Hamilton function

This article deals with the Hamilton function in theoretical mechanics. See the Hamilton function (control theory) for its meaning in the theory of optimal control.

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

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

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 : ${\displaystyle t}$

${\displaystyle {\mathcal {H}}\neq {\mathcal {H}}(t)\Rightarrow {\frac {\mathrm {d} {\mathcal {H}}}{\mathrm {d} t}}={\frac {\partial {\mathcal {H}}}{\partial t}}=0\Rightarrow {\mathcal {H}}=konst.}$

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 .