Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena.
Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.
Overview
Phase space coordinates (p,q) and Hamiltonian H
Let be a mechanical system with the configuration space and the smooth Lagrangian Select a standard coordinate system on The quantities are called momenta. (Also generalized momenta, conjugate momenta, and canonical momenta). For a time instant the Legendre transformation of is defined as the map which is assumed to have a smooth inverse For a system with degrees of freedom, the Lagrangian mechanics defines the energy function
The Legendre transform of turns into a function known as the . The Hamiltonian satisfies
which implies that
where the velocities are found from the (-dimensional) equation which, by assumption, is uniquely solvable for The (-dimensional) pair is called phase space coordinates. (Also canonical coordinates).
From Euler–Lagrange equation to Hamilton's equations
In phase space coordinates the (-dimensional) Euler–Lagrange equation
becomes Hamilton's equations in dimensions
From stationary action principle to Hamilton's equations
Let be the set of smooth paths for which and The action functional is defined via
where and (see above). A path is a stationary point of (and hence is an equation of motion) if and only if the path in phase space coordinates obeys the Hamilton's equations.
Basic physical interpretation
A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one nonrelativistic particle of mass . The value of the Hamiltonian is the total energy of the system, in this case the sum of kinetic and potential energy, traditionally denoted and , respectively. Here is the momentum and is the space coordinate. Then
is a function of alone, while is a function of alone (i.e., and are scleronomic).
In this example, the time derivative of is the velocity, and so the first Hamilton equation means that the particle's velocity equals the derivative of its kinetic energy with respect to its momentum. The time derivative of the momentum equals the Newtonian force, and so the second Hamilton equation means that the force equals the negative gradient of potential energy.
Example
A spherical pendulum consists of a mass m moving without friction on the surface of a sphere. The only forces acting on the mass are the reaction from the sphere and gravity. Spherical coordinates are used to describe the position of the mass in terms of (r, θ, φ), where is fixed, . The Lagrangian for this system is
Thus the Hamiltonian is
where
and
In terms of coordinates and momenta, the Hamiltonian reads
Hamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations,
Momentum , which corresponds to the vertical component of angular momentum , is a constant of motion. That is a consequence of the rotational symmetry of the system around the vertical axis. Being absent from the Hamiltonian, azimuth is a cyclic coordinate, which implies conservation of its conjugate momentum.
Deriving Hamilton's equations
Hamilton's equations can be derived by a calculation with the Lagrangian , generalized positions , and generalized velocities , where . Here we work off-shell, meaning are independent coordinates in phase space, not constrained to follow any equations of motion (in particular, is not a derivative of ). The total differential of the Lagrangian is: The generalized momentum coordinates were defined as , so we may rewrite the equation as:
After rearranging, one obtains:
The term in parentheses on the left-hand side is just the Hamiltonian defined previously, therefore:
One may also calculate the total differential of the Hamiltonian with respect to coordinates instead of , yielding:
One may now equate these two expressions for , one in terms of , the other in terms of :
Since these calculations are off-shell, one can equate the respective coefficients of on the two sides:
On-shell, one substitutes parametric functions which define a trajectory in phase space with velocities , obeying Lagrange's equations: