Hamilton-Jacobi formalism

The aim of the Hamilton-Jacobi formalism (named after the mathematicians William Rowan Hamilton and Carl Gustav Jakob Jacobi ) of Classical Mechanics is to transform Hamilton's equations of motion by means of a special canonical transformation

${\displaystyle (q,p)\rightarrow (q',p')}$

to simplify. This creates a new Hamilton function that is identical to zero:

${\displaystyle {\tilde {H}}(q',p',t)=0}$

The consequence of this is that both the transformed generalized spatial coordinates and their canonically conjugated pulse coordinates are conserved quantities, i.e. all dynamic quantities in the new Hamilton function are cyclic coordinates : ${\displaystyle q'}$ ${\displaystyle p'}$

${\displaystyle {\begin{aligned}{\frac {\partial {\tilde {H}}}{\partial p'_{k}}}&={\dot {q}}'_{k}=0\quad \Leftrightarrow \quad q'_{k}=\mathrm {const} \\-{\frac {\partial {\tilde {H}}}{\partial q'_{k}}}&={\dot {p}}'_{k}=0\quad \Leftrightarrow \quad p'_{k}=\mathrm {const} .\end{aligned}}}$

These transformed equations of motion are trivial, the problem shifts instead to finding a suitable generator . By adding its partial derivative with respect to time to the untransformed Hamilton function, one obtains the transformed Hamilton function: ${\displaystyle S}$

${\displaystyle {\tilde {H}}(q',p',t)=H(q,p,t)+{\frac {\partial S}{\partial t}}=0.}$

A generating function is specifically selected, which depends on the old location coordinates and the new (constant) pulses , so that ${\displaystyle S(q,p',t)}$ ${\displaystyle q}$ ${\displaystyle p'}$

${\displaystyle p_{k}={\frac {\partial S(q_{k},p'_{k},t)}{\partial q_{k}}}\ ,\quad q'_{k}={\frac {\partial S(q_{k},p'_{k},t)}{\partial p'_{k}}}.}$

Inserted in results in the Hamilton-Jacobi differential equation for : ${\displaystyle {\tilde {H}}=0}$ ${\displaystyle S}$

${\displaystyle H\!\left(q_{k},{\frac {\partial {S}}{\partial q_{k}}},t\right)+{\frac {\partial S}{\partial t}}=0}$

It is a partial differential equation in the variables and for the Hamiltonian action function (the use of the term “ action ” is justified below). ${\displaystyle q_{k}}$ ${\displaystyle t}$ ${\displaystyle S}$

Derivation of the Hamilton-Jacobi equation from the action integral

For the concrete derivation of this differential equation one considers the action functional

${\displaystyle S[q](t)=\int _{0}^{t}L(s,q(s),{\dot {q}}(s))ds}$

with the Lagrange function . The total time derivative of this gives the Lagrange function, ie ${\displaystyle L}$

${\displaystyle {\frac {dS}{dt}}=L}$.

You can see but as a function of the coordinates and on, then for the total time differential ${\displaystyle S}$ ${\displaystyle q}$ ${\displaystyle t}$

${\displaystyle {\frac {dS}{dt}}={\frac {\partial S}{\partial t}}+\sum {\frac {\partial S}{\partial q_{k}}}{\frac {dq_{k}}{dt}}={\frac {\partial S}{\partial t}}+\sum {\frac {\partial S}{\partial q_{k}}}{\dot {q_{k}}}}$.

The partial coordinate derivative results together with the Euler-Lagrange equations

${\displaystyle {\frac {\partial S}{\partial q_{k}}}=\int _{0}^{t}{\frac {\partial L}{\partial q_{k}}}ds=\int _{0}^{t}{\frac {d}{ds}}{\frac {\partial L}{\partial {\dot {q_{k}}}}}ds={\frac {\partial L}{\partial {\dot {q_{k}}}}}=p_{k}}$

with the canonical impulses . By comparing the total time derivatives of , one thus obtains ${\displaystyle p_{k}}$ ${\displaystyle S}$

${\displaystyle {\frac {dS}{dt}}=L={\frac {\partial S}{\partial t}}+\sum p_{k}{\dot {q_{k}}}}$,

from which the asserted equation follows immediately after the definition of the Hamilton function .

Hamilton-Jacobi formalism for a Hamilton function that is not explicitly time-dependent

For conservative systems (i.e. not explicitly time-dependent:) a generating function is constructed for the original Hamilton function, which depends on the old impulses and locations , which transforms it into a new Hamilton function, which only depends on the new (constant) Impulses depends ${\displaystyle H}$ ${\displaystyle H(q,p)\neq H(t)}$ ${\displaystyle S(q,p')}$

${\displaystyle H(q,p)\Rightarrow {\tilde {H}}(p')}$

The new impulses are constants of movement:

${\displaystyle {\dot {p}}'=-{\frac {\partial {\tilde {H}}(p')}{\partial q'}}=0\Leftrightarrow p'=\mathrm {const} ,}$

the new locations only change linearly over time:

${\displaystyle {\dot {q}}'={\frac {\partial {\tilde {H}}(p')}{\partial p'}}=C\Leftrightarrow q'=Ct+b}$ With ${\displaystyle C,b=\mathrm {const} .}$

For apply must ${\displaystyle S(q,p')}$

${\displaystyle p={\frac {\partial S(q,p')}{\partial q}},}$

${\displaystyle q'={\frac {\partial S(q,p')}{\partial p'}}}$

Inserted into the Hamilton function results in the Hamilton-Jacobi differential equation for conservative systems: ${\displaystyle S(q,p')}$

${\displaystyle H(q,p)\Rightarrow H\left(q,{\frac {\partial S(q,p')}{\partial q}}\right)={\tilde {H}}(p').}$

To illustrate , the total derivative with respect to time is calculated ${\displaystyle S}$

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}\,S(q,p')&={\frac {\partial S}{\partial q}}{\dot {q}}+{\frac {\partial S}{\partial p'}}{\dot {p}}'\\&=p{\dot {q}}+q'{\dot {p}}'\\&=p{\dot {q}}\quad \quad \quad \mathrm {wegen} \;{\dot {p}}'=0.\end{aligned}}}$

If we now use the Lagrange equations of motion (with Lagrangian function , where the kinetic energy is the potential): ${\displaystyle L=T-V}$ ${\displaystyle T}$ ${\displaystyle V(q)}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}S(q,p')={\frac {\partial L}{\partial {\dot {q}}}}{\dot {q}}={\frac {\partial T}{\partial {\dot {q}}}}{\dot {q}}=2T}$.

The temporal integration delivers

${\displaystyle S=\int _{t_{1}}^{t_{2}}2T\ \mathrm {d} t=W,}$

therefore is identical with the effect integral. ${\displaystyle S(q,p')}$

Example: The one-dimensional harmonic oscillator

Be any potential. The Hamilton function is ${\displaystyle U=U(q)}$

${\displaystyle H(p,q)={\frac {p^{2}}{2m}}+U(q),}$

the Hamilton-Jacobi equation

${\displaystyle {\frac {1}{2m}}\left({\frac {\partial S(q,p')}{\partial q}}\right)^{2}+U(q)={\tilde {H}}=E.}$

In the one-dimensional oscillator, the only constant is motion. Since it also has to be constant, one sets what is possible for all conservative systems. ${\displaystyle {\tilde {H}}}$ ${\displaystyle p'}$ ${\displaystyle p'={\tilde {H}}=E}$

${\displaystyle \left({\frac {\partial S(q,p')}{\partial q}}\right)^{2}+2mU(q)=2mp'}$

Integrating follows

${\displaystyle S(q,p')={\sqrt {2m}}\int _{q_{0}}^{q}{\sqrt {(p'-U({\tilde {q}}))}}\,\mathrm {d} {\tilde {q}},}$

With ${\displaystyle q'={\frac {\partial S(q,p')}{\partial p'}}}$

${\displaystyle q'={\frac {m}{\sqrt {2m}}}\int _{q_{0}}^{q}{\frac {\mathrm {d} {\tilde {q}}}{\sqrt {p'-U({\tilde {q}})}}}.}$

Because of Hamilton's equation of motion, the following also applies

${\displaystyle {\dot {q}}'={\frac {\partial {\tilde {H}}(p')}{\partial p'}}={\frac {\partial E}{\partial p'}}={\frac {\partial p'}{\partial p'}}=1,}$

${\displaystyle \Rightarrow q'=t-{t_{0}}.}$

In order to be able to display the movement in and , the old coordinates must be transformed back ${\displaystyle p(t)}$ ${\displaystyle q(t)}$

${\displaystyle p(t)={\frac {\partial S(q,p')}{\partial q}}={\sqrt {2m(p'-U(q))}},}$

${\displaystyle q'=t-{t_{0}}={\frac {m}{\sqrt {2m}}}\int _{q_{0}}^{q}{\frac {\mathrm {d} {\tilde {q}}}{\sqrt {E-U({\tilde {q}})}}}.}$

For the special case of the harmonic oscillator, we get with ${\displaystyle U(q)={\frac {1}{2}}aq^{2}}$

${\displaystyle p(t)={\sqrt {2m\left(E-{\frac {1}{2}}aq^{2}\right)}},}$

${\displaystyle q'=t-{t_{0}}={\frac {m}{\sqrt {2m}}}\int _{q_{0}}^{q}{\frac {\mathrm {d} {\tilde {q}}}{\sqrt {E-{\frac {1}{2}}a{\tilde {q}}^{2}}}}.}$

Thus (in case ) ${\displaystyle q_{0}=0}$

${\displaystyle t-{t_{0}}={\sqrt {\frac {m}{a}}}\arcsin {\sqrt {\frac {a}{2E}}}q}$

and ultimately

${\displaystyle q(t)={\sqrt {\frac {2E}{a}}}\sin {\sqrt {\frac {a}{m}}}(t-{t_{0})},}$

${\displaystyle p(t)={\sqrt {2mE}}\cos {\sqrt {\frac {a}{m}}}(t-{t_{0}}).}$

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 .