The Hamiltonian function (also Hamiltonian, after William Rowan Hamilton ) of a system of particles is, if there are no rheonomic (i.e. time-dependent) constraints , the total energy as a function of the locations and momenta of the particles and possibly time. It is a Legendre transform of the Lagrange function of the system. Instead of using the location and pulse coordinates , the functional relationship can also be expressed using the generalized location coordinates and generalized pulse coordinates.
![{\ displaystyle p = (p_ {1}, p_ {2}, \ dotsc, p_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2dbce003dd791566977e0ec3e514cd9aac13023)
definition
The Hamilton function is defined by
![{\ displaystyle {\ mathcal {H}} (q, p, t): = \ left \ {\ sum _ {i = 1} ^ {n} {\ dot {q}} _ {i} p_ {i} \ right \} - {\ mathcal {L}} (q, {\ dot {q}}, t), {\ text {with}} {\ dot {q}} = {\ dot {q}} (q , p, t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/538cbff55828e8a71022846b10e4e6ddd2da9dec)
and depends on
- time ,
![t](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
- the generalized coordinates and
![{\ displaystyle q = (q_ {1}, q_ {2}, \ dotsc, q_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4209bad4064ff28a07f5d7586f8a948811545d7)
- the generalized impulses .
![{\ displaystyle p = (p_ {1}, p_ {2}, \ dotsc, p_ {n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2dbce003dd791566977e0ec3e514cd9aac13023)
It is derived from a Legendre transformation of the Lagrange function with respect to the generalized velocities, which depends on the generalized coordinates and their velocities :
![{\ displaystyle {\ mathcal {L}} (t, q, {\ dot {q}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fa1b98a7de93906fb88b16ae99d93e619403879)
![{\ displaystyle {\ dot {q}} = ({\ dot {q}} _ {1}, {\ dot {q}} _ {2}, \ dotsc, {\ dot {q}} _ {n} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee1ce497009be7b3cfbf55063a0a6b6f159b84e)
![{\ displaystyle {\ mathcal {H}} (t, q, p) = \ left \ {\ sum _ {i = 1} ^ {n} {\ dot {q}} _ {i} \, p_ {i } \ right \} - {\ mathcal {L}} (t, q, {\ dot {q}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59724a36bc14db5f219c31587be3812752c0b303)
Here, on the right side with the speeds those functions
![{\ dot q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/399dc6b6e91a780c89824ccc26b4453b289e4387)
![{\ displaystyle {\ dot {q}} (t, q, p)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/480b849d75647526a21e11dcee77b7f72cdf30f7)
meant, which one gets when one takes the definition of generalized impulses
![p_ {i}: = {\ frac {\ partial {\ mathcal L}} {\ partial {\ dot q} _ {i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1668e3cc82b13b6a78ac2b6de1e85dd87e7dc250)
dissolves according to the speeds.
properties
Derivation
The total differential of the Hamilton function is:
![{\ displaystyle \ mathrm {d} {\ mathcal {H}} = \ sum _ {i = 1} ^ {n} {\ frac {\ partial {\ mathcal {H}}} {\ partial q_ {i}} } \ mathrm {d} q_ {i} + \ sum _ {i = 1} ^ {n} {\ frac {\ partial {\ mathcal {H}}} {\ partial p_ {i}}} \ mathrm {d } p_ {i} + {\ frac {\ partial {\ mathcal {H}}} {\ partial t}} \ mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7708a087a373a3e61adccd7bbe1508122b857b)
Based on the product rule , one obtains
![{\ displaystyle \ mathrm {d} {\ mathcal {H}} = \ sum _ {i = 1} ^ {n} \ left (p_ {i} \ mathrm {d} {\ dot {q}} _ {i } + {\ dot {q}} _ {i} \ mathrm {d} p_ {i} - {\ frac {\ partial {\ mathcal {L}}} {\ partial q_ {i}}} \ mathrm {d } q_ {i} - {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {q}} _ {i}}} \ mathrm {d} {\ dot {q}} _ { i} \ right) - {\ frac {\ partial {\ mathcal {L}}} {\ partial t}} \ mathrm {d} t,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4670472684153ab71d1aa12a058a669777a8115e)
where, due to the definition of the generalized momentum, the first and last terms in the brackets have the sum 0, so that:
![{\ frac {\ partial {\ mathcal L}} {\ partial {\ dot {q}} _ {i}}} = p_ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bdd58f147f8024cd29800832f5774c7505129e1)
![{\ displaystyle \ mathrm {d} {\ mathcal {H}} = \ sum _ {i = 1} ^ {n} \ left ({\ dot {q}} _ {i} \ mathrm {d} p_ {i } - {\ frac {\ partial {\ mathcal {L}}} {\ partial q_ {i}}} \ mathrm {d} q_ {i} \ right) - {\ frac {\ partial {\ mathcal {L} }} {\ partial t}} \ mathrm {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d609e6c58be540bce3591d42e2cdb6202c27ac7f)
With the above notation of the total differential, the partial derivatives of the Hamilton function follow :
![{\ frac {\ partial {\ mathcal H}} {\ partial p_ {i}}} = {\ dot {q}} _ {i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11912cc12f32c0b83bce14ee52e9d5c186cfbac5)
![{\ frac {\ partial {\ mathcal H}} {\ partial q_ {i}}} = - {\ frac {\ partial {\ mathcal L}} {\ partial q_ {i}}} = - {\ dot { pi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45e2e4514a8faeb15cbcc5cfae72d3f3244400ad)
![{\ frac {\ partial {\ mathcal H}} {\ partial t}} = - {\ frac {\ partial {\ mathcal L}} {\ partial t}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe03eb46210b459be26b476580cab5d111898a0)
Conservation size
The total derivative of the Hamilton function with respect to time is identical to the partial:
![{\ displaystyle {\ begin {aligned} {\ frac {\ mathrm {d} {\ mathcal {H}}} {\ mathrm {d} t}} & = \ sum _ {i = 1} ^ {f} \ left ({\ frac {\ partial {\ mathcal {H}}} {\ partial p_ {i}}} {\ dot {p}} _ {i} + {\ frac {\ partial {\ mathcal {H}} } {\ partial q_ {i}}} {\ dot {q}} _ {i} \ right) + {\ frac {\ partial {\ mathcal {H}}} {\ partial t}} \\ & = \ sum _ {i = 1} ^ {f} \ left ({\ dot {q}} _ {i} {\ dot {p}} _ {i} - {\ dot {p}} _ {i} {\ dot {q}} _ {i} \ right) + {\ frac {\ partial {\ mathcal {H}}} {\ partial t}} \\ & = {\ frac {\ partial {\ mathcal {H}} } {\ partial t}} \ end {aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e578f46e5c8a06b98ea00b508d524fe4dc317d24)
So if the Hamilton function does not explicitly depend on time , its value is a conserved quantity :
![t](https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560)
![{\ displaystyle {\ mathcal {H}} \ neq {\ mathcal {H}} (t) \ Rightarrow {\ frac {\ mathrm {d} {\ mathcal {H}}} {\ mathrm {d} t}} = {\ frac {\ partial {\ mathcal {H}}} {\ partial t}} = 0 \ Rightarrow {\ mathcal {H}} = const.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa386b8bf264136b823b2f6dcb41de12cd06e5e)
Implications
The Hamilton function determines the temporal development of the particle locations and impulses using Hamilton's equations of motion :
![{\ displaystyle {\ dot {q}} _ {k} = {\ frac {\ partial {\ mathcal {H}}} {\ partial p_ {k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb23c229c31acc91963ee58b81b914179abe1bb2)
![{\ displaystyle {\ dot {p}} _ {k} = - {\ frac {\ partial {\ mathcal {H}}} {\ partial q_ {k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48620c2e15615cc7dcbb5cbddd1ace3b4f98648f)
The Hamilton operator also determines the time evolution in quantum mechanics . In many cases it is obtained from the Hamilton function by canonical quantization by reading the algebraic expression for as a function of operators and that satisfy the canonical commutation relations.
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
![p](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
Examples
Mass point
For a particle of mass that moves non-relativistically in a potential , the Hamilton function is composed of kinetic and potential energy:
![m](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc)
![V](https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845)
![{\ displaystyle {\ mathcal {H}} (t, {\ vec {q}}, {\ vec {p}}) = {\ frac {{\ vec {p}} ^ {2}} {2 \, m}} + V ({\ vec {q}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb1e9c1e6ce6875c56ebd7fd62e2bb0def92748)
For a relativistic, free particle with the energy-momentum relationship
![{\ displaystyle E ^ {2} - {\ vec {p}} ^ {2} \, c ^ {2} = m ^ {2} \, c ^ {4}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b52761554808eb9ae2bf678f8943bf9ff58b1ba)
applies to the Hamilton function
![{\ displaystyle {\ mathcal {H}} (t, {\ vec {q}}, {\ vec {p}}) = {\ sqrt {m ^ {2} \, c ^ {4} + {\ vec {p}} ^ {2} \, c ^ {2}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81641eec319e43d03cf3965471179be83465f6d5)
For the free relativistic particle with the Lagrangian function
![{\ displaystyle {\ mathcal {L}} = - m \, c ^ {2} {\ sqrt {1 - {\ dot {\ vec {q}}} ^ {2} / c ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6005d1d3a99d4fe09b833d97727069f741acab38)
the generalized momentum depends according to
![p = {\ frac {\ partial {\ mathcal L}} {\ partial {\ dot q}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e86180c13270ba46448b7d8a2b6663c61c641646)
![{\ displaystyle {\ vec {p}} = {\ frac {m {\ dot {\ vec {q}}}} {\ sqrt {1 - {\ dot {\ vec {q}}} ^ {2} / c ^ {2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43e0779badf398e8187a9dab4298cea952772d04)
on the speed. Conversely, the speed is therefore the function
![{\ displaystyle {\ dot {\ vec {q}}} = {\ frac {{\ vec {p}} \, c ^ {2}} {\ sqrt {m ^ {2} \, c ^ {4} + {\ vec {p}} ^ {2} \, c ^ {2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b520a6a9bdefada90687ccb31cf721db4d4cf2d)
of the momentum.
Harmonic oscillator
The Hamilton function of a one-dimensional harmonic oscillator is given by:
![{\ displaystyle {\ mathcal {H}} (x, p) = {\ dot {x}} p - {\ mathcal {L}} (x, {\ dot {x}}) = {\ frac {p ^ {2}} {2m}} + {\ frac {m} {2}} \ omega _ {0} ^ {2} x ^ {2} = T + V = E}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a152f4c8929fb66bae91be14c4e95d6a33df944f)
Charged particle in an electromagnetic field
In Cartesian coordinates ( ), the Lagrange function of a particle of charge moving through an electromagnetic field is
![{\ displaystyle {\ vec {q}} = {\ vec {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8538c9168c845e7a95af6aa7d368f7ccff0df2)
![q](https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d)
![{\ displaystyle {\ mathcal {L}} = {\ frac {1} {2}} m {\ dot {\ vec {x}}} ^ {2} + q \ left ({\ dot {\ vec {x }}} \ cdot {\ vec {A}} \ right) -q \ phi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34098e4340f22a77ff41cc74c85661c37b71973d)
Here is the electrical potential and the vector potential of the magnetic field. The canonical impulse is
![\ phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4)
![{\ vec {A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/391292ffadc65b0cde3e96f23afcdb811619dd95)
![{\ displaystyle {\ vec {p}} = {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {\ vec {x}}}}} = m {\ dot {\ vec { x}}} + q {\ vec {A}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6e80cdd9d58dc01474362f7424b7c2d24086216)
This equation can be rearranged so that the speed is expressed in terms of momentum:
![{\ displaystyle {\ dot {\ vec {x}}} = {\ frac {1} {m}} \ left ({\ vec {p}} - q {\ vec {A}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b11fbdef33c61fcd579f52b19c6375419d8949)
If the expression for and is used in the definition of the Hamilton function, this results in:
![{\ displaystyle {\ dot {\ vec {x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ed483ca94407c179bfb9a9e7b2818a14fc830dc)
![{\ vec {p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84fee53c81592db54e0fe6c6f9eba002bb1dc74b)
![{\ displaystyle {\ mathcal {H}} = {\ dot {\ vec {x}}} \ cdot {\ vec {p}} - {\ mathcal {L}} = {\ frac {1} {2m}} \ left ({\ vec {p}} - q {\ vec {A}} \ right) ^ {2} + q \ phi}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e902728835379d2155f12ca159586a452a4419a)
literature
- Herbert Goldstein, Charles P. Poole, Jr., John L. Safko: Classical Mechanics . 3. Edition. Wiley-VCH, Weinheim 2006, ISBN 3-527-40589-5 .
- Wolfgang Nolting: Basic Course Theoretical Physics 2. Analytical Mechanics . 7th edition. Springer, Heidelberg 2006, ISBN 3-540-30660-9 .