The
Hamiltonian for a system of discrete particles is a function of their
generalized coordinates and conjugate momenta, and possibly, time. For continua and fields, Hamiltonian mechanics is unsuitable but can be extended by considering a large number of point masses, and taking the continuous limit, that is, infinitely many particles forming a continuum or field. Since each point mass has one or more
degrees of freedom, the field formulation has infinitely many degrees of freedom.
One scalar field
The Hamiltonian density is the continuous analogue for fields; it is a function of the fields, the conjugate "momentum" fields, and possibly the space and time coordinates themselves. For one
scalar fieldφ(x, t), the Hamiltonian density is defined from the
Lagrangian density by[nb 1]
with ∇ the
"del" or "nabla" operator, x is the
position vector of some point in space, and t is
time. The Lagrangian density is a function of the fields in the system, their space and time derivatives, and possibly the space and time coordinates themselves. It is the field analogue to the Lagrangian function for a system of discrete particles described by generalized coordinates.
As in Hamiltonian mechanics where every generalized coordinate has a corresponding generalized momentum, the field φ(x, t) has a conjugate momentum fieldπ(x, t), defined as the partial derivative of the Lagrangian density with respect to the time derivative of the field,
in which the overdot[nb 2] denotes a
partial time derivative ∂/∂t, not a
total time derivative d/dt.
Many scalar fields
For many fields φi(x, t) and their conjugates πi(x, t) the Hamiltonian density is a function of them all:
where each conjugate field is defined with respect to its field,
In general, for any number of fields, the
volume integral of the Hamiltonian density gives the Hamiltonian, in three spatial dimensions:
The Hamiltonian density is the Hamiltonian per unit spatial volume. The corresponding
dimension is [energy][length]−3, in
SI units Joules per metre cubed, J m−3.
The fields φi and conjugates πi form an infinite dimensional
phase space, because fields have an infinite number of degrees of freedom.
Poisson bracket
For two functions which depend on the fields φi and πi, their spatial derivatives, and the space and time coordinates,
and the fields are zero on the boundary of the volume the integrals are taken over, the field theoretic
Poisson bracket is defined as (not to be confused with the
commutator from quantum mechanics).[1]
Under the same conditions of vanishing fields on the surface, the following result holds for the time evolution of A (similarly for B):
which can be found from the total time derivative of A,
integration by parts, and using the above Poisson bracket.
Explicit time-independence
The following results are true if the Lagrangian and Hamiltonian densities are explicitly time-independent (they can still have implicit time-dependence via the fields and their derivatives),
Kinetic and potential energy densities
The Hamiltonian density is the total energy density, the sum of the kinetic energy density () and the potential energy density (),
Continuity equation
Taking the partial time derivative of the definition of the Hamiltonian density above, and using the
chain rule for
implicit differentiation and the definition of the conjugate momentum field, gives the
continuity equation:
in which the Hamiltonian density can be interpreted as the energy density, and
the energy flux, or flow of energy per unit time per unit surface area.
Relativistic field theory
Covariant Hamiltonian field theory is the
relativistic formulation of Hamiltonian field theory.
Hamiltonian field theory usually means the symplectic
Hamiltonian formalism when applied to
classical field theory, that takes the form of the instantaneous Hamiltonian formalism on an infinite-dimensional
phase space, and where
canonical coordinates are field functions at some instant of time.[2] This Hamiltonian formalism is applied to
quantization of fields, e.g., in quantum
gauge theory. In Covariant Hamiltonian field theory,
canonical momentapμi corresponds to derivatives of fields with respect to all world coordinates xμ.[3] Covariant Hamilton equations are equivalent to the
Euler–Lagrange equations in the case of hyperregular
Lagrangians. Covariant Hamiltonian field theory is developed in the Hamilton–De Donder,[4] polysymplectic,[5] multisymplectic[6] and k-symplectic[7] variants. A phase space of covariant Hamiltonian field theory is a finite-dimensional
polysymplectic or
multisymplectic manifold.
^It is a standard abuse of notation to abbreviate all the derivatives and coordinates in the Lagrangian density as follows:
The μ is an index which takes values 0 (for the time coordinate), and 1, 2, 3 (for the spatial coordinates), so strictly only one derivative or coordinate would be present. In general, all the spatial and time derivatives will appear in the Lagrangian density, for example in Cartesian coordinates, the Lagrangian density has the full form:
Here we write the same thing, but using ∇ to abbreviate all spatial derivatives as a vector.
^This is standard notation in this context, most of the literature does not explicitly mention it is a partial derivative. In general total and partial time derivatives of a function are not the same.
^Gotay, M., A multisymplectic framework for classical field theory and the calculus of variations. II. Space + time decomposition, in "Mechanics, Analysis and Geometry: 200 Years after Lagrange" (North Holland, 1991).
^Krupkova, O., Hamiltonian field theory, J. Geom. Phys. 43 (2002) 93.
^Giachetta, G., Mangiarotti, L.,
Sardanashvily, G., Covariant Hamiltonian equations for field theory, J. Phys. A32 (1999) 6629;
arXiv:
hep-th/9904062.
^Echeverria-Enriquez, A., Munos-Lecanda, M., Roman-Roy, N., Geometry of multisymplectic Hamiltonian first-order field theories, J. Math. Phys. 41 (2002) 7402.
^Rey, A., Roman-Roy, N. Saldago, M., Gunther's formalism (k-symplectic formalism) in classical field theory: Skinner-Rusk approach and the evolution operator, J. Math. Phys. 46 (2005) 052901.
Goldstein, Herbert (1980). "Chapter 12: Continuous Systems and Fields". Classical Mechanics (2nd ed.). San Francisco, CA: Addison Wesley. pp. 562–565.
ISBN0201029189.