Physics Lournal

Hamiltonian Formalism

The Hamiltonian Formalism describes mechanic trajectories in phase-space (qualitative space), while Lagrangian Mechanics use trajectories in configuration-space.

phase

A Phase Space is a mathematical tool that allows us to grasp important aspects of complicated systems.

Intuitive

The choice of variables (coordinates), can make a problem in mechanics much more tractable.

A lever can be modeled as as a near infinity of atoms, or as a lever arm, with one position coordinate (angle, zero), and the condition of rigidity (perhaps applied to the infinite atoms to generate the illusory solid).

Generally we assume the second to be more efficient, but Hamilton, realized the most sparse description is not always the best, and additional coordinates may yield different insights.

Hamilton brought in a doubling of number coordinates in any mechanics, but it’s important not to think of this as doubling of a number, i.e going from 5-10, but a doubling of dimensionality, as in going from children, to boys to girls.

This is an amazing analogy.

Hamiltons Allegory of The Course:

Consider the difference between these two situations.

On Saturday, we drive 100 balls of golf, and then go to the hole, and count up the number of balls we found there.

We count the number of balls after arrival here.

We can reconstruct the history of all of our driven balls here.

On Sunday, we don’t drive any balls, but we calculate the trajectory of a 100, given conditions of angle and speed, and determine whether or not the golf ball would land.

We calculate the trajectory of a given ball, and know where it arrives and when.

Here, we forego the ability to construct history, in exchange for a proportion corresponding to the number of arrivals.

These two results can be said to exist in different ‘spaces’.

Saturday, is phase-space, whereas Sunday is configuration-space.

phase-space allows us to obtain qualitative information, about more golfballs, in one go. Less from more.

Deduction in physics!

Concrete

One novel ingredient of the Hamiltonian lies in the variables one uses to describe a physical system.

Prior to this, the positions of particles were primary, and the velocities were simply the change of position as a function of time.

Recall that when specifying the initial state of a Newtonian system, we need the positions and velocities of all particles, in order that the subsequent behavior be determinate.

With the Hamiltonian, we select the momenta of the particles rather than velocity.

momenta is plural for momentum.

The momentum of a particle is $velocity\; x\; mass.$

This seems a small change, but the position and momentum of each particle are treated as independent quantities, thus one assumes, at first, that the momenta of particles have nothing to do with the rates of change of their position variables.

In this formulation, we have now have two equations:

One equation that tells us how the momenta of particles change with time, and the other tells us how the positions change with time.

In each case, the rates of change are determined by the various values of the respective variables.

Roughly, the first set of Hamiltonian equations states Newton's second law, that the rate of change of momentum $->$ force, with the second set telling us what the momenta actually are, in terms of the velocities.

Rate of change of position $->\; momentum + mass.$

The laws of motion of Galilei-Newton were described in terms of accelerations, rates of change of position (sometimes called second-order-equations).

Now we need only talk about the rates of change of things, rather than the rates of change of the rates of change of things.

All of these are derived from the Hamiltonian function, $H$ , which expresses the total energy of a system in terms of all of the position and momentum variables.

The formula provides an elegant and symmetrical description of mechanics.

$H\; :=\;$ $(\Large \dot{p}_i = - \frac{\partial H}{\partial x_i}, \quad \dot{x}_i = \frac{\partial H}{\partial p_i} ).$

The dot appearing on the left hand side of the equations indicates that the rate of change is respective of time.

Here, the index $i$ is distinguishing different momentum coordinates: $p_1, p_2, p_3, p_4,\ldots$

and different position coordinates $x_1, x_2, x_3, x_4,\ldots$ .

For any $n$ unconstrained particles, we shall have $3n$ momentum coordinates, and $3n$ position coordinates (one for each spatial dimension).

The $\partial$ symbol refers to partial-differentiation.

This is the partial derivative of f with respect to x. Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "partial".

The $x_i$ and $p_i$ coordinates aren't restricted to ordinary Cartesian coordinates, with $x_i `s$ , being ordinary distances, measured in 3 different directions at right angles.

Referenced in

Classical Mechanics

Hamiltonian Formalism

On Dirac: Glory from Perseverance

The Hamiltonian, is a mathematical operator that describes the sum total of energies in a system, both kinetic and potential.

Formalisms

Hamiltonian Formalism: Where we describe the evolution of our system as a trajectory in phase-space.

Hamiltonian Formalism

Hamiltonian Formalism

The Hamiltonian Formalism describes mechanic trajectories in phase-space (qualitative space), while Lagrangian Mechanics use trajectories in configuration-space.

A Phase Space is a mathematical tool that allows us to grasp important aspects of complicated systems.

Intuitive

The choice of variables (coordinates), can make a problem in mechanics much more tractable.

A lever can be modeled as as a near infinity of atoms, or as a lever arm, with one position coordinate (angle, zero), and the condition of rigidity (perhaps applied to the infinite atoms to generate the illusory solid).

Generally we assume the second to be more efficient, but Hamilton, realized the most sparse description is not always the best, and additional coordinates may yield different insights.

Hamilton brought in a doubling of number coordinates in any mechanics, but it’s important not to think of this as doubling of a number, i.e going from 5-10, but a doubling of dimensionality, as in going from children, to boys to girls.

The further criteria, however, is that these two variables (boys and girls), must be dynamically related.

Hamiltons Allegory of The Course:

Consider the difference between these two situations.

On Saturday, we drive 100 balls of golf, and then go to the hole, and count up the number of balls we found there.

On Sunday, we don’t drive any balls, but we calculate the trajectory of a 100, given conditions of angle and speed, and determine whether or not the golf ball would land.

These two results can be said to exist in different ‘spaces’.

Saturday, is phase-space, whereas Sunday is configuration-space.

phase-space allows us to obtain qualitative information, about more golfballs, in one go. Less from more.

Deduction in physics!

Concrete

One novel ingredient of the Hamiltonian lies in the variables one uses to describe a physical system.

Prior to this, the positions of particles were primary, and the velocities were simply the change of position as a function of time.

Recall that when specifying the initial state of a Newtonian system, we need the positions and velocities of all particles, in order that the subsequent behavior be determinate.

With the Hamiltonian, we select the momenta of the particles rather than velocity.

momenta is plural for momentum.

The momentum of a particle is velocity x mass.velocity\; x\; mass.velocityxmass.

This seems a small change, but the position and momentum of each particle are treated as independent quantities, thus one assumes, at first, that the momenta of particles have nothing to do with the rates of change of their position variables.

In this formulation, we have now have two equations:

One equation that tells us how the momenta of particles change with time, and the other tells us how the positions change with time.

In each case, the rates of change are determined by the various values of the respective variables.

Roughly, the first set of Hamiltonian equations states Newton's second law, that the rate of change of momentum −>->−> force, with the second set telling us what the momenta actually are, in terms of the velocities.

The laws of motion of Galilei-Newton were described in terms of accelerations, rates of change of position (sometimes called second-order-equations).

Now we need only talk about the rates of change of things, rather than the rates of change of the rates of change of things.

All of these are derived from the Hamiltonian function, HHH, which expresses the total energy of a system in terms of all of the position and momentum variables.

The formula provides an elegant and symmetrical description of mechanics.

The dot appearing on the left hand side of the equations indicates that the rate of change is respective of time.

Here, the index iii is distinguishing different momentum coordinates: p1,p2,p3,p4,…p_1, p_2, p_3, p_4,\ldotsp1​,p2​,p3​,p4​,…

and different position coordinates x1,x2,x3,x4,…x_1, x_2, x_3, x_4,\ldotsx1​,x2​,x3​,x4​,….

For any nnn unconstrained particles, we shall have 3n3n3n momentum coordinates, and 3n3n3n position coordinates (one for each spatial dimension).

The ∂\partial∂ symbol refers to partial-differentiation.

This is the partial derivative of f with respect to x. Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "partial".

The xix_ixi​ and pip_ipi​ coordinates aren't restricted to ordinary Cartesian coordinates, with xi‘sx_i `sxi​‘s, being ordinary distances, measured in 3 different directions at right angles.

Referenced in

Classical Mechanics

Hamiltonian Formalism

On Dirac: Glory from Perseverance

The Hamiltonian, is a mathematical operator that describes the sum total of energies in a system, both kinetic and potential.

Formalisms

Hamiltonian Formalism: Where we describe the evolution of our system as a trajectory in phase-space.

The momentum of a particle is $velocity\; x\; mass.$

Roughly, the first set of Hamiltonian equations states Newton's second law, that the rate of change of momentum $->$ force, with the second set telling us what the momenta actually are, in terms of the velocities.

All of these are derived from the Hamiltonian function, $H$ , which expresses the total energy of a system in terms of all of the position and momentum variables.

Here, the index $i$ is distinguishing different momentum coordinates: $p_1, p_2, p_3, p_4,\ldots$

and different position coordinates $x_1, x_2, x_3, x_4,\ldots$ .

For any $n$ unconstrained particles, we shall have $3n$ momentum coordinates, and $3n$ position coordinates (one for each spatial dimension).

The $\partial$ symbol refers to partial-differentiation.

The $x_i$ and $p_i$ coordinates aren't restricted to ordinary Cartesian coordinates, with $x_i `s$ , being ordinary distances, measured in 3 different directions at right angles.