# Hamilton function

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. ${\ displaystyle {\ mathcal {H}} ({\ vec {q}} _ {1}, {\ vec {q}} _ {2}, \ ldots, {\ vec {p}} _ {1}, {\ vec {p}} _ {2}, \ ldots, t)}$ ${\ displaystyle q = (q_ {1}, q_ {2}, \ dotsc, q_ {n})}$ ${\ displaystyle p = (p_ {1}, p_ {2}, \ dotsc, p_ {n})}$ ## definition

The Hamilton function is defined by

${\ displaystyle {\ mathcal {H}} (q, p, t): = \ left \ {\ sum _ {i = 1} ^ {n} {\ dot {q}} _ {i} p_ {i} \ right \} - {\ mathcal {L}} (q, {\ dot {q}}, t), {\ text {with}} {\ dot {q}} = {\ dot {q}} (q , p, t)}$ and depends on

• time ,${\ displaystyle t}$ • the generalized coordinates and${\ displaystyle q = (q_ {1}, q_ {2}, \ dotsc, q_ {n})}$ • the generalized impulses .${\ displaystyle p = (p_ {1}, p_ {2}, \ dotsc, p_ {n})}$ 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 : ${\ displaystyle {\ mathcal {L}} (t, q, {\ dot {q}})}$ ${\ displaystyle {\ dot {q}} = ({\ dot {q}} _ {1}, {\ dot {q}} _ {2}, \ dotsc, {\ dot {q}} _ {n} )}$ ${\ displaystyle {\ mathcal {H}} (t, q, p) = \ left \ {\ sum _ {i = 1} ^ {n} {\ dot {q}} _ {i} \, p_ {i } \ right \} - {\ mathcal {L}} (t, q, {\ dot {q}})}$ Here, on the right side with the speeds those functions ${\ displaystyle {\ dot {q}}}$ ${\ displaystyle {\ dot {q}} (t, q, p)}$ meant, which one gets when one takes the definition of generalized impulses

${\ displaystyle p_ {i}: = {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {q}} _ {i}}}}$ dissolves according to the speeds.

## properties

### Derivation

The total differential of the Hamilton function is:

${\ displaystyle \ mathrm {d} {\ mathcal {H}} = \ sum _ {i = 1} ^ {n} {\ frac {\ partial {\ mathcal {H}}} {\ partial q_ {i}} } \ mathrm {d} q_ {i} + \ sum _ {i = 1} ^ {n} {\ frac {\ partial {\ mathcal {H}}} {\ partial p_ {i}}} \ mathrm {d } p_ {i} + {\ frac {\ partial {\ mathcal {H}}} {\ partial t}} \ mathrm {d} t}$ Based on the product rule , one obtains

${\ displaystyle \ mathrm {d} {\ mathcal {H}} = \ sum _ {i = 1} ^ {n} \ left (p_ {i} \ mathrm {d} {\ dot {q}} _ {i } + {\ dot {q}} _ {i} \ mathrm {d} p_ {i} - {\ frac {\ partial {\ mathcal {L}}} {\ partial q_ {i}}} \ mathrm {d } q_ {i} - {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {q}} _ {i}}} \ mathrm {d} {\ dot {q}} _ { i} \ right) - {\ frac {\ partial {\ mathcal {L}}} {\ partial t}} \ mathrm {d} t,}$ where, due to the definition of the generalized momentum, the first and last terms in the brackets have the sum 0, so that: ${\ displaystyle {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {q}} _ {i}}} = p_ {i}}$ ${\ displaystyle \ mathrm {d} {\ mathcal {H}} = \ sum _ {i = 1} ^ {n} \ left ({\ dot {q}} _ {i} \ mathrm {d} p_ {i } - {\ frac {\ partial {\ mathcal {L}}} {\ partial q_ {i}}} \ mathrm {d} q_ {i} \ right) - {\ frac {\ partial {\ mathcal {L} }} {\ partial t}} \ mathrm {d} t}$ With the above notation of the total differential, the partial derivatives of the Hamilton function follow :

${\ displaystyle {\ frac {\ partial {\ mathcal {H}}} {\ partial p_ {i}}} = {\ dot {q}} _ {i}}$ ${\ displaystyle {\ frac {\ partial {\ mathcal {H}}} {\ partial q_ {i}}} = - {\ frac {\ partial {\ mathcal {L}}} {\ partial q_ {i}} } = - {\ dot {p}} _ {i}}$ ${\ displaystyle {\ frac {\ partial {\ mathcal {H}}} {\ partial t}} = - {\ frac {\ partial {\ mathcal {L}}} {\ partial t}}}$ ### 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}}} So if the Hamilton function does not explicitly depend on time , its value is a conserved quantity : ${\ displaystyle t}$ ${\ displaystyle {\ mathcal {H}} \ neq {\ mathcal {H}} (t) \ Rightarrow {\ frac {\ mathrm {d} {\ mathcal {H}}} {\ mathrm {d} t}} = {\ frac {\ partial {\ mathcal {H}}} {\ partial t}} = 0 \ Rightarrow {\ mathcal {H}} = const.}$ ### Implications

The Hamilton function determines the temporal development of the particle locations and impulses using Hamilton's equations of motion :

${\ displaystyle {\ dot {q}} _ {k} = {\ frac {\ partial {\ mathcal {H}}} {\ partial p_ {k}}}}$ ${\ displaystyle {\ dot {p}} _ {k} = - {\ frac {\ partial {\ mathcal {H}}} {\ partial q_ {k}}}}$ 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. ${\ displaystyle {\ mathcal {H}} (t, q, p)}$ ${\ displaystyle q}$ ${\ displaystyle p}$ ## 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: ${\ displaystyle m}$ ${\ displaystyle V}$ ${\ displaystyle {\ mathcal {H}} (t, {\ vec {q}}, {\ vec {p}}) = {\ frac {{\ vec {p}} ^ {2}} {2 \, m}} + V ({\ vec {q}})}$ For a relativistic, free particle with the energy-momentum relationship

${\ displaystyle E ^ {2} - {\ vec {p}} ^ {2} \, c ^ {2} = m ^ {2} \, c ^ {4}}$ applies to the Hamilton function

${\ displaystyle {\ mathcal {H}} (t, {\ vec {q}}, {\ vec {p}}) = {\ sqrt {m ^ {2} \, c ^ {4} + {\ vec {p}} ^ {2} \, c ^ {2}}}.}$ For the free relativistic particle with the Lagrangian function

${\ displaystyle {\ mathcal {L}} = - m \, c ^ {2} {\ sqrt {1 - {\ dot {\ vec {q}}} ^ {2} / c ^ {2}}}}$ the generalized momentum depends according to ${\ displaystyle p = {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {q}}}}}$ ${\ displaystyle {\ vec {p}} = {\ frac {m {\ dot {\ vec {q}}}} {\ sqrt {1 - {\ dot {\ vec {q}}} ^ {2} / c ^ {2}}}}}$ on the speed. Conversely, the speed is therefore the function

${\ displaystyle {\ dot {\ vec {q}}} = {\ frac {{\ vec {p}} \, c ^ {2}} {\ sqrt {m ^ {2} \, c ^ {4} + {\ vec {p}} ^ {2} \, c ^ {2}}}}}$ of the momentum.

### Harmonic oscillator

The Hamilton function of a one-dimensional harmonic oscillator is given by:

${\ displaystyle {\ mathcal {H}} (x, p) = {\ dot {x}} p - {\ mathcal {L}} (x, {\ dot {x}}) = {\ frac {p ^ {2}} {2m}} + {\ frac {m} {2}} \ omega _ {0} ^ {2} x ^ {2} = T + V = E}$ ### Charged particle in an electromagnetic field

In Cartesian coordinates ( ), the Lagrange function of a particle of charge moving through an electromagnetic field is ${\ displaystyle {\ vec {q}} = {\ vec {x}}}$ ${\ displaystyle q}$ ${\ displaystyle {\ mathcal {L}} = {\ frac {1} {2}} m {\ dot {\ vec {x}}} ^ {2} + q \ left ({\ dot {\ vec {x }}} \ cdot {\ vec {A}} \ right) -q \ phi}$ Here is the electrical potential and the vector potential of the magnetic field. The canonical impulse is ${\ displaystyle \ phi}$ ${\ displaystyle {\ vec {A}}}$ ${\ displaystyle {\ vec {p}} = {\ frac {\ partial {\ mathcal {L}}} {\ partial {\ dot {\ vec {x}}}}} = m {\ dot {\ vec { x}}} + q {\ vec {A}}}$ This equation can be rearranged so that the speed is expressed in terms of momentum:

${\ displaystyle {\ dot {\ vec {x}}} = {\ frac {1} {m}} \ left ({\ vec {p}} - q {\ vec {A}} \ right)}$ If the expression for and is used in the definition of the Hamilton function, this results in: ${\ displaystyle {\ dot {\ vec {x}}}}$ ${\ displaystyle {\ vec {p}}}$ ${\ displaystyle {\ mathcal {H}} = {\ dot {\ vec {x}}} \ cdot {\ vec {p}} - {\ mathcal {L}} = {\ frac {1} {2m}} \ left ({\ vec {p}} - q {\ vec {A}} \ right) ^ {2} + q \ phi}$ ## 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 .