Hamiltonian mechanics

See also classical Hamiltonian system.

The states of a system of Classical Mechanics are determined by generalized variables and their derivatives. In other words, we work in the tangent bundle of the configuration space, or the jet bundle of order 1. That is part of the formulation of Lagrangian Mechanics. The Legendre transform allows us to move everything to the cotangent bundle (phase space). The states of the system are determined by the original variables and generalized momenta. It may seem cumbersome, but it has advantages in geometric interpretation. The systems (dynamical systems) with this origin are called classical Hamiltonian systems. Some of them are called integrable systems. Important example: harmonic oscillator.

The Hamiltonian mechanics formalism can be summarized as follows (modified from "Review of CM and QM by Terence Tao:

1. The physical system has a phase space $Ω$ of states $\vec{x}$ (often parameterized by position variables $q$ and momentum variables $p$ ), which mathematically corresponds to a symplectic manifold with a symplectic form $ω$ (e.g., $ω = d p \land d q$ for position and momentum coordinates).

2. The complete state of the system at any time $t$ is represented by a point $\vec{x} (t)$ in the phase space $Ω$ .

3. Every physical observable $A$ (e.g., energy, momentum, position) is associated with a function $A$ that maps the phase space $Ω$ to the range of the observable (e.g., for real observables, $A$ maps $Ω$ to $R$ ). Measuring the observable $A$ at time $t$ yields the measurement $A (\vec{x} (t))$ .

4. The Hamiltonian $H : Ω \to R$ is a special observable that governs the evolution of the state $\vec{x} (t)$ over time through Hamiltonian equations of motion. In terms of position and momentum coordinates $\vec{x} (t) = (q_{i} (t), p_{i} (t))_{i = 1}^{n}$ , these equations are given by:

\frac{\partial p_{i}}{\partial t} = - \frac{\partial H}{\partial q_{i}}, \frac{\partial q_{i}}{\partial t} = \frac{\partial H}{\partial p_{i}} .

More abstractly, using the symplectic form $ω$ , the equations of motion can be written as:

\frac{\partial \vec{x} (t)}{\partial t} = - \nabla_{ω} H (\vec{x} (t)), (2)

where $\nabla_{ω} H$ is the symplectic gradient of $H$ .

Hamilton's equations of motion can also be expressed in a dual form using observables $A$ as Poisson's equations of motion:

\frac{\partial A (\vec{x} (t))}{\partial t} = - {H, A} (\vec{x} (t)),

where ${H, A} = \nabla_{ω} H \cdot \nabla A$ is the induced Poisson bracket (here we consider an arbitrary Riemannian metric defined on $Ω$ in order to define $\nabla A$ and the dot product. See relation symplectic form and Riemannian metric). In a more abstract form, Poisson's equation can be written as:

\frac{\partial A}{\partial t} = - {H, A}, (3)

where ${H, A}$ represents the Poisson bracket.

In the formalism above, we assume the system is in a pure state at each time $t$ , occupying a single point $\vec{x} (t)$ in phase space. However, mixed states can also be considered, where the state of the system at time $t$ is described by a probability distribution $ρ (t, \vec{x}) d \vec{x}$ on the phase space. Measuring an observable $A$ at time $t$ becomes a random variable, and its expectation $⟨ A ⟩$ is given by:

⟨ A ⟩ (t) = \int_{Ω} A (\vec{x}) ρ (t, \vec{x}) d \vec{x}, (4)

The equation of motion for a mixed state $ρ$ is given by the advection equation:

\frac{\partial ρ}{\partial t} = div (ρ \nabla_{ω} H)

using the same vector field $- \nabla_{ω} H$ as in equation (2). This equation can also be derived from equations (3), (4), and a duality argument.

Pure states can be seen as a special case of mixed states, where the probability distribution $ρ (t, \vec{x}) d \vec{x}$ is a Dirac delta $δ_{\vec{x} (t)} (\vec{x})$ . Mixed states can be thought of as continuous averages of pure states, or equivalently, the space of mixed states is the convex hull of the space of pure states.

Consider a system of 2 particles with a joint phase space $Ω = Ω_{1} \times Ω_{2}$ , where $Ω_{1}$ and $Ω_{2}$ are the individual one-particle phase spaces. A pure joint state is represented by a point $x = ({\vec{x}}_{1}, {\vec{x}}_{2})$ in $Ω$ , where ${\vec{x}}_{1}$ and ${\vec{x}}_{2}$ represent the states of the first and second particles, respectively. If the joint Hamiltonian $H : Ω \to R$ splits as:

H ({\vec{x}}_{1}, {\vec{x}}_{2}) = H_{1} ({\vec{x}}_{1}) + H_{2} ({\vec{x}}_{2}),

then the equations of motion for the first and second particles are completely decoupled, with no interactions between them. However, in practice, the joint Hamiltonian contains coupling terms between ${\vec{x}}_{1}$ and ${\vec{x}}_{2}$ that prevent total decoupling. For instance, the joint Hamiltonian may be:

H ({\vec{x}}_{1}, {\vec{x}}_{2}) = \frac{| p_{1} |^{2}}{2 m_{1}} + \frac{| p_{2} |^{2}}{2 m_{2}} + V (q_{1} - q_{2}),

where ${\vec{x}}_{1} = (q_{1}, p_{1})$ and ${\vec{x}}_{2} = (q_{2}, p_{2})$ are position and momentum coordinates, $m_{1}$ and $m_{2}$ are mass constants, and $V (q_{1} - q_{2})$ represents the interaction potential depending on the spatial separation $q_{1} - q_{2}$ between the particles.

Similarly, a mixed joint state is a joint probability distribution $ρ ({\vec{x}}_{1}, {\vec{x}}_{2}) d {\vec{x}}_{1} d {\vec{x}}_{2}$ on the product state space. To obtain the (mixed) state of an individual particle, we consider marginal distributions such as:

ρ_{1} ({\vec{x}}_{1}) = \int_{Ω_{2}} ρ ({\vec{x}}_{1}, {\vec{x}}_{2}) d {\vec{x}}_{2}

for the first particle or

ρ_{2} ({\vec{x}}_{2}) = \int_{Ω_{1}} ρ ({\vec{x}}_{1}, {\vec{x}}_{2}) d {\vec{x}}_{1}

for the second particle. For $N$ -particle systems, if the joint distribution of $N$ distinct particles is given by $ρ ({\vec{x}}_{1}, \dots, {\vec{x}}_{N}) d {\vec{x}}_{1} \dots d {\vec{x}}_{N}$ , then the distribution of the first particle is:

ρ_{1} ({\vec{x}}_{1}) = \int_{Ω_{2} \times \dots \times Ω_{N}} ρ ({\vec{x}}_{1}, {\vec{x}}_{2}, \dots, {\vec{x}}_{N}) d {\vec{x}}_{2} \dots d {\vec{x}}_{N},

and the distribution of the first two particles is:

ρ_{12} ({\vec{x}}_{1}, {\vec{x}}_{2}) = \int_{Ω_{3} \times \dots \times Ω_{N}} ρ ({\vec{x}}_{1}, {\vec{x}}_{2}, \dots, {\vec{x}}_{N}) d {\vec{x}}_{3} \dots d {\vec{x}}_{N},

and so on.

A typical Hamiltonian for such systems can take the form:

H ({\vec{x}}_{1}, \dots, {\vec{x}}_{N}) = \sum_{j = 1}^{N} \frac{| p_{j} |^{2}}{2 m_{j}} + \sum_{1 \leq j < k \leq N} V_{j k} (q_{j} - q_{k}),

which combines single-particle Hamiltonians $H_{j}$ and interaction perturbations. When momenta $p_{j}$ and masses $m_{j}$ are normalized to be of size $O (1)$ and the potential $V_{j k}$ has an average value (i.e., $L^{1}$ norm) of $O (1)$ , the first sum has size $O (N)$ and the second sum has size $O (N^{2})$ . In order to balance the two components and obtain more interesting limiting dynamics as $N \to \infty$ , we introduce a normalization factor of $\frac{1}{N}$ on the right-hand side, resulting in the Hamiltonian:

$H ({\vec{x}}_{1}, \dots, {\vec{x}}_{N}) = \sum_{j = 1}^{N} \frac{| p_{j} |^{2}}{2 m_{j}} + \frac{1}{N} \sum_{1 \leq j < k \leq N} V_{j k} (q_{j} - q_{k}) .$

Now consider a system of $N$ indistinguishable particles, where all the state spaces $Ω_{1} = \dots = Ω_{N}$ are identical, and observables (including the Hamiltonian) are symmetric functions of the product space $Ω = Ω_{1}^{N}$ (i.e., invariant under the action of the symmetric group $S_{N}$ ). In this case, we can average over the symmetric group (without affecting physical observables) and assume that all mixed states $ρ$ are also symmetric. However, this comes at the cost of mostly giving up pure states $({\vec{x}}_{1}, \dots, {\vec{x}}_{N})$ , as they are symmetric only in exceptional cases where ${\vec{x}}_{1} = \dots = {\vec{x}}_{N}$ .

A typical example of a symmetric Hamiltonian is:

$H ({\vec{x}}_{1}, \dots, {\vec{x}}_{N}) = \sum_{j = 1}^{N} \frac{| p_{j} |^{2}}{2 m} + \frac{1}{N} \sum_{1 \leq j < k \leq N} V (q_{j} - q_{k}),$

where $V$ is an even function (implying all particles have the same individual Hamiltonian and interact with other particles using the same potential). In many physical systems, it is natural to consider short-range interaction potentials, where the interaction between $q_{j}$ and $q_{k}$ is localized to the region $q_{j} - q_{k} = O (r)$ for some small $r$ . This is modeled by Hamiltonians of the form:

H ({\vec{x}}_{1}, \dots, {\vec{x}}_{N}) = \sum_{j = 1}^{N} H ({\vec{x}}_{j}) + \frac{1}{N} \sum_{1 \leq j < k \leq N} \frac{1}{r^{d}} V (\frac{{\vec{x}}_{j} - {\vec{x}}_{k}}{r}),

where $d$ is the ambient dimension of each particle (typically 3 in physical models). The factor of $\frac{1}{r^{d}}$ normalizes the interaction potential's $L^{1}$ norm to be $O (1)$ . An interesting limit arises when $r$ approaches zero as $N$ goes to infinity with $r = N^{- β}$ , where $β$ is some power law. For example, one can consider $N$ particles of "radius" $r$ bouncing around in a box, which is a basic model for classical gases.

A symmetric mixed state example is a factored state:

$ρ ({\vec{x}}_{1}, \dots, {\vec{x}}_{N}) = ρ_{1} ({\vec{x}}_{1}) \dots ρ_{1} ({\vec{x}}_{N}),$

where $ρ_{1}$ is a single-particle probability density function. In the absence of interaction terms in the Hamiltonian, the factored state property is preserved by Hamilton's equations of motion, with $ρ_{1}$ evolving according to the one-particle equation. However, with interactions, the factored nature may be lost over time.

Old stuff

Symplectic form

See symplectic form.

Hamiltonian vector fields

See Hamiltonian vector fields.

This symplectic form has a main use: to convert the differential of the hamiltonian, $d H$ , into a vector field called the symplectic gradient: the only vector field $X_{H}$ such that $ω (X_{H}, -) = d H$ . If we have also a Riemannian metric $g$ , it turns out that this symplectic gradient and the usual gradient are orthogonal: remember that given a function $H$ the gradient is the only vector field $\nabla H$ such that $d H = g (\nabla H, -)$ . Therefore

g (\nabla H, X_{H}) = d H (X_{H}) = ω (X_{H}, X_{H}) = 0

It's important to observe that:

1. $i_{X_{H}} ω = d H$

2. $L_{X_{H}} ω = 0$ . It is proven by Cartan's formula.

3. $L_{X_{H}} ω^{n} = 0$ . This $2 n$ - form is a volume form. So the flow over the Hamiltonian vector field conserve the volume. This fact is known as Liouville theorem.

More things about:

$T^{*} Q$ is a space that represents the states of the system. The function $H : T^{*} Q \mapsto R$ determines the dynamic of the system. $H$ produces a vector field, $X$ , the symplectic gradient, that yields the dynamic itself. Its flow, $ϕ$ , is again the dynamic.

$X$ is tangent to hypersurfaces $H = c t e$ :

X (H) = d H (X) = i_{X} (ω) (X) = 0

(maybe with a minus sign?)

So $H$ is constant along the $ϕ$ curves:

X (H) = \frac{d}{d t} H \circ ϕ

The one-parameter group $ϕ$ is the evolution of the system. This privileged parameter $t$ will be called time.

Imagine other function $F : T^{*} Q \mapsto R$ . Possibly you want to know how it changes with _time_. Fix a initial point $P$ :

D = \frac{d}{d t} (F \circ ϕ) = X (F) = d F (X)

But observe that if we take $Y$ such that $i_{Y} ω = - d F$ then

d F (X) = - i_{Y} ω (X) = - ω (Y, X) =

= ω (X, Y) = i_{X} ω (Y) = - d H (Y) = - Y (H)

So the interesting quantity $D$ can be computed in two ways

X (F) = - Y (H)

This motivate a new definition. Remove, temporarily, the special paper to $H$ . For the functions $F, H$ we define the Poisson brackets as the new function:

{F, H} = X (F) = - Y (H) = ω (X, Y)

It is satisfied:

${q_{i}, p_{j}} = δ_{i j}$

Observe that if we leave a slot ${-, H}$ we have an operator that acts over functions. In fact, is the vector field $X$ ! So the Poisson bracket let us to find the symplectic gradient directly, without the use of $d$ and $ω$ . More explicitly

X_{f} = {-, f}

From $d ω = 0$ it can be deduced the Jacobi identity:

{f, {g, h}} = {{f, h}, g} + {h, {f, g}}

for any three functions $f, g, h$ .

Leaving an slot instead of $f$ we get

{-, {g, h}} = [{-, g}, {-, h}]

expression that can be rewritten as

X_{{g, h}} = [X_{g}, X_{h}] = X_{ω (X_{g}, X_{h})}

Taking the Poisson bracket of any function with a moment $p_{i}$ gives us how the function varies along the associated coordinate $q_{i}$ :

{F, p_{i}} = X_{p_{i}} (F)

but $X_{p_{i}}$ is just $\partial q_{i}$

${f, g}$ can be interpreted as the area of the paralelogram made of $X_{f}$ and $X_{g}$

{f, g} = X_{g} (f) = d f (X_{g}) = \nabla f \cdot X_{g}

Any function $f \in C^{\infty} (T^{*} Q)$ is called an observable. They constitute a commutative algebra that is, in fact, a Poisson algebra due to the existence of the Poisson bracket ${-, -}$ . If you fix any $f \in C^{\infty} (T^{*} Q)$ then you get a one-parameter local group of transformations for $C^{\infty} (T^{*} Q)$ in the following way.

First, you have the vector field $X_{f}$ given by

X_{f} = {-, f}

Then, the flow theorem for vector fields let you assure that there exists the local flow

ϕ_{s} : T^{*} Q \mapsto T^{*} Q

such that

1. $ϕ_{0} = i d$

1. $\frac{d}{d s} ϕ_{s} (m) |_{s = 0} = X_{f}$

1. $ϕ_{s} \circ ϕ_{t} = ϕ_{s + t}$

This can be seen also as a flow of observables in CM.

________________________________________

Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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