.e.,
Cancelling out the ‘s,
Deduce further from (A1-2) () is that
Let’s examine (1-1) qualitatively first.
We see that the SI model has two critical points:
This indicates that in the presence of any initial infectious hosts, the entire population will be infected in time. The rate of infection is at its peak when .
Fig. 1-2 confirms that the higher the number of initial infectious hosts(), the sooner the entire population becomes infected ()
The SI model does not take into consideration any medical practice in combating the spread of infectious disease. It is pessimistic and unrealistic.
An improved model is the SIR model. The assumptions are
(A2-1) See (A1-1)
(A2-2) See (A1-2)
(A2-3) See (A1-3)
(A2-4) Number of individuals recovered from the disease in unit time is . The recovered are without immunity. They can be infected again.
By (A2-1) – (A2-4), the modified model is
The new model has two critical points:
Without solving (2-1), we extract from it the following qualitative behavior:
The cases are illustrated by solving (2-1) analytically using Omega CAS Explorer (see Fig. 2-1,2-2,2-3)
Fig. 2-1 approaches asymptotically.
Fig. 2-2 approaches asymptotically.
Fig. 2-3 approaches asymptotically.
Fig. 2-4 ‘s monotonicity depends on .
Fig. 2-4 shows that for , if , then increases on a convex curve. Otherwise, increases on a concave curve first. The curve turns convex after reaches . However, monotonically decreases along a concave curve.
Fig. 2-5 monotonically decrease.
Fig 2-5 illustrates the case .
We also have:
Fig. 2-6 monotonically decrease.
From these results we may draw the following conclusion:
If , the monotonicity of depends on the level of . Otherwise (), will decrease and approach to since the rate of recovery from medical treatment is at least on par with the rate of infection
This model is only valid for modeling infectious disease with no immunity such as common cold, dysentery. Those who recovered from such disease become the susceptible and can be infected again.
However, for many disease such as smallpox, measles, the recovered is immunized and therefore, falls in a category that is neither infectious nor susceptible. To model this type of disease, a new mathematical model is needed.
Enter the Kermack-McKendrick model of infectious disease with immunity.
There are three assumptions:
(A3-1) The total population does not change.
(A3-2) Let and denote the percentage of the infectious, susceptible and recovered respectively. At . The recovered are the individuals who have been infected and then recovered from the disease. They will not be infected again or transmit the infection to others.
(A3-3) is the unit time number of encounters with the infectious, the unit time recoveries from the disease.
For the recovered, we have
This system of differential equations appears to defy any attempts to obtain an analytic solution (i.e., no solution can be expressed in terms of known function).
Numerical treatments for two sets of given and are depicted in Fig. 3-1 and Fig. 3-2.
However, it is only the rigorous analysis in general terms gives the correct interpretations and insights into the model.
To this end, we let
It has the following qualitatives:
[3-2] reaches its maximum.
The analytical solution to
(see Fig. 3-3) is
is a critical point of (3-1-1) and (3-1-2).
all points on the s-axis of the s-i phase plane are critical points of (3-1-1) and (3-1-2).
By a theorem of qualitative theory of ordinary differential equations (see Fred Brauer and John Nohel: The Qualitative Theory of Ordinary Differential Equations, p. 192, Lemma 5.2),
Since , together, (3-5) and implies
Clearly, is a critical point of (3-1-1) and (3-1-2). Lemma 5.2 thus ensures
It follows that
To the list ([3-1]-[3-3]) , we now add:
And so, for all ,
In fact, for all finite ,
Thus, the orbits of (3-1-1) and (3-1-2) have the form illustrated in Fig. 3-7.
What we see is that as time advances, moves along the curve (3-3) in the direction of decreasing Consequently, if is less than , then decreases monotonically to , and decreases to . Therefore, if a small group of infectious is introduced into the population with the susceptibles , with , the disease will die out rapidly. On the other hand, if is greater than , then increases as decreases to . Only after attaining its maximum at starts to decrease when the number of susceptibles falls below the threshold value
We therefore conclude:
An epidemic will occur only if the number of susceptibles in a population exceeds the threshold value .
It means a larger is preferred.
To increase , the recovery rate is boosted through adequate medical care and treatment. Meanwhile, is reduced by quarantine and social distancing.
In addition to increase , we can also decrease through immunizing the population.
If the number of susceptibles is initially greater than, but close to the threshold value :
and is very small compared to :
we can estimate the number of individuals who ultimately contracted the disease.
From (3-1-4), we have
Given (3-9), we deduce from it that
(3-1-2) / (3-1-3) gives
After substituting (3-15) in (3-1-3),
In view of the fact that
we approximate the term in (3-16) with a Taylor expression up to the second order (see Fig. 3-9)
The result is an approximation of equation (3-16):
It can be solved analytically (see Fig. 3-10).
As a result,
It follows from
that (3-18) yields
Namely, the size of the epidemic is roughly . Consequently, by (3-13),
The above analysis leads to the following threshold theorem of epidemiology:
(a) An epidemic occurs if and only if exceeds the threshold .
(b) If and , then after the epidemic, the number of susceptible individuals is reduced by an amount approximately , namely, .
We can also obtain (b) without solving for :
From (3-3), as ,
When is small compared to (see (3-17)), we approximate with a truncated Taylor series after two terms. Namely,
Solving for yields
Exercise-1 For the Kermack-McKendrick model, show that ,