Analytical Mechanics - Hamiltonian Mechanics

Hamiltonian Mechanics

Hamiltonian and Hamilton's equations

The Legendre transformation of the Lagrangian replaces the generalized coordinates and velocities (q, q̇) with (q, p); the generalized coordinates and the generalized momenta conjugate to the generalized coordinates:

and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):

where • denotes the dot product, also leading to Hamilton's equations:

which are now a set of 2N 1st-order ordinary differential equations, one for each q_i(t) and p_i(t). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:

which is often considered one of Hamilton's equation of motion additionally to the others. The generalized momenta can be written in terms of the generalized forces in the same way as Newton's 2nd law:

Generalized momentum space

Analogous to the configuration space, the set of all momenta is the momentum space (technically in this context; generalized momentum space):

"Momentum space" also refers to "k-space"; the set of all wave vectors (given by De Broglie relations) as used in quantum mechanics and theory of waves: this is not referred to in this context.

Phase space

The set of all positions and momenta form the phase space;

that is, the cartesian product × of the configuration space and generalized momentum space.

A particular solution to Hamilton's equations is called a phase path, i.e. a particular curve (q(t),p(t)), subject to the required initial conditions. The set of all phase paths, i.e. general solution to the differential equations, is the phase portrait:

Likewise, the phase space can be defined more deeply using topological manifolds and the cotangent bundle.

Principle of least action

The Hamiltonian formulation is more general, allowing time-varying energy, identifying the path followed to be the one with stationary action. This is known as the principle of least action:

holding the departure t₁ and arrival t₂ times fixed. The term action has various meanings. This definition is only one, and corresponds specifically to an integral of the Lagrangian of the system. The term "path" or "trajectory" refers to the time evolution of the system as a path through configeration space, i.e. q(t) tracing out a path in . The path for which action is least is the path taken by the system.

From this principle, all equations of motion in classical mechanics can be derived. Generalizations of these approaches underlie the path integral formulation of quantum mechanics, and is used for calculating geodesic motion in general relativity.