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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

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 mehanics

These generalized coordinates p and q represent degrees of freedom a system can have. Degrees of freedom is number of independent parameters that uniquely characterize a system. Generalized coordinates fix 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 function 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

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 will look like this :
phase space

The total number of particles is n. Sif there is 10 particles the phase space will be 6X10 = 60 dimensional. If 20 , 120 dimensional and so on. Here is a short video explaining classical hamiltonian mechanics

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

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 , by this principle, lazy.
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. 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
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
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.

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.

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.
sine rule

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