Hamilton equations

The Hamilton equations are a very powerful formulation of the equations of analytical mechanics. They are fundamental through their general role in physics and are at the foundation of the discovery and formulation of quantum mechanics.

Sir William Rowan Hamilton (1805-1865).

These equations were initially limited to mechanical systems having a finite number of degrees of freedom, such as the positions and velocities of particles in a gas or the angles and rotation speeds describing a gyroscope; they can now be extended to describe continuous systems with an infinite number of degrees of freedom. This is the case with the equations of an electromagnetic or gravitational field which can be cast into the form known as a Hamiltonian.

A simple example will be sufficient to understand the meaning of these equations.

Consider a set of N particles in a gas which will then have 3N position coordinates $q_{i}$ and 3N velocity coordinates $\dot{q_{i}}$ where the dot denotes the total derivative of the preceding coordinate with respect to time, i.e. $\dot{q_{i}}=\frac{dq_{i}}{dt}$ .

We know that in the Lagrange formulation of the equations of the mechanics of a particle system, the differential equations of movement of these particles are given by:

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q_i}})- \frac{\partial L}{\partial q_i}=0$

Where the so-called Lagrangian function $L$ for these particles in a potential V is written

$L (q_{i}, \dot{q_{i}})=\sum_{i=1}^{3n} \frac{m_{i} {\dot{q_{i}}}^2}{2} - V(q_{i})$

The masses $m_i$ are obviously the same for a particle and its three position coordinates. The first term on the right in the equation is a sum of kinetic energies.

Joseph Louis Lagrange (25 January 1736, Turin - 10 April 1813, Paris)

The Lagrangian formulation is invariant to a change in the coordinates of the system $q_{i}$ which is what makes it so powerful as it allows us to reduce large classes of differential equations expressed in Cartesian, spherical etc. coordinates to a few fundamental cases that we know how to solve.

Moreover, it can be used to prove that the theorems of conservation of energy, momentum etc. are automatically valid for these large classes of differential equations since, by proving conservation laws in a given coordinate system, we can be sure that they will be valid in other coordinate systems or in other mechanical systems that can be cast into equations in Lagrangian form.

The validity of the laws of conservation such as for energy or momentum, initially established for material points, will also be valid for electromagnetic fields if the Maxwell equations can be cast in Lagrangian form. And indeed they can.

The implications are actually very far reaching and have repercussions in quantum mechanics and in quantum field theory through Noether's theorem relating the invariance of the Lagrangian to certain symmetry transformations and the establishment of conservation laws.

Thus, if we write a differential equation with variables that have no direct relationship with the positions of a particle system but deriving from a Lagrangian that can take the form of a particle system invariant to translation in time, then we can define a function $H$ that corresponds to an energy and that will satisfy a conservation law.

The Lagrange formulation can be extended as follows. Another variable is introduced $p_{i}$ defined by $p_{i}=\frac{\partial L }{\partial \dot{q_{i}}}$ . A so-called Hamiltonian function can then be built $H$ , deduced from the previous Lagrangian by the Legendre transformation:

$H= \sum_{i=1}^{3N} \dot{q_{i}}p_{i} - L$

The system of Lagrangian second order differential equations can then be written in the form of a system of first order differential equations:

$\dot{p_{i}}=-\frac{\partial H}{\partial q_{i}}$ $\dot{q_{i}}=\frac{\partial H}{\partial p_{i}}$

These are the Hamilton equations and they were to be of capital importance in the hands of Heisenberg, Born, Jordan and Dirac for the discovery and the formulation of matrix mechanics and its final extension, quantum mechanics. These physicists discovered that the passage from classical mechanics to quantum mechanics can be made by replacing the above quantities by matrices.

Here again of course, the Hamiltonian function $H(q_i,p_i)$ expresses the total energy of the mechanical system being considered.

Several important consequences arise from these equations which are also the basis of the Gibbs, Boltzmann and Einstein statistical mechanical formulation.

For example, these equations reduce the movement of N material points (particles of a gas, stars in a galaxy, etc.) to a single abstract point in a 6N dimension space called the phase space of the Hamiltonian mechanical system. The coordinates of this point are defined by 3N pairs of coordinates called conjugate coordinates $q_{i}$ et $p_{i}$ .

These same equations are invariant to a large class of coordinate changes called canonical transformations of the previous conjugate variables. They are written:

$Q_j=f_j(q_i,p_i, t)$

$P_j=g_j(q_i,p_i, t)$

This gives rise to a powerful integration technique for the equations of motion for a mechanical system as was initially demonstrated in celestial mechanics. This is why Bohr and especially Sommerfeld used it to study and construct atomic models with orbits of electrons around the nucleus of an atom.

Siméon Denis Poisson (21 June 1781 - 25 April 1840 )

One part of these integration techniques uses a development of the Hamilton equations involving the Poisson equations in analytical mechanics.

Consider any two functions:

$F(q_i,p_i, t)$

$G(q_i,p_i, t)$

conjugate coordinates of a mechanical system in Hamiltonian form.

Poisson then introduced the following construction:

$[F , G]=\sum_{i=1}^{3N}\frac{\partial F}{\partial q_{i}} \frac{\partial G}{\partial p_{i}} - \frac{\partial F}{\partial p_{i}} \frac{\partial G}{\partial q_{i}}$

called the Poisson bracket.

Differential equations can then be written in the form:

$\frac{dF}{dt}=[F,H]+\frac{\partial F}{\partial t}$

Where $H(q_i,p_i)$ is always the total energy of a Hamiltonian mechanical system.

In particular, if $F(q_i,p_i, t)$ is equal to $q_{i}$ or $p_{i}$ , the previous Hamilton equations are obtained.

The equations and Poisson brackets are the basis of the fundamental equations of matrix mechanics, the Heisenberg equations.

In particular, the Poisson brackets are canonical invariants because they do not change form during a canonical transformation. They have the following remarkable property:

$[q_{i},p_{j}]=\delta_{ij}$

with $\delta_{ij}=1$ if $i=j$ and 0 otherwise.

Transposed in the form of matrices, or operators, and using Planck's constant, this relationship is the key to the quantification of mechanical systems as shown by Born and Dirac.

In the form of operators this relation becomes:

$[\hat x_j, \hat p_k]=i\hbar\delta_{jk}$ $=\hat x_j \hat p_k- \hat p_k \hat x_j$

This time $i$ is not an index but the usual imaginary number.

If therefore $F(q_i,p_i)$ no longer depends implicitly on time, and

$[F, H]=0$

then from the Poisson equations, we have

$\frac{dF}{dt}=0$

The quantity $F(q_i,p_i)$ is therefore constant in time and this is how the laws of conservation in the Hamiltonian formalism are obtained.

In particular, this is the conservation of energy obtained by setting $F(q_i,p_i)$ equal to $H(q_i,p_i)$ .

Finally, there is an equation that is highly important in Hamiltonian mechanics because it connects the formalisms of Hamilton and Lagrange.

This is the Hamilton-Jacobi equation.

Carl Gustav Jacobi (1804 - 1851)

The Lagrange equations can be differentiated starting from the condition that the function $S$

$S=\int_{t1}^{t2} L (q_{i}, \dot{q_{i}})dt$

is an extremum for all variations (in a precise sense) of the coordinates of positions and velocities of a mechanical system having a Lagrangian, between two dates separating the states of motion of the mechanical system being considered.

This function $S$ is called the action and it allows the following equations to be written:

$H(q_i,p_i)=-\frac {\partial S}{\partial t}$