ⓘ Hamiltons principle. In physics, Hamiltons principle is William Rowan Hamiltons formulation of the principle of stationary action. It states that the dynamics o ..


ⓘ Hamiltons principle

In physics, Hamiltons principle is William Rowan Hamiltons formulation of the principle of stationary action. It states that the dynamics of a physical system is determined by a variational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamiltons principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.


1. Mathematical formulation

Hamiltons principle states that the true evolution q t of a system described by N generalized coordinates q = between two specified states q 1 = q t 1 and q 2 = q t 2 at two specified times t 1 and t 2 is a stationary point a point where the variation is zero of the action functional

S _{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\;\left{\boldsymbol {\varepsilon }}\cdot {\frac {\partial L}{\partial \mathbf {q} }}-{\boldsymbol {\varepsilon }}\cdot {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right\,dt}

The boundary conditions ε t 1 = ε t 2 = d e f 0 {\displaystyle {\boldsymbol {\varepsilon }}t_{1}={\boldsymbol {\varepsilon }}t_{2}\ {\stackrel {\mathrm {def} }{=}}\ 0} causes the first term to vanish

δ S = ∫ t 1 t 2 ε ⋅ ∂ L ∂ q − d t ∂ L ∂ q d t {\displaystyle \delta {\mathcal {S}}=\int _{t_{1}}^{t_{2}}\;{\boldsymbol {\varepsilon }}\cdot \left{\frac {\partial L}{\partial \mathbf {q} }}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right\,dt}

Hamiltons principle requires that this first-order change δ S {\displaystyle \delta {\mathcal {S}}} is zero for all possible perturbations ε t, i.e., the true path is a stationary point of the action functional S {\displaystyle {\mathcal {S}}} either a minimum, maximum or saddle point. This requirement can be satisfied if and only if

These equations are called the Euler–Lagrange equations for the variational problem.


1.1. Mathematical formulation Canonical momenta and constants of motion

The conjugate momentum p k for a generalized coordinate q k is defined by the equation

p k = d e f ∂ L ∂ q k {\displaystyle p_{k}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\partial L}{\partial {\dot {q}}_{k}}}}.

An important special case of the Euler–Lagrange equation occurs when L does not contain a generalized coordinate q k explicitly,

∂ L ∂ q k = 0 ⇒ d t ∂ L ∂ q k = 0 ⇒ d p k d t = 0, {\displaystyle {\frac {\partial L}{\partial q_{k}}}=0\quad \Rightarrow \quad {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{k}}}=0\quad \Rightarrow \quad {\frac {dp_{k}}{dt}}=0\,}

that is, the conjugate momentum is a constant of the motion.

In such cases, the coordinate q k is called a cyclic coordinate. For example, if we use polar coordinates t, r, θ to describe the planar motion of a particle, and if L does not depend on θ, the conjugate momentum is the conserved angular momentum.


1.2. Mathematical formulation Example: Free particle in polar coordinates

Trivial examples help to appreciate the use of the action principle via the Euler–Lagrange equations. A free particle mass m and velocity v in Euclidean space moves in a straight line. Using the Euler–Lagrange equations, this can be shown in polar coordinates as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy

L = 1 2 m v 2 = 1 2 m x 2 + y 2 {\displaystyle L={\frac {1}{2}}mv^{2}={\frac {1}{2}}m\left{\dot {x}}^{2}+{\dot {y}}^{2}\right}

in orthonormal x, y coordinates, where the dot represents differentiation with respect to the curve parameter usually the time, t. Therefore, upon application of the Euler–Lagrange equations,

d t ∂ L ∂ x − ∂ L ∂ x = 0 ⇒ m x ¨ = 0 {\displaystyle {\frac {d}{dt}}\left{\frac {\partial L}{\partial {\dot {x}}}}\right-{\frac {\partial L}{\partial x}}=0\qquad \Rightarrow \qquad m{\ddot {x}}=0}

And likewise for y. Thus the Euler–Lagrange formulation can be used to derive Newtons laws.

In polar coordinates r, φ the kinetic energy and hence the Lagrangian becomes

L = 1 2 m r 2 + r 2 φ 2. {\displaystyle L={\frac {1}{2}}m\left{\dot {r}}^{2}+r^{2}{\dot {\varphi }}^{2}\right.}

The radial r and φ components of the Euler–Lagrange equations become, respectively

d t ∂ L ∂ r − ∂ L ∂ r = 0 ⇒ r ¨ − r φ 2 = 0 {\displaystyle {\frac {d}{dt}}\left{\frac {\partial L}{\partial {\dot {r}}}}\right-{\frac {\partial L}{\partial r}}=0\qquad \Rightarrow \qquad {\ddot {r}}-r{\dot {\varphi }}^{2}=0} d t ∂ L ∂ φ − ∂ L ∂ φ = 0 ⇒ φ ¨ + 2 r φ = 0. {\displaystyle {\frac {d}{dt}}\left{\frac {\partial L}{\partial {\dot {\varphi }}}}\right-{\frac {\partial L}{\partial \varphi }}=0\qquad \Rightarrow \qquad {\ddot {\varphi }}+{\frac {2}{r}}{\dot {r}}{\dot {\varphi }}=0.}

The solution of these two equations is given by

r = a t + b 2 + c 2 {\displaystyle r={\sqrt {at+b^{2}+c^{2}}}} φ = tan − 1 ⁡ a t + b c + d {\displaystyle \varphi =\tan ^{-1}\left{\frac {at+b}{c}}\right+d}

for a set of constants a, b, c, d determined by initial conditions. Thus, indeed, the solution is a straight line given in polar coordinates: a is the velocity, c is the distance of the closest approach to the origin, and d is the angle of motion.


2. Applied to deformable bodies

Hamiltons principle is an important variational principle in elastodynamics. As opposed to a system composed of rigid bodies, deformable bodies have an infinite number of degrees of freedom and occupy continuous regions of space; consequently, the state of the system is described by using continuous functions of space and time. The extended Hamilton Principle for such bodies is given by

∫ t 1 t 2 dt=0.}

This is called Hamiltons principle and it is invariant under coordinate transformations.


3. Comparison with Maupertuis principle

Hamiltons principle and Maupertuis principle are occasionally confused and both have been called incorrectly the principle of least action. They differ in three important ways:

  • their definition of the action.
Maupertuis principle uses an integral over the generalized coordinates known as the abbreviated action or reduced action S 0 = d e f ∫ p ⋅ d q {\displaystyle {\mathcal {S}}_{0}\ {\stackrel {\mathrm {def} }{=}}\ \int \mathbf {p} \cdot d\mathbf {q} } where p = are the conjugate momenta defined above. By contrast, Hamiltons principle uses S {\displaystyle {\mathcal {S}}}, the integral of the Lagrangian over time.
  • the solution that they determine.
Hamiltons principle determines the trajectory q t as a function of time, whereas Maupertuis principle determines only the shape of the trajectory in the generalized coordinates. For example, Maupertuis principle determines the shape of the ellipse on which a particle moves under the influence of an inverse-square central force such as gravity, but does not describe per se how the particle moves along that trajectory. However, this time parameterization may be determined from the trajectory itself in subsequent calculations using the conservation of energy. By contrast, Hamiltons principle directly specifies the motion along the ellipse as a function of time.
  • .and the constraints on the variation.
Maupertuis principle requires that the two endpoint states q 1 and q 2 be given and that energy be conserved along every trajectory same energy for each trajectory. This forces the endpoint times to be varied as well. By contrast, Hamiltons principle does not require the conservation of energy, but does require that the endpoint times t 1 and t 2 be specified as well as the endpoint states q 1 and q 2.

4.1. Action principle for fields Classical field theory

The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravity.

The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle.

The path of a body in a gravitational field i.e. free fall in space time, a so-called geodesic can be found using the action principle.


4.2. Action principle for fields Quantum mechanics and quantum field theory

In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all imaginable paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, that gives the probability amplitudes of the various outcomes.

Although equivalent in classical mechanics with Newtons laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynmans path integral formulation of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwells equations can be derived as conditions of stationary action.

  • Morse theory. Maupertuis principle and Hamilton s principle exemplify the principle of stationary action. The action principle is preceded by earlier ideas
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  • Spinoza. Some philosophers have associated the principle of sufficient reason with ex nihilo nihil fit Hamilton identified the laws of inference modus ponens
  • physics, particularly in discussions of gravitation theories, Mach s principle or Mach s conjecture is the name given by Einstein to an imprecise hypothesis