Hamilton's principle

The Hamiltonian principle of theoretical mechanics is an extremal principle . Physical fields and particles then assume an extreme (i.e. largest or smallest) value for a certain size. This evaluation is called effect, mathematically the effect is a functional , hence the term effect-functional . In many cases, the effect turns out to be not minimal, but only “stationary” (i.e. extreme). That is why some textbook authors also call the principle the principle of stationary action . Some authors also call Hamilton's principle the principle of the smallest action , which, however - as explained above - is not precise.

One example is Fermat's principle , according to which a light beam in a medium traverses all conceivable paths from the starting point to the end point on the path with the shortest transit time.

The Newtonian equations of motion follow from the Hamiltonian principle if the action is suitably selected. But also the Maxwell equations of electrodynamics and the Einstein equations of general relativity can be traced back to a principle of smallest effect.

history

In 1746, Pierre Maupertuis was the first to speak of a generally applicable principle of nature, extreme or optimal (see also Ockham's razor ). Leonhard Euler and Joseph Lagrange clarified in the middle of the eighteenth century that such a principle meant the validity of Euler-Lagrange equations . The Lagrangian formulation of mechanics comes from 1788. In 1834 William Hamilton formulated the principle named after him.

Max Planck interpreted it as an indication that all natural processes are targeted. It is a sign of a purpose of the world beyond the human sensory and cognitive apparatus.

Richard Feynman showed in the 1940's that Hamilton's principle just this results in quantum field theory that all possible paths are (also non-targeted) and allowed integrated up to. Paths with an extreme effect are superimposed constructively and those deviating from them are destructive, so that nature ultimately appears goal-oriented.

Mathematical description

In mechanics, the effect is the time integral over the so-called Lagrangian function

${\displaystyle L(t,\mathbf {x} ,\mathbf {v} ).}$

The Lagrangian is a function of time , place and speed . For example, in Newtonian mechanics the Lagrangian function of a particle of mass moving in potential is the difference between kinetic and potential energy : ${\displaystyle t}$ ${\displaystyle \mathbf {x} }$ ${\displaystyle \mathbf {v} }$ ${\displaystyle m}$ ${\displaystyle V(t,\mathbf {x} )}$

${\displaystyle L(t,\mathbf {x} ,\mathbf {v} )={\frac {1}{2}}m\mathbf {v} ^{2}-V(t,\mathbf {x} ),}$

In relativistic mechanics, the Lagrangian is a free particle

${\displaystyle L(t,\mathbf {x} ,\mathbf {v} )=-mc^{2}{\sqrt {1-\mathbf {v} ^{2}/c^{2}}}.}$

Each path that is traversed from a starting point to an end point over time is assigned the following value: ${\displaystyle \Gamma :t\mapsto \mathbf {x} (t)}$ ${\displaystyle t}$ ${\displaystyle {\underline {\mathbf {x} }}=\mathbf {x} (t_{1})}$ ${\displaystyle {\overline {\mathbf {x} }}=\mathbf {x} (t_{2})}$

${\displaystyle S[\Gamma ]=\int _{t_{1}}^{t_{2}}L{\bigl (}t,\mathbf {x} (t),\mathbf {v} (t){\bigr )}\mathrm {d} t.}$

The effect is therefore the dimension of energy times time. ${\displaystyle S}$

Hamilton's principle now states that of all conceivable paths that initially run through and finally through , those paths in nature are followed that have a stationary effect. For the physically traversed paths, the first variation of the effect disappears : ${\displaystyle {\underline {\mathbf {x} }}}$ ${\displaystyle {\overline {\mathbf {x} }}}$

${\displaystyle \delta S=0.}$

They therefore satisfy the Euler-Lagrange equation

${\displaystyle {\frac {\partial L}{\partial x}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial v}}=0.}$

For example, Newton's equations of motion result for the non-relativistic motion of a particle in the potential

${\displaystyle -\operatorname {grad} V-m{\ddot {x}}=0.}$

For a free relativistic particle, however, the momentum is independent of time:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {m\mathbf {v} }{\sqrt {1-\mathbf {v} ^{2}/c^{2}}}}=0.}$

Hamilton's principle for fields

In field theory , however, the behavior of fields is examined, i. H. how they change and interact with their environment.

Put into the Hamilton principle

${\displaystyle \delta \int _{t_{1}}^{t_{2}}L\,dt=0}$

the Lagrangian density

${\displaystyle L=\int d^{3}r{\mathcal {L}}\left(\phi ,{\frac {\partial \phi }{\partial t}},{\frac {\partial \phi }{\partial x}},{\frac {\partial \phi }{\partial y}},{\frac {\partial \phi }{\partial z}},t\right)}$  with a field ${\displaystyle \,\phi =\phi (x,y,z,t)}$

one gets the Hamilton principle for fields, with

${\displaystyle \delta \int _{t_{1}}^{t_{2}}dt\int d^{3}r\,\,{\mathcal {L}}=0\,.}$

With the obvious identification

${\displaystyle q:\,=\left({\frac {\partial \phi }{\partial t}},{\frac {\partial \phi }{\partial x}},{\frac {\partial \phi }{\partial y}},{\frac {\partial \phi }{\partial z}}\right)}$

the integrand can now be used as

${\displaystyle \sum _{\nu =0}^{3}\,\delta q_{\nu }\,{\frac {\delta {\mathcal {L}}}{\delta q_{\nu }}}\ =\sum _{\nu }\,\delta {\frac {\partial \phi }{\partial q_{\nu }}}\cdot \left(\,-{\frac {\partial }{\partial q_{\nu }}}{\frac {\partial {\mathcal {L}}}{\partial {\frac {\partial \phi }{\partial q_{\nu }}}}}+\,{\frac {\delta {\mathcal {L}}}{\frac {\partial \phi }{\partial q_{\nu }}}}\,\right)}$

to be written.

One recognizes that this formulation is particularly interesting for the theory of relativity , since it is integrated over time and place . Analogous to the usual Hamilton principle, the Lagrangian equations for fields can be determined from this modified version.

Connection to quantum mechanics

If one develops quantum mechanics starting with the path integral formalism , it quickly becomes clear why minimizing the effect of describing classical particle trajectories is so effective. Here it is true that the effect for orbits, which one usually encounters in daily life, is very large measured in Planck's quantum of action, which is often the case due to the large mass of macroscopic objects. Thus, the exponential function in the path integral that contains the effect is a very rapidly oscillating function. The main contribution to the path integral is now provided by terms for which the effect is stationary. It is very important to note that only the requirement for stationarity follows and not a requirement for a minimum value. This also offers the appropriate justification for the fact that it is usually not checked whether the extreme values ​​obtained by minimizing the effect are actually minimum values, because one actually only needs extreme values ​​to obtain a classic description.

properties

Since the operating principle is independent of the coordinate system used, the Euler-Lagrange equations can be investigated in coordinates that are appropriate to the respective problem and, for example, use spherical coordinates when it comes to the movement in the rotationally invariant gravitational field of the sun. This simplifies the solution of the equation.

In addition, constraints can easily be taken into account when mechanical devices restrict the free movement of the mass points, such as the suspension of a spherical pendulum.

Above all, however, in this formulation of the equations of motion, the Noether theorem can be proven, which says that every symmetry of the effect has a conservation quantity and, conversely, every conservation quantity has a symmetry of the effect.

The conserved quantities, in turn, are decisive for whether the equations of motion can be solved by integrals over given functions .

References and footnotes

1. ↑ Continuum and Contact Mechanics , Kai Willner, Springer-Verlag, 2003, p. 288.
2. ↑ Carsten Könneker: Drawing boundaries - or crossing them? Preface to the subject area "Reason and Faith", Spectrum of Science, January 2012.
3. ↑ For a particle of mass in a gravitational field with potential energy , Einstein's general theory of relativity results in the lowest order with respect to : what fits with Taylor expansion with respect to and exactly . ${\displaystyle m}$ ${\displaystyle \phi \,}$ ${\displaystyle \phi \,}$ ${\displaystyle L(t,\mathbf {x} ,\mathbf {v} )\cong -mc^{2}{\sqrt {1-\mathbf {v} ^{2}/c^{2}+{\frac {2\phi }{mc^{2}}}}}}$ ${\displaystyle v^{2}}$ ${\displaystyle \phi \,}$ ${\displaystyle L=T-V}$
4. ↑ for the derivation see L. Landau , JM Lifschitz : Textbook of Theoretical Physics - Volume 1: Mechanics . 14th edition. Harri Deutsch, Frankfurt am Main 2007, ISBN 978-3-8171-1326-2 , pp. 3 f .

Web links

Wikisource: de Maupertuis: Accord de différentes loix de la nature qui avoient jusqu'ici paru incompatibles  - Sources and full texts (French)