Principle of action

In physics, the principle of action is an assertion about the nature of the movement or trajectory of an object or more generally the temporal evolution of a physical system, subjected to predetermined actions.

According to this principle there exists a scalar function defined by an invariant integral called the action integral, such that, over the "path" system time, this function takes extreme values. For example, in classical mechanics the real trajectory that a particle will follow is precisely the one that yields a stationary value of the action. The action is a scalar physical magnitude, representable by a number, with dimensions of energy · time. The principle is a simple, general, and powerful theory for predicting motion in all areas of physics. Extensions of the principle of action describe relativistic mechanics, quantum mechanics, electromagnetism.

The principle is also called the principle of stationary action and principle of least action or principle of least action (although this form is less general and in fact for certain systems it is incorrect to speak of least action). Restricted to classical mechanics, the principle admits a particular formulation known as Hamilton's principle.

History

The principle of least action was first formulated by Maupertuis in 1746 and later developed (from 1748 onwards) by the mathematicians Euler, Lagrange, and Hamilton. Maupertuis arrived at this principle from the feeling that the very perfection of the universe requires a certain economy in nature and is opposed to any unnecessary expenditure of energy. Natural movements should use some amount to a minimum. It was only necessary to find that amount, and this he proceeded to do. It was the product of the duration (time) of movement within a system by the "vis viva" (violence or living force) or twice what we now call the kinetic energy of the system. Euler (in "Reflexions sur quelques lois générales de la nature.", 1748) adopts the principle of least action, calling quantity "effort". Its expression corresponds to what we would now call potential energy, so that its statement of least action in statics is equivalent to the principle that a system of bodies at rest will assume a configuration that minimizes its total potential energy.

Importance in modern physics

The principle of action arose in the context of classical mechanics, as a generalization of Newton's laws. In fact, in inertial systems, the principle of least action and Newton's laws are equivalent. However, the greater ease of generalizing the action principle makes it preferable in certain types of complex applications, which makes the principle occupy a central role in modern physics. In fact, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics or field theory. Feynman's formulation of quantum mechanics is based on a stationary action principle, using path integrals. Maxwell's equations can be derived as conditions of a stationary action.

Scheme of the curvature of space-time around a source of gravity force.

Many problems in physics can be represented and solved in the form of a principle of action, such as finding the fastest way to descend onto the beach to reach a drowning person. Water running down slopes seeks the steepest slope, the quickest way down, and water running in a basin is distributed so that its surface is as low as possible. Light finds the fastest path through an optical system (Fermat's principle of shortest time). The path of a body in a gravitational field (ie, free fall in space-time, a so-called geodesic) can be found using the action principle.

Symmetries in a physical situation can best be dealt with by the action principle, together with the Euler-Lagrange equations that are derived from the action principle. For example, Emmy Noether's theorem assigns that all continuous symmetry in a physical situation corresponds to a conservation law. This deep connection, however, requires assuming the principle of action.

In classical (non-relativistic, non-quantum) mechanics, the correct choice of action can be derived from Newton's laws of motion. Conversely, the action principle proves Newton's equation of motion given the correct choice of action. Therefore, in classical mechanics the principle of action is equivalent to Newton's equation of motion. The use of the principle of action is often simpler than the direct use of Newton's equation of motion. The principle of action is a scalar theory, with derivations and applications using elementary calculus.

The principle of action in classical mechanics

Newton's laws of movement can be established in several alternative ways. One of them is the Lagrangian formalism, also called lagrangian mechanics. That we will set it in general coordinates, so that we can use cartesins, polar or spherical, as required by the system to treat. If we denote the trajectory of a particle based on time t Like q(t)with a speed ${displaystyle {dot {q}}(t)}$, then lagrangian is a function dependent on these quantities and possibly also explicitly of the time:

${displaystyle L(q(t),{dot {q}}(t),t)}$

the integral action S is the temporary integral of the lagrangian between a given starting point ${displaystyle q(t_{1})}$ in time ${displaystyle t_{1}$ and a given final point ${displaystyle q(t_{2}}}}$ in time ${displaystyle t_{2}}$

${displaystyle S=int _{t_{1}}{t_{2}}L(q(t),{dot {q}}(t),t)dt}$

In lagrangiana mechanics, the trajectory of an object is derived finding the path for which the integral action S is stationary (a minimum or a stretch point). The integral action is a functional (a function depending on a function, in this case ${displaystyle q(t)}$).

For a system with conservative forces (forces that can be described in terms of a potential, such as the gravitational force and not the frictional forces), choosing a Lagrangian as the kinetic energy minus the potential energy gives rise to Newton's correct laws of mechanics (note that the sum of the kinetic and potential energy is the total energy of the system).

Euler-Lagrange equations for the action integral

One-dimensional case

The stationary point of an integral along a path is equivalent to a system of differential equations, called the Euler-Lagrange equations. This can be seen as follows where we restrict ourselves to one coordinate only. The extension to more coordinates is straightforward.

Suppose we have an action integral S of a constituent L which depends on the coordinates ${displaystyle x(t)}$ and ${displaystyle {dot {x}}(t)}$their derivatives with respect to t:

${displaystyle S=int _{t_{1}}^{t_{2}}{;L(x,{dot {x}}(t))dt}$

consider a second curve ${displaystyle x_{1}(t)}$ that begins and ends at the same points as the first curve, and assumes that the distance between the two curves is small everywhere: ${displaystyle epsilon (t)=x_{1}(t)-x(t)}$ It's small. In the beginning and in the end we have ${displaystyle epsilon (t_{1})=epsilon (t_{2})=0}$.

The difference between the integrals along curve one and along curve two is:

${displaystyle delta S=int _{t_{1}}{t_{2};(L(x+epsilon{dot {x}}{dot {epsilon}{epsilon}{x}{dot {x}{x}{dot {x}{x1}{x}{x}{x}{x}{dot {x}{x}}}}}}}}}{x(x(x(x)}}}{x(x(x(x)}}}}{x(x)}{x)}}{x)}{x(x(x)}{x(x)}}}{x)}{x)}{x)}}{x)}{x)}}{x(x(x(x(x)}}{x)}{x)}}{x)}{x(x(x(x(x(x(x(x(x)}}}}}}}}}}}}}}}}{$

where we used the first extension of the order L in ε and ${displaystyle {dot {epsilon}}}$. Now use partial integration in the past term and use the conditions ${displaystyle epsilon (t_{1})=epsilon (t_{2})=0}$ to find:

${displaystyle delta S=int _{t_{1}}{t_{2};left(epsilon {partial l over partial x}-epsilon {d over dt}{partial l over partial {dot {x}}}{right)dt}$

S reaches a stationary point (an extremum), ie. δS = 0 for each ε. Note that this is the only requirement: the extremum could be a minimum, a saddle point, or formally equal to a maximum. δS = 0 for every ε if and only if

${displaystyle {partial l over partial x^{a}}-{d over dt{partial l over partial {dot {x}}^{a}}}=0;;;{mbox{ecuciones de Euler-Lagrange}}}}}$

where we have replaced ${displaystyle x^{a}a=0,1,2,3}$ for ${displaystyle x}$, since this must be worth for every coordinate. This system of equations is called the equations of Euler-Lagrange for the variable problem. An important simple consequence of these equations is that if L does not explicitly contain the coordinate xI mean,

Yeah. ${displaystyle {partial L/partial x}=0}$ then. ${displaystyle {partial L/partial {dot {x}}}}}$ It's constant.

Such coordinate x is called a cyclical coordinate Sand ${displaystyle {partial L}/{partial {dot {x}}}}}$

is called the conjugate momentum, which is conserved. For example if L does not depend on time, the associated constant of motion (the conjugate momentum) is called the energy. If we use spherical coordinates t, r, φ, θ and L do not depend on φ, the conjugate momentum is (conserved) angular momentum.

Example: The free particle in polar coordinates

Trivial examples help to appreciate the use of the principle of action via the equations Euler-Lagrange. A free particle (masa m and speed v) in an euclidian space moves in a straight line. Using the Euler-Lagrange equations, this can be shown in polar coordinates as follows. In the absence of potential, lagrangian is simply equal to kinetic energy ${displaystyle mv^{2}/2={dot {x}}{^{2}/2+{dot {y}}{2}{2}/2}$ in ${displaystyle (x,y)}$ Orthonormal coordinates, where the point represents the differentiation with respect to the curve parameter (usually the time t). In polar coordinates (r, φ) kinetic energy and therefore the grenagian becomes

${displaystyle l={frac {1}{2}}}left({dot {r}{2}{2}{2}{dot {phi }{2}{2}right)}}$

radial components ${displaystyle phi }$ of the Euler-Lagrange equations are converted, respectively

${displaystyle {frac {d}{dt}}{left({frac {partial l}{partial {dot {r}}}}}{right)}{frac {partial l}{partial r}}=0qquad Rightarrow qquad {ddot {r}{dot}{$

${displaystyle {frac {d;}{dt}}{left({frac {partial l}{partial l}{dot {phi }}{right}{frac}{partial l}{partial phi }}}{qquad Rightarrow qquad {ddot {$

that the solution of these two equations is given

${displaystyle {begin{cases}rcos phi =at+brsin phi =ct+dend{cases}}qquad {begin{cases}r={sqrt {(at+b){2}{2}(ct+d)^{2}{2}{phi =arctancase left({frac}{$

for a system of constants ${displaystyle a,b,c,d}$ determined by initial conditions. So, actually, the solution is a straight line given in polar coordinates.

N-dimensional case

In the case n-dimensional the action associated with a physical field ${displaystyle phi _{r}^{alpha },}$ lagrangian is a density over the n-dimensional space, and therefore the action is an integral over a n-dimensional domain:

${displaystyle S[phi _{r}^{alpha }]equiv int _{M}{mathcal {L}}{O}{Big (}{big (}{phi _{r^}{alpha }{x),{partial phi _{r}{alpha }{big}{big},x{$

Given certain contour conditions on the edge of a region ${displaystyle Vsubset M}$, then the equations of the movement are given by the equations of Euler-Lagrange:

${displaystyle {frac {partial }{partial x{mu }}}{left({frac {partial {mathcal {L}{O}}}{partial}{partial}{mu }{phi _{r}{alphah}}}}{right}{frac {partial {math}{$

Incidentally, the left side is the functional derivative of the action with respect to ${displaystyle phi _{r}^{alpha }}}$.

A Note on Formalism

The formalisms above are valid in classical mechanics in a very restrictive sense of the term. More generally, an action is a functional from the configuration space to the real numbers, and in general, it need not necessarily be even an integral because non-local actions are possible. The configuration space need not necessarily be a function space because we could have things like noncommutative geometry.