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hamiltonian mechanics


hamiltonian mechanics

"The facts are relative but the law is absolute"

Special theory of relativity   |   General theory of relativity   |   Tensor calculus

Quantum mechanics

Schrodinger's equation   |   Matrix mechanics

Lagrange has perhaps done more than any other to give extent and harmony to such deductive researches by showing that the most varied consequences … may be derived from one radical formula, the beauty of the method so suiting the dignity of the results as to make his great work a kind of scientific poem.
W. R. Hamilton

This website is mine. I am a scientist. I am a writer too. I am trying to build this website for mass people and education. My goal is to make everybody aware of science and technology. I have tried my best to share my knowledge and experience here. But at this moment I have to give a lot of effort and money to others , which I can hardly manage. If you like reading my website, I will be happy and if not , please do not go away from it. Ads are displayed on front page. If you click on it, I will get some money. Thus my wrtiting will be worthful. One click can make both of us happy. Your contribution can change the world. If you invest in learning and education , you will be rewarded in future. Thank you

Classical mechanics

Classical physics is based on Newton's laws of motion and universal laws of gravitation. The laws of motion are alaways applicable where motion and accerelation are involved. It is very surprising that some fundamental processes are able to describe our universe completely or at least some parts of it. Newton's laws of motion are some such fundamental facts , which can be explained in a simple way with little mathematics. All the physical laws are more or less related to these laws. Einstein was able to discover relativity because he mastered all the other thoeries prior to him. The laws of motion are as follows:
a) If no external force is applied then a moving body will always move in a stright line and static body will remain static forever. This law holds everywhere and everything obeys it. But we will see later that some modification of this law is needed as Einstein 's theory of relativity suggested. Straight lines will need to be generalized to be geodesics. Ok let us break it down :
Newton thought that a material object is inertial object when no force acting on it so it's inertia opposes the change of its momentum or velocity. So he generalized his first law accordingly. But gravity seems to rule the universe and is everywhere. Force of gravity affect s every object. So it is better to represent gravity as a property of spacetime. So the ide a of geodesic arose. Geodesic is nothing but straight line in curved spacetime. For more details you can find discussion about it in General relativity.
b) The magnitude of the applied force is proprtional to rate of change of momentum of a body and the change of momentum occurs in the direction where the force is applied. This law is very fundamental like other three laws. The concept of force and mass are just defined using this law.
c) Every action has equal and opposite reaction. This is the case with everything. when we sit on a chair, the chair pushes up upward with a force equal and opposite of the force that we exert on the chair. When we fire a rifle the bullet and rifle experiences forces that are equal and opposite of each other. There are numerous other examples that we can give to demonstrate the third law.
Based on these laws of motion Newton gave the idea of his ClockWork universe. If a ball is thrown into space it will fall to earth after some time. If we know the intial position and velocity of the ball we can completely determine its trajectory in space. So Newton thought that if positions of every particle and forces acting on them are known, an intelligent being , given sufficient time, will able to calculate the state of the universe in later time. Past , present and future will be completely determined. This is also known as determinism. The evolution of our universe can be traced forward and bakward in time with certainty. There is no randomness in such determinism. With the advent of quantum mechanics determinism started to fall apart. The physical phenomena in atomic level become random like throwing a dice. A long discussion is needed to understand quantum mechanics, which at the moment best left. Although some kind of determinism is still present in quantum theory. The claasical phenomena is nothing but the statistical average of a large class of quantum phenomena. This fact is exactly a reflection of Bond's correspondence principle.
Another revolution happened in classical physics when Kepler gave his laws of planetary motion. Kepler's law can be derived using Newton's law of gravitation. Kepler first gave mathematical descriptions of planetary motion. He explained planetary motion is three laws :
1) Every planet revolves around the sun in elliptic orbit.
2) Each planet sweeps equal area in equal time. That is, in the plane on which the planets orbit the sun the areas covered by the planet in equal times are equal.
3) Square of the period of the orbit ( T^2 ) is proportional to the cube of the distance D^3. D is the semi-major axis of the ellipse.

Concepts in hamiltonian Mechanics

Classical hamiltonian mechanics is also known as hamiltonian mechanics. William hamilton first formulated a new mechanics which is now called Hamiltonian mechanics , starting from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. In Newtonian mechanics, the time evolution is obtained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time-evolutions of both position and velocity are computed. In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and inserting it in the Hamilton's equations. Hamiltonian and Lagrangan are denoted by H and L which represent energy of a physical system. Hamiltonian H is related to Lagrangian L by certain relation, which are both reformulation of classical mechanics with more flexibility. Hamiltonian is a function of generalized coordinates like position q, momentum p and time t. It is the total energy of a system, which is algebric sum of potential and kinetic energy.

hamiltonian mechanics

These generalized coordinates p and q represent degrees of freedom a system can have. Degrees of freedom is number of independent parameter that uniquely characterize a system. Generalized coordinates fix the configuration of a system so that dynamics of the system can be determined from them. For example, a pendulum swinging in one direction can be modeled with a single degree of freedom, namely, the angle θ. A ball which is rolling on the floor has two degrees of freedom : one is the direction x of translation and other is related to angular velocity of the ball, namely θ . So in this case the system has two types of energy : one is kinetic energy and other is rotational energy.
Time evolution of a system can be determined from two equations related to Hamiltonian H:

hamiltonian mechanics
The total differential of Hamiltonian is given by the relation


Hamiltonian

Thus , total differential is the algebric sum of sum of changes of individual components that changes due to change in each coordinate and total change in time coordinate. In the relaion above we used Legendre transform


hamiltonian mechanics

which gives the relation between hamiltonian and Laglangian. Lagrangian L is a function of position q, derivative of position dq/dt and time t. And it is straight forward to find relationship between Hamiltonian and Lagrangian. Hamiltonian and Lagrangian are both dependable on kinetic and potential energy.


hamiltonian mechanics

In Newtonian mechanics, the time evolution is attained by computing the total force being exerted on each particle of the system, and from Newton's second law, the time-evolutions of both position and velocity are computed. In contrast, in Hamiltonian mechanics, the time evolution is obtained by computing the Hamiltonian of the system in the generalized coordinates and plugging it in the Hamilton's equations. This approach is equivalent to the one used in Lagrangian mechanics. In fact, as is shown below, the Hamiltonian is the Legendre transform of the Lagrangian when holding q and t fixed and defining p as the dual variable, and thus both method give the same equations for the same generalized momentum. The main inspiration to use Hamiltonian mechanics instead of Lagrangian mechanics comes from the symplectic structure of Hamiltonian systems.
While Hamiltonian mechanics can be used to describe simple systems such as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time, its strength is shown in more complex dynamic systems, such as planetary orbits in celestial mechanics. The more degrees of freedom the system possesses, the more complicated its time evolution is and, in most cases, it becomes chaotic.


In quantum mechanics, Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system . It is usually denoted by H, also Ȟ or Ĥ. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of supreme importance in many formulations of quantum theory.


hamiltonian mechanics

The concept of phase space is related to Hamiltonian mechanics. The phase space plot specifies the state of a dynamical system at any given moment. Phase space is the space which axes are determined by the generalized coordinates ( p, q) . So if a system which has six degrees of freedom has N particles, the phase space will have 6N ( 3 position q coordinates and 3 momentum p coordinates for each particle) dimensions. Each state of any such dynamical system is a unique point in phase space. One dimensional phase space is called phase line and two dimensional phase space is called phase plane.

A 6n dimensional phase space of Hamiltonian mechanics will look like this :
phase space

The total number of particles is n. If there is 10 particles the phase space will be 6X10 = 60 dimensional. If 20 , 120 dimensional and so on.

Action

Action is a physical quatity. It is usually expressed as the integral over time when a system evolve through a path in the time interval over which integral is taken. It is numerically equivalent to energy multiplied by time.
action hamiltonian mechanics

L is the lagrangian of the system. Action has many applications in both claasical and modern physics. It can be used to derive many laws of nature. The most useful application is called priciple of least action. This principle states that for a physical process the action is minimum. As an example, when a body moves through two events in spacetime interval the body sees as the time between the events. For a clock which moves without constraints between two events in spaceitme, the time between the events that the clock shows becomes maximum. If the clock were constrainted to travel by some other route and were also present at the same two events, the time would be lesser. This is kind of cosmic laziness. Our universe is , according to this principle, very lazy. This is kind of cosmic boredoom. Everything in the universe tried to follow a path which takes least energy.
principle of least action

The reason that the time is maximum , not minimum is that spacetime interval is always timelike in relativity. Before going into more details slight different formulation of action can be stated: In this formulation action is a functional which takes a function as an argument and returns a scalar:


hamiltonian mechanics

q(t) is generalized coordinate of the system. It is a function of time itself. Generalised coordinates can be called degrees of freedom too. The Lagrangian is a function of q , dq/dt and t. Principle of least action is the condition when the variation of Lagrangian vanishes. In the language of calculus this is similar to finding maximum or minimum values of a function. The maximum or minimum values of a function occurs when it becomes stationary at the value of independent variable. The condition is that first order change of action is zero. In case of ordinary one variable function, the slope of curve becomes zero at the point where the function becomes extreme.

hamiltonian mechanics

When the value of action is minimum for a path that an object follows, the actions corresponding to small paths that make up the whole path are also minimum. Otherwise the total action would not be minimum. Same interpretation can be given for maximum or minimum value for a functional. In case of functional we are interested in the function that extremizes the functional. This is called Hamiltonian Principle. The mathematical expression is
action hamiltonian mechanics
So the condition is that variation of action is zero. First order change of action is zero , to be precise. This condition leads to an interesting equation known as Euler-Lagrange equations. The latter has many applications relating to calculus of variation. For the moment I just mention the equation , which is necessarily a differential equation. The reason is that the equation contains lagrangian which is a function of a function.
euler-lagrange equation
Where f is a function of Y`, and Y. Y , on the other hand is a function of independent variable x. Euler - lagrange equation plays a very important role in deducing many theories of physics. Brachistochrone problem can be solved using this equation. This is a problem of calculus of variation , which states "what is the curve of a bead sliding on a frictionless wire under influence of gravity, that minimizes the time". So the problem is to find the curve of quickest descent.
Johann Bernoulli's solution divides the plane into strips and he assumes that the particle follows a straight line in each strip. The path is then piecewise linear. The problem is to determine the angle of the straight line segment in each strip and to do this he appeals to Fermat's principle , namely that light always follows the shortest possible time of travel. If v is the velocity in one strip at angle α to the vertical and u in the velocity in the next strip at angle β to the vertical then, according to the usual sine law

v/sin α = u/sin β.
hamiltonian mechanics
In the limit, as the strips become infinitely thin, the line segments tend to a curve where at each point the angle the line segment made with the vertical becomes the angle the tangent to the curve makes with the vertical. If v is the velocity at (x, y) and α is the angle the tangent makes with the vertical then the curve satisfies
v/sin α = constant.
Galileo had shown that the velocity v satisfies
v = √(2gy)
(where g is the acceleration due to gravity) and substituting for v gives the equation of the curve as
y/sin α = constant or y = k2 sin2α
Use y' = dy/dx = cot α and sin2α = 1/(1 + cot2α) = 1/(1 + y'2) to get
y(1 + y'2) = 2h
for a constant h (= 1/2 k2)).

The cycloid x(t) = h(t - sin t), y(t) = h(1 - cos t) satisfies this equation. To see this note that

y' = dy/dx = dy/dt . dt/dx = -(sin t)/(1 - cos t)
so
y(1+y'2) = h(1 - cos t)(1+sin2t/(1-cos t)2)

= h(1 - cos t + sin2t/(1-cos t))

= h((1 - cos t)2+ sin2t)/(1 - cos t)

= h(2 - 2cos t)/(1-cos t) = 2h

Classical hamiltonian mechanics is very powerful method used in both theorectical and practical problems. It has saved a lot of labours of physicists . Using classical mechanics of Newton it would be very hard to solve specific problem.

Applications

Simple harmonic oscillator can be analyzed through principles of hamiltonian mechanics. The hamiltonian will be the sum of potential and kinetic energy.

hamiltonian mechanics

The action of such operation on some function u describing the harmonic motion will produce an energy eigen value.
hamiltonian mechanics

Notes

Every triangle satisfies some identities involving lengths of it's and sines and cosines of the angles that it posseses. The law involving sines is called sine law and that involving cosine is called cosine law.
hamiltonian mechanics

FUNCTIONS
In mathematics, a function was originally the idealization of how a varying quantity depends on another varying quantity. For example, the position of a planet is a function of time parameter. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of function was formalized at the end of the 19th century in terms of Cantor's set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that assigns each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the converse domain of the function. If the function is called f, this relation is denoted y = f(x) (read f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).
A function is uniquely represented by its graph which is the set of all ordered pairs (x, f(x)). When the domain and the codomain are sets of numbers, each such pair may be considered as the Cartesian coordinates of a point in the plane. In general, these points form a curve, which is also called the graph of the function. This is a practical representation of the function, which is commonly used everywhere. For example, graphs of functions are commonly used in newspapers for representing the evolution of price indexes and stock market indexes
Functions are widely used in science, and in most fields of mathematics. Their role is so important that it has been said that they are "the central objects of investigation" in most fields of mathematics.
Schematic depiction of a function described metaphorically as a "machine" or "black box" that for each input yields a corresponding output The red curve is the graph of a function, because any vertical line has exactly one crossing point with the curve. A function that associates any of the four colored shapes to its color. The sole purpose of explaining function theory is that hamiltonian mechanics is based on function named hamiltonian.

a function is a relation and conversely a relation is said to be a function. Every relation has a domain and a converse domain. The class of terms which has to some relation with a term or others is called the domain of a relation. The class of terms which has some relation to some term or others is called the converse domain. Both the domain and the converse domain are together called the field of the relation. For example spouse is a relation which domain consists of all the man as its re ferents and all the woman as its converse domain. The term from which a relation proceeds is called the referent and the term to which the relation proceeds is called relata. The relational arithmetic is vital to mathematical logic. Som e terms and properties must now be explained:
A relation is said to be reflextive when there is a term x such that xRx or the relation R hold between the term x with itself.
A relation is symmetrical when there are terms x and y such that if xRy , then yRx
A relation is transitive when a term y exists between x and z such that if xRy and yRz the xRz.
In every kind of relation xRy means relation R hold between x and y. The converse relation R^ holds between y and x. With the defintion of this relation R we can find the relative product of two relations. The relative product of two relation is defined as the logical product of two relations , which is itself a transitive relative. So when xRy and yRz then xRz. For example Grandfather is the relative product of both the relations son and father. So z is grandfather of x means there is some person y so x is the son of y and z is the father of y. A very useful relation in pure mathematics is the one-to-one relation. We all know what this relation is. To put it simply, a relation is one-to-one when there is two terms x and x` such that f(x) != f(x`) . Using one-to-one relation we can define a progression.
A progression is a one-to-one relation so that just one term belongs to domain but not to converse domain and all the domain is just the posterity of this one term.
Motion

Much been written concerning the laws of motion, the possibility of dispensing with Causality in Dynamics, the relativity of motion and other kindred questions. But there are several preliminary questions, of great difficulty and importance, concerning which little has been said. Yet these questions, speaking logically, must be settled before the more complex problems usually discussed can be attacked with any hope of success. Most of the relevant modern philosophical literature will illustrate the truth of these remarks: the theories suggested usually repose on a common dogmatic basis, and can be easily seen to be unsatisfactory. So long as an author confines himself to demolishing his opponents, he is irrefutable; when he constructs his own theory, he exposes himself, as a rule, to a similar demolition by the next author. Under these circumstances, we must seek some different path, whose by-ways remain unexplained. “Back to Newton” is the watchword of reform in this matter. Newton’s scholium to the definitions contains arguments which are unrefuted, and so far as I know, irrefutable: they have been before the world two hundred years, and it is time they were refuted or accepted. Being unequal to the former, I have adopted the latter alternative. The concept of motion is logically subsequent to that of occupying a place at a time, and also to that of change. Motion is the occupation, by one entity, of a continuous series of places at a continuous series of times. Change is the difference, in respect of truth or falsehood, between a proposition concerning an entity and a time T and a proposition concerning the same entity and another time T', provided that the two propositions differ only by the fact that T occurs in the one where T' occurs in the other. Change is continuous when the propositions of the above kind form a continuous series correlated with a continuous series of moments. Change thus always involves (1) a fixed entity, (2) a three-cornered relation between this entity, another entity, and some but not all, of the moments of time. This is its bare minimum. Mere existence at some but not all moments constitutes change on this definition. Consider pleasure, for example. This, we know, exists at some moments, and we may suppose that there are moments when it does not exist. Thus there is a relation between pleasure, existence, and some moments, which does not subsist between pleasure, existence, and other moments. According to the definition, therefore, pleasure changes in passing from existence to nonexistence or vice versâ. This shows that the definition requires emendation, if it is to accord with usage. Usage does not permit us to speak of change except where what changes is an existent throughout, or is at least a class-concept one of whose particulars always exists. Thus we should say, in the case of pleasure, that my mind is what changes when the pleasure ceases to exist. On the other hand, if my pleasure is of different magnitudes at different times, we should say the pleasure changes its amount, though we agreed in Part III that not pleasure, but only particular amounts of pleasure, are capable of existence. Similarly we should say that colour changes, meaning that there are different colours at different times in some connection; though not colour, but only particular shades of colour, can exist. And generally, where both the class-concept and the particulars are simple, usage would allow us to say, if a series of particulars exists at a continuous series of times, that the classconcept changes. Indeed it seems better to regard this as the only kind of change, and to regard as unchanging a term which itself exists throughout a given period of time. But if we are to do this, we must say that wholes consisting of existent parts do not exist, or else that a whole cannot preserve its identity if any of its parts be changed. The latter is the correct alternative, but some subtlety is required to maintain it. Thus people say they change their minds; they say that the mind changes when pleasure ceases to exist in it. If this expression is to be correct, the mind must not be the sum of its constituents. For if it were the sum of all its constituents throughout time, it would be evidently unchanging; if it were the sum of its constituents at one time, it would lose its identity as soon as a former constituent ceased to exist or a new one began to exist. Thus if the mind is anything, and if it can change, it must be something persistent and constant, to which all constituents of a psychical state have one and the same relation. Personal identity could be constituted by the persistence of this term, to which all a person’s states (and nothing else) would have a fixed relation. The change of mind would then consist merely in the fact that these states are not the same at all times.

Infinitesimal calculus

The Infinitesimal Calculus is the traditional name for the differential and integral calculus together, and as such I have retained it; although, as we shall shortly see, there is no allusion to, or implication of, the infinitesimal in any part of this branch of mathematics. The philosophical theory of the Calculus has been, ever since the subject was invented, in a somewhat disgraceful condition. Leibniz himself—who, one would have supposed, should have been competent to give a correct account of his own invention—had ideas, upon this topic, which can only be described as extremely crude. He appears to have held that, if metaphysical subtleties are left aside, the Calculus is only approximate, but is justified practically by the fact that the errors to which it gives rise are less than those of observation.* When he was thinking of Dynamics, his belief in the actual infinitesimal hindered him from discovering that the Calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which, in his philosophy, infinite division was supposed to lead.† And in his mathematical expositions of the subject, he avoided giving careful proofs, contenting himself with the enumeration of rules.‡ At other times, it is true, he definitely rejects infinitesimals as philosophically valid;§ but he failed to show how, without the use of infinitesimals, the results obtained by means of the Calculus could yet be exact, and not approximate. In this respect, Newton is preferable to Leibniz: his Lemmas* give the true foundation of the Calculus in the doctrine of limits, and, assuming the continuity of space and time in Cantor’s sense, they give valid proofs of its rules so far as spatio-temporal magnitudes are concerned. But Newton was, of course, entirely ignorant of the fact that his Lemmas depend upon the modern theory of continuity; moreover, the appeal to time and change, which appears in the word fluxion, and to space, which appears in the Lemmas, was wholly unnecessary, and served merely to hide the fact that no definition of continuity had been given. Whether Leibniz avoided this error, seems highly doubtful: it is at any rate certain that, in his first published account of the Calculus, he defined the differential coefficient by means of the tangent to a curve. And by his emphasis on the infinitesimal, he gave a wrong direction to speculation as to the Calculus, which misled all mathematicians before Weierstrass (with the exception, perhaps, of De Morgan), and all philosophers down to the present day. It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations, as is natural, are as yet little known among philosophers, except in France.† Philosophical works on the subject, such as Cohen’s Princip der Infinitesimalmethode und seine Geschichte,‡ are vitiated, as regards the constructive theory, by an undue mysticism, inherited from Kant, and leading to such results as the identification of intensive magnitude with the extensive infinitesimal. § I shall examine in the next chapter the conception of the infinitesimal, which is essential to all philosophical theories of the Calculus hiterto propounded. For the present, I am only concerned to give the constructive theory as it results from modern mathematics.
The differential coefficient depends essentially upon the notion of a continuous function of a continuous variable. The notion to be defined is not purely ordinal; on the contrary, it is applicable, in the first instance, only to series of numbers, and thence, by extension, to series in which distances or stretches are numerically measureable. But first of all we must define a continuous function.
We have already seen (Chap. 32) what is meant by a function of a variable, and what is meant by a continuous variable (Chap. 36). If the function is one-valued, and is only ordered by correlation with the variable, then, when the variable is continuous, there is no sense in asking whether the function is continuous; for such a series by correlation is always ordinally similar to its less than a, will differ by less than ε; in popular language, the value of the function does not make any sudden jumps as x approaches a from the left. Under similar circumstances, f(x) will have a limit as it approaches a from the right. But these two limits, even when both exist, need not be equal either to each other or to f(a), the value of the function when x = a. The precise condition for a determinate finite limit may be thus stated:*

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This website is mine. I am a self-claimed scientist. I am a writer too. I am trying to build this website for mass people and education. My goal is to make everybody aware of science and technology. I have tried my best to share my knowledge and experience here. But at this moment I have to give a lot of effort and money to others , which I can hardly manage. If you like reading my website, I will be happy and if not , please do not go away from it. Ads are displayed on front page. If you click on it, I will get some money. Thus my wrtiting will be worthful. One click can make both of us happy. Your contribution can change the world. If you invest in learning and education , you will be rewarded in future. Thank you
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Reference materials:

Lagrangian Mecahnics and Hamiltonian
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
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