Inphysics,Hamiltonian mechanics is a reformulation ofLagrangian mechanics that emerged in 1833. Introduced bySir William Rowan Hamilton,[1] Hamiltonian mechanics replaces (generalized) velocities used in Lagrangian mechanics with (generalized)momenta. Both theories provide interpretations ofclassical mechanics and describe the same physical phenomena.
Let be amechanical system withconfiguration space and smoothLagrangian Select a standard coordinate system on The quantities are calledmomenta. (Alsogeneralized momenta,conjugate momenta, andcanonical momenta). For a time instant theLegendre 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 theenergy function
The Legendre transform of turns into a function known as theHamiltonian. The Hamiltonian satisfieswhich implies thatwhere the velocities are found from the (-dimensional) equation which, by assumption, is uniquely solvable for. The (-dimensional) pair is calledphase space coordinates. (Alsocanonical coordinates).
From Euler–Lagrange equation to Hamilton's equations
Let be the set of smooth paths for which and Theaction functional is defined viawhere, and (see above). A path is astationary point of (and hence is an equation of motion) if and only if the path in phase space coordinates obeys the Hamilton equations.
A simple interpretation of Hamiltonian mechanics comes from its application on a one-dimensional system consisting of one nonrelativistic particle of massm. The value of the Hamiltonian is the total energy of the system, in this case the sum ofkinetic andpotential energy, traditionally denotedT andV, respectively. Herep is the momentummv andq is the space coordinate. ThenT is a function ofp alone, whileV is a function ofq alone (i.e.,T andV arescleronomic).
In this example, the time derivative ofq 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 momentump equals theNewtonian force, and so the second Hamilton equation means that the force equals the negativegradient of potential energy.
A spherical pendulum consists of amassm moving withoutfriction on the surface of asphere. The onlyforces acting on the mass are thereaction from the sphere andgravity.Spherical coordinates are used to describe the position of the mass in terms of(r,θ,φ), wherer is fixed,r =ℓ.
Thus the Hamiltonian iswhereandIn terms of coordinates and momenta, the Hamiltonian readsHamilton's equations give the time evolution of coordinates and conjugate momenta in four first-order differential equations,Momentum, which corresponds to the vertical component ofangular 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 acyclic coordinate, which implies conservation of its conjugate momentum.
Hamilton's equations can be derived by a calculation with theLagrangian, generalized positionsqi, and generalized velocities⋅qi, where.[3] Here we workoff-shell, meaning,, are independent coordinates in phase space, not constrained to follow any equations of motion (in particular, is not a derivative of). Thetotal 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, obeyingLagrange's equations:
Rearranging and writing in terms of the on-shell gives:
Thus Lagrange's equations are equivalent to Hamilton's equations:
In the case of time-independent and, i.e., Hamilton's equations consist of2n first-orderdifferential equations, while Lagrange's equations consist ofn second-order equations. Hamilton's equations usually do not reduce the difficulty of finding explicit solutions, but important theoretical results can be derived from them, because coordinates and momenta are independent variables with nearly symmetric roles.
Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate does not occur in the Hamiltonian (i.e. acyclic coordinate), the corresponding momentum coordinate is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set. This effectively reduces the problem fromn coordinates to(n − 1) coordinates: this is the basis ofsymplectic reduction in geometry. In the Lagrangian framework, the conservation of momentum also follows immediately, however all the generalized velocities still occur in the Lagrangian, and a system of equations inn coordinates still has to be solved.[4]
The value of the Hamiltonian is the total energy of the system if and only if the energy function has the same property. (See definition of).[clarification needed]
when, form a solution of Hamilton's equations. Indeed, and everything but the final term cancels out.
does not change underpoint transformations, i.e. smooth changes of space coordinates. (Follows from the invariance of the energy function under point transformations. The invariance of can be established directly).
if and only if.A coordinate for which the last equation holds is calledcyclic (orignorable). Every cyclic coordinate reduces the number of degrees of freedom by, causes the corresponding momentum to be conserved, and makes Hamilton's equationseasier to solve.
In its application to a given system, the Hamiltonian is often taken to be
where is the kinetic energy and is the potential energy. Using this relation can be simpler than first calculating the Lagrangian, and then deriving the Hamiltonian from the Lagrangian. However, the relation is not true for all systems.
The relation holds true for nonrelativistic systems when all of the following conditions are satisfied[5][6]
where is time, is the number of degrees of freedom of the system, and each is an arbitrary scalar function of.
In words, this means that the relation holds true if does not contain time as an explicit variable (it isscleronomic), does not contain generalised velocity as an explicit variable, and each term of is quadratic in generalised velocity.
Preliminary to this proof, it is important to address an ambiguity in the related mathematical notation. While a change of variables can be used to equate,it is important to note that.In this case, the right hand side always evaluates to 0. To perform a change of variables inside of a partial derivative, themultivariable chain rule should be used. Hence, to avoid ambiguity, the function arguments of any term inside of a partial derivative should be stated.
Additionally, this proof uses the notation to imply that.
Proof
Starting from definitions of the Hamiltonian, generalized momenta, and Lagrangian for an degrees of freedom system
Substituting the generalized momenta into the Hamiltonian gives
Substituting the Lagrangian into the result gives
Now assume that
and also assume that
Applying these assumptions results in
Next assume that T is of the form
where each is an arbitrary scalar function of.
Differentiating this with respect to,, gives
Splitting the summation, evaluating the partial derivative, and rejoining the summation gives
For a system of point masses, the requirement for to be quadratic in generalised velocity is always satisfied for the case where, which is a requirement for anyway.
Proof
Consider the kinetic energy for a system of N point masses. If it is assumed that, then it can be shown that (SeeScleronomous § Application). Therefore, the kinetic energy is
The chain rule for many variables can be used to expand the velocity
If the conditions for are satisfied, then conservation of the Hamiltonian implies conservation of energy. This requires the additional condition that does not contain time as an explicit variable.
In summary, the requirements for to be satisfied for a nonrelativistic system are[5][6]
is a homogeneous quadratic function in
Regarding extensions to the Euler-Lagrange formulation which use dissipation functions (SeeLagrangian mechanics § Extensions to include non-conservative forces), e.g. theRayleigh dissipation function, energy is not conserved when a dissipation function has effect. It is possible to explain the link between this and the former requirements by relating the extended and conventional Euler-Lagrange equations: grouping the extended terms into the potential function produces a velocity dependent potential. Hence, the requirements are not satisfied when a dissipation function has effect.
Hamiltonian of a charged particle in an electromagnetic field
Undergauge transformation:wheref(r,t) is any scalar function of space and time. The aforementioned Lagrangian, the canonical momenta, and the Hamiltonian transform like:which still produces the same Hamilton's equation:
In quantum mechanics, thewave function will also undergo alocalU(1) group transformation[7] during the Gauge Transformation, which implies that all physical results must be invariant under local U(1) transformations.
Relativistic charged particle in an electromagnetic field
An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum,, is
This has the advantage that kinetic momentum can be measured experimentally whereas canonical momentum cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of therelativistic energy (kinetic+rest),, plus thepotential energy,.
As aclosednondegeneratesymplectic2-formω. According toDarboux's theorem, in a small neighbourhood around any point onM there exist suitable local coordinates (canonical orsymplectic coordinates) in which thesymplectic form becomes:The form induces anatural isomorphism of thetangent space with thecotangent space:. This is done by mapping a vector to the 1-form, where for all. Due to thebilinearity and non-degeneracy of, and the fact that, the mapping is indeed alinear isomorphism. This isomorphism isnatural in that it does not change with change of coordinates on Repeating over all, we end up with an isomorphism between the infinite-dimensional space of smooth vector fields and that of smooth 1-forms. For every and,
(In algebraic terms, one would say that the-modules and are isomorphic). If, then, for every fixed,, and. is known as aHamiltonian vector field. The respective differential equation onis calledHamilton's equation. Here and is the (time-dependent) value of the vector field at.
A Hamiltonian system may be understood as afiber bundleE overtimeR, with thefiberEt being the position space at timet ∈R. The Lagrangian is thus a function on thejet bundleJ overE; taking the fiberwiseLegendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber att is thecotangent spaceT∗Et, which comes equipped with a naturalsymplectic form, and this latter function is the Hamiltonian. The correspondence between Lagrangian and Hamiltonian mechanics is achieved with thetautological one-form.
The Hamiltonian vector field induces aHamiltonian flow on the manifold. This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called "the time"); in other words, anisotopy ofsymplectomorphisms, starting with the identity. ByLiouville's theorem, each symplectomorphism preserves thevolume form on thephase space. The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called "the Hamiltonian mechanics" of the Hamiltonian system.
The symplectic structure induces aPoisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of aLie algebra.
IfF andG are smooth functions onM then the smooth functionω(J(dF),J(dG)) is properly defined; it is called aPoisson bracket of functionsF andG and is denoted{F,G}. The Poisson bracket has the following properties:
non-degeneracy: if the pointx onM is not critical forF then a smooth functionG exists such that.
Given a functionfif there is aprobability distributionρ, then (since the phase space velocity has zero divergence and probability is conserved) its convective derivative can be shown to be zero and so
A Hamiltonian may have multiple conserved quantitiesGi. If the symplectic manifold has dimension2n and there aren functionally independent conserved quantitiesGi which are in involution (i.e.,{Gi,Gj} = 0), then the Hamiltonian isLiouville integrable. TheLiouville–Arnold theorem says that, locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism into a new Hamiltonian with the conserved quantitiesGi as coordinates; the new coordinates are calledaction–angle coordinates. The transformed Hamiltonian depends only on theGi, and hence the equations of motion have the simple formfor some functionF.[9] There is an entire field focusing on small deviations from integrable systems governed by theKAM theorem.
The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems arechaotic; concepts of measure, completeness, integrability and stability are poorly defined.
An important special case consists of those Hamiltonians that arequadratic forms, that is, Hamiltonians that can be written aswhere⟨ , ⟩q is a smoothly varyinginner product on thefibersT∗ qQ, thecotangent space to the pointq in theconfiguration space, sometimes called a cometric. This Hamiltonian consists entirely of the kinetic term.
When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every pointq of the configuration space manifoldQ, so that therank of the cometric is less than the dimension of the manifoldQ, one has asub-Riemannian manifold.
The Hamiltonian in this case is known as asub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice versa. This implies that everysub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian. The existence of sub-Riemannian geodesics is given by theChow–Rashevskii theorem.
The continuous, real-valuedHeisenberg group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given bypz is not involved in the Hamiltonian.
Hamilton's equations above work well forclassical mechanics, but not forquantum mechanics, since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time. However, the equations can be further generalized to then be extended to apply to quantum mechanics as well as to classical mechanics, through the deformation of thePoisson algebra overp andq to the algebra ofMoyal brackets.
Specifically, the more general form of the Hamilton's equation readswheref is some function ofp andq, andH is the Hamiltonian. To find out the rules for evaluating aPoisson bracket without resorting to differential equations, seeLie algebra; a Poisson bracket is the name for the Lie bracket in aPoisson algebra. These Poisson brackets can then be extended toMoyal brackets comporting to an inequivalent Lie algebra, as proven byHilbrand J. Groenewold, and thereby describe quantum mechanical diffusion in phase space (SeePhase space formulation andWigner–Weyl transform). This more algebraic approach not only permits ultimately extendingprobability distributions inphase space toWigner quasi-probability distributions, but, at the mere Poisson bracket classical setting, also provides more power in helping analyze the relevantconserved quantities in a system.
