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Action (physics)

In physics, the action is a fundamental scalar quantity defined as the integral over time of the L, which is the difference between the T and V of a : S = \int_{t_1}^{t_2} L \, dt = \int_{t_1}^{t_2} (T - V) \, dt. For a single particle in one dimension, this takes the explicit form S = \int_{t_1}^{t_2} \left[ \frac{1}{2} m \left( \frac{dx}{dt} \right)^2 - V(x) \right] dt, where m is the and V(x) is the potential. The principle of least action asserts that the actual path followed by the between two fixed points in time and configuration space is the one that makes , meaning the first variation \delta S = 0, typically corresponding to a minimum or value of S. This provides a unified framework for deriving the in , equivalent to Newton's laws for conservative systems. The concept of action traces its origins to the 18th century, with Pierre-Louis Maupertuis introducing the principle of least in 1744 as a teleological law inspired by of least time in , positing that nature acts to minimize a proportional to times times . Leonhard Euler refined this idea mathematically in the 1740s and 1750s, extending it to variational problems and collaborating with Maupertuis to apply it to mechanical systems. further developed the formulation in the 1780s by expressing mechanics in terms of , leading to the Euler-Lagrange equations derived from \delta S = 0. consolidated the modern version in the 1830s, framing the principle as a geometric analogy to optics in phase space and introducing the Hamiltonian as the Legendre transform of the Lagrangian, which facilitated applications to quantum mechanics. Mathematically, applying the principle involves the : for a path x(t) + \eta(t) with \eta(t_1) = \eta(t_2) = 0, the condition \delta S = 0 yields the Euler-Lagrange equation \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}} \right) - \frac{\partial L}{\partial x} = 0, which for L = T - V reproduces Newton's second law m \ddot{x} = -\frac{\partial V}{\partial x}. In , the action for a is S = -m c^2 \int_{t_1}^{t_2} \sqrt{1 - v^2/c^2} \, dt, where m is the rest mass and c is the , leading to the equations of . For systems with electromagnetic fields, the action includes interaction terms like -q \int (\phi - \mathbf{v} \cdot \mathbf{A}) \, dt, where q is charge, \phi is , and \mathbf{A} is . The principle of least action extends beyond , serving as the cornerstone for and formulations that underpin field theories, , and . In , it connects to path integrals, where the amplitude for a process is the sum over all paths weighted by e^{iS/\hbar}, with the classical path dominating when S \gg \hbar. Symmetries of the action lead to conservation laws via , such as from time-translation invariance. This versatility makes the action a powerful tool for unifying diverse physical laws, from particle dynamics to cosmology.

Introduction

Core concept

In physics, the action S is a scalar defined as the time of the L, which for classical mechanical s is given by the difference between T and V: S = \int_{t_1}^{t_2} L \, dt = \int_{t_1}^{t_2} (T - V) \, dt. This quantifies the dynamics of a along a specific path in configuration space. The units of action are those of energy multiplied by time, corresponding to joule-seconds (J·s) in the (SI). This dimensional structure aligns action with other fundamental quantities, such as the reduced Planck's constant \hbar, which also carries units of J·s. Physically, the action serves as a measure of the "effort" expended by a system or the "cost" associated with a particular path in variational principles, where the actual trajectory of motion corresponds to the path that makes the action stationary (typically a minimum or ). This interpretation underscores its role in determining the laws of motion through optimization rather than direct force balances. The term "" derives from the Latin actio, meaning "a doing," "performing," or "motion," rooted in the agere ("to do, , set in motion"). This reflects the concept's emphasis on dynamic processes in .

Illustrative example

To illustrate the concept of in , consider a particle of m falling vertically under a constant V(y) = m g y, where y is the height coordinate (positive upward) and g = 9.8 \, \mathrm{m/s^2} is the . The is given by the time integral of the , S = \int_{t_1}^{t_2} \left[ \frac{1}{2} m \dot{y}^2 - m g y \right] dt, with fixed endpoints in position and time. A simple numerical case involves the particle starting from rest at height h = 4.9 \, \mathrm{m} (y(0) = 4.9 \, \mathrm{m}, \dot{y}(0) = 0) and reaching y(T) = 0 after fixed time T = 1 \, \mathrm{s}. The actual physical path, derived from Newton's second law \ddot{y} = -g, is the straight-line trajectory y(t) = 4.9 - 4.9 t^2, with velocity \dot{y}(t) = -9.8 t. Substituting into the Lagrangian yields L(t) = \frac{1}{2} m (9.8 t)^2 - m g (4.9 - 4.9 t^2) = m (48.02 t^2 - 48.02 + 48.02 t^2) = m (96.04 t^2 - 48.02). Integrating gives the action S = m \int_0^1 (96.04 t^2 - 48.02) \, dt = m \left[ \frac{96.04}{3} t^3 - 48.02 t \right]_0^1 = m (32.01 - 48.02) = -16.01 m (in units of \mathrm{J \cdot s}, assuming m in kg). This path extremizes the action according to Hamilton's principle. To demonstrate minimization, compare this to a nearby non-physical path that deviates horizontally while preserving the same y(t) and endpoints (e.g., adding a small lateral x(t) = a \sin(2 \pi t) with a = 1 \, \mathrm{m}, so x(0) = x(1) = 0). The now includes a component \dot{x}(t) = 2 \pi a \cos(2 \pi t), increasing the term to \frac{1}{2} m \left[ \dot{y}^2 + \dot{x}^2 \right] while the potential remains unchanged. The additional contribution to is \Delta S = \int_0^1 \frac{1}{2} m [2 \pi a \cos(2 \pi t)]^2 \, dt = \frac{1}{2} m (4 \pi^2 a^2) \int_0^1 \cos^2(2 \pi t) \, dt = \frac{1}{2} m (4 \pi^2 a^2) \cdot \frac{1}{2} = m \pi^2 a^2 \approx 9.87 m. Thus, the total action is S' \approx -16.01 m + 9.87 m = -6.14 m, which is greater (less negative) than S. The extra from the deviation raises the action, confirming that the straight vertical path minimizes it. Numerical simulations of random paths converging to this minimum via optimization further verify this for gravitational systems. For a more general case like under (initial height 500 m, mass 5 kg, flight time 10 s, small initial horizontal velocity), the minimizing path is a parabola, with action computed similarly along the Newtonian trajectory; deviations increase S due to unbalanced kinetic and potential contributions.

Quantum significance

In 1900, introduced the quantum hypothesis to resolve the in , proposing that the energy of oscillators is exchanged in discrete packets proportional to the frequency ν, given by E = h ν, where h is a universal constant. This marked the birth of , with Planck's constant h representing the fundamental scale at which energy quantization occurs. Planck later described h as the "quantum of action," reflecting its dimensions of (energy × time), which ties it directly to the classical action in physics. In , action is quantized in units of h, or more precisely, the reduced Planck's constant ħ = h / 2π serves as the fundamental quantum of and phase, with physical processes involving changes in action that are multiples of ħ. This quantization underscores the action's central role, bridging classical variational principles to discrete quantum transitions. A key manifestation appears in the semiclassical approximation, particularly the Wentzel–Kramers–Brillouin (WKB) method, where the quantum wavefunction ψ(x) is approximated as ψ(x) ≈ C exp(i S(x) / ħ) in regions away from turning points, with S(x) being the classical action. This exponential phase factor e^{i S / ħ} encodes the classical action's influence on quantum interference, justifying the method's validity when the de Broglie wavelength varies slowly. The classical limit emerges when typical actions S in macroscopic systems greatly exceed ħ, typically by factors of 10^{30} or more, rendering quantum fluctuations negligible and recovering deterministic classical trajectories. For instance, the action for a baseball's flight is on the order of 10^2 J s, vastly larger than ħ ≈ 10^{-34} J s, explaining why quantum effects are imperceptible at everyday scales.

Historical context

Pre-19th century origins

The roots of the concept of action in physics trace back to ancient philosophical ideas about the efficiency and purposefulness of natural motion. In Aristotelian , nature was understood as an internal principle governing motion and rest, ensuring that changes occur in the most direct and goal-oriented manner possible. Aristotle posited that every natural process is directed toward a final cause, or , which explains why objects move toward their proper place—such as heavy elements descending and light elements ascending—without unnecessary detours, embodying a kind of in the . This teleological framework emphasized efficient causation as the means by which forms actualize potentials, avoiding wasteful or violent motions in favor of those aligned with an entity's inherent . Medieval thinkers, building on Aristotelian foundations, further explored these ideas through qualitative analyses of motion, often in the context of theological and mechanical inquiries. Scholars like Jean Buridan and in the 14th century developed impetus theory to explain , advancing qualitative analyses of natural motion. Their work reflected a persistent view of the as an ordered system where divine economy minimized deviations, influencing later interpretations of natural efficiency. A significant precursor to the principle of least action emerged in with Pierre de Fermat's formulation in the 1650s, known as the principle of least time. Fermat argued that rays propagate between two points along the path that minimizes travel time, unifying and under a single optimizing rule and suggesting nature's preference for the most expedient route. This idea, derived from earlier geometric observations, carried teleological implications by implying that "chooses" efficient paths, akin to a purposeful in propagation. In 1744, Pierre-Louis Moreau de Maupertuis explicitly introduced the principle of least , extending Fermat's optical insight to both and mechanical s, asserting that , in all its operations, acts by the simplest means and thus minimizes a quantity he termed "." Maupertuis's formulation posited that the true path of a —whether a ray or a particle—renders this stationary, reflecting a profound metaphysical commitment to divine . His controversial teleological interpretation framed the principle as evidence of purposeful design in the universe, where embeds efficiency into natural laws, sparking debates among contemporaries about the role of final causes in physics. Leonhard Euler advanced these ideas in the 1740s by developing the , providing a systematic method to determine optimal paths that minimize quantities like action. In his 1744 treatise Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Euler transformed Maupertuis's qualitative principle into a rigorous geometric approach, solving problems of brachistochrones and isoperimetrics by considering variations in curvilinear paths. This work emphasized nature's frugality, aligning with teleological views while laying groundwork for later mechanical applications. These 18th-century developments bridged philosophical origins to more formal theories in the following century.

19th-century formalization

In the late 18th century, advanced the mathematical foundations of in his seminal 1788 treatise Mécanique Analytique, where he integrated the concept of action into as a derived from the more fundamental axiom of . Although action—defined through integrals involving and path elements—was not the central pillar of his framework, Lagrange employed it to derive for systems with fixed endpoints, emphasizing its role in unifying diverse mechanical problems under a single variational approach. This formalization marked a shift toward rigorous , building on earlier ideas but prioritizing coordinate-based methods over geometric intuitions. The pivotal 19th-century developments occurred through William Rowan 's contributions in the 1830s, which elevated to a core dynamical quantity. In his 1834 paper "On a General Method in Dynamics," presented to the Royal Society, introduced the as the over time for systems with specified initial and final coordinates, reducing the study of motions in conservative systems to solving a single . 's 1835 follow-up essay expanded this by defining the principal function, which captured 's variation across time-dependent paths, enabling canonical transformations that simplified complex dynamical problems. These innovations were immediately applied to , where demonstrated their utility in for two-body interactions and planetary orbits, providing tools to handle small deviations from Keplerian motion with greater precision. Further generalization came from in the early 1840s, whose lectures at the (1842–1844, edited and published in 1866 as Vorlesungen über Dynamik) extended Hamilton's formalism to multivariable systems and time-dependent forces. Jacobi reformulated the principle of least action in a coordinate-independent manner, introducing transformations that decoupled equations for multi-particle systems and emphasized the action's minimality as a universal variational law. His work facilitated applications to advanced , notably the , where Jacobi's methods using elliptic coordinates and transformations offered new insights into orbital stability and periodic solutions beyond Hamilton's initial scope. These 19th-century advancements solidified action as a foundational in , bridging variational calculus with practical computations in astronomy.

Core definitions

Action functional

In classical mechanics, the action functional for a discrete system described by q(t) is defined as the integral of the L(q, \dot{q}, t) over a time interval from t_1 to t_2: S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt. This formulation, introduced by in his 1834 and 1835 papers on a general method in dynamics, encapsulates the system's dynamics within the framework of the calculus of variations. A key property of the action functional is its stationarity with respect to small variations \delta q(t) in the path q(t), provided these variations satisfy the fixed-endpoint conditions \delta q(t_1) = 0 and \delta q(t_2) = 0. This means that for the physical , the first-order change in vanishes: \delta S = 0. The fixed-endpoint boundary conditions arise naturally in the variational setup, as they eliminate additional boundary terms in the expression for \delta S, ensuring that stationarity depends solely on the interior path. Endpoint variations are thus constrained to zero to focus the principle on paths connecting specified initial and final configurations in configuration space. For numerical computation, the continuous integral can be discretized by dividing the time interval into N steps of size h, approximating the functional as a : S_d \approx \sum_{k=0}^{N-1} L_d(q_k, q_{k+1}), where q_k \approx q(t_k) with t_k = t_1 + k h, and L_d(q_k, q_{k+1}) is a discrete Lagrangian approximating L over each subinterval, often via quadrature (e.g., midpoint rule with L_d = h L\left( \frac{q_k + q_{k+1}}{2}, \frac{q_{k+1} - q_k}{h} \right)). This approach underlies variational integrators that preserve the structure of the original system.

Abbreviated action

The abbreviated action, also referred to as the reduced or short action, is a functional defined for systems with fixed total E, given by the S = \int_{q_1}^{q_2} \mathbf{p} \cdot d\mathbf{q}, where the is taken along a in configuration space from initial coordinates \mathbf{q}_1 to final coordinates \mathbf{q}_2, and \mathbf{p} is the conjugate to \mathbf{q}./09%3A_Maupertuis_Principle_-_Minimum_Action_Path_at_Fixed_Energy/9.03%3A_The_Abbreviated_Action) This form, often denoted as S_0 or \bar{S}, arises in the context of varying paths that conserve and connect the specified endpoints, with the physical extremizing S. Unlike the full integral over time, the abbreviated action is independent of the temporal parameterization, focusing instead on the spatial in projected onto configuration space. The abbreviated action is connected to the standard action functional S_{\text{full}} = \int_{t_1}^{t_2} L \, dt through a Legendre transform, which shifts the variational parameter from time t to E: specifically, \bar{S}(q_1, q_2, E) = S_{\text{full}}(q_1, q_2, t) + E t, where t is the duration of the path satisfying the energy constraint. This transformation ensures that the extremum condition for \bar{S} with respect to paths at fixed E yields the same trajectory as minimizing S_{\text{full}} over time-parameterized paths, but it simplifies calculations by eliminating explicit time dependence for conservative systems. For a single particle of m moving in a one-dimensional conservative potential V(q), the p = \sqrt{2m(E - V(q))} allows an explicit expression for the abbreviated action: S = \int_{q_1}^{q_2} \sqrt{2m \left( E - V(q) \right)} \, dq, which represents the spatial arc length in an effective geometry defined by the potential./09%3A_Maupertuis_Principle_-_Minimum_Action_Path_at_Fixed_Energy/9.03%3A_The_Abbreviated_Action) This construct finds significant use in geodesic problems on Riemannian manifolds, where extremizing the abbreviated action corresponds to finding shortest paths (geodesics) that minimize the integrated momentum along the curve, analogous to straight lines in flat space but adapted to curved geometries via the metric tensor. In optics, the abbreviated action parallels Fermat's principle, with the integral \int n \, ds (optical path length, where n is the refractive index) serving as an effective momentum integral that determines light ray trajectories through varying media, providing a corpuscular interpretation of wave propagation.

Hamilton's characteristic function

Hamilton's characteristic function, denoted W(\mathbf{q}, \boldsymbol{\alpha}; t), is defined as the value of the action functional S evaluated along the true dynamical path connecting a fixed configuration \boldsymbol{\alpha} at time zero to the final \mathbf{q} at time t. This function encapsulates the integral of the over the extremal trajectory, providing a that links initial and final states in . In the context of canonical transformations, W serves as a type-2 generating function F_2(\mathbf{q}, \boldsymbol{\alpha}, t), facilitating a transformation from the old phase space variables (\mathbf{q}, \mathbf{p}) to new variables (\boldsymbol{\alpha}, \mathbf{P}), where \boldsymbol{\alpha} represents constant initial coordinates and \mathbf{P} are the corresponding constant momenta. This transformation simplifies the equations of motion by rendering the new Hamiltonian dependent only on the constants \mathbf{P}, thereby integrating the system directly. The partial derivatives of W yield the momenta associated with the endpoints of the path: the final momentum \mathbf{p} = \frac{\partial W}{\partial \mathbf{q}} and the negative of the initial momentum -\mathbf{P} = \frac{\partial W}{\partial \boldsymbol{\alpha}}. These relations arise from the , ensuring that variations in the extremize the action while preserving the symplectic structure of . For conservative systems where the Hamiltonian is independent of time, the characteristic function separates into a time-independent part and a linear time term: W(\mathbf{q}, \boldsymbol{\alpha}; t) = \tilde{W}(\mathbf{q}, \boldsymbol{\alpha}) - E t, with E the constant total . Here, \tilde{W} is the abbreviated , corresponding to the \int \mathbf{p} \cdot d\mathbf{q} along the path, which simplifies the Hamilton-Jacobi to a time-independent form H\left(\mathbf{q}, \frac{\partial \tilde{W}}{\partial \mathbf{q}}\right) = E.

Generalized coordinate action

In , systems with multiple are described using q_1, q_2, \dots, q_n, which provide a minimal set of parameters to specify the , allowing the action to be expressed in a compact form. The action S for such a system is defined as the time integral of the L, given by S = \int_{t_1}^{t_2} L(q_1, \dots, q_n, \dot{q}_1, \dots, \dot{q}_n, t) \, dt, where the typically takes the form L = T - V, with T the and V the , both expressed in terms of the generalized coordinates and their time derivatives. This formulation generalizes the single-particle action by accommodating arbitrary coordinate choices, such as angles or arc lengths, that simplify the description of constrained or complex motions. The T in is a homogeneous in the velocities, reflecting the inertial properties of the system: T = \frac{1}{2} \sum_{i,j=1}^n m_{ij}(q_1, \dots, q_n, t) \dot{q}_i \dot{q}_j, where m_{ij} are elements of the (or in coordinate space), which may depend on the coordinates and time to account for variable geometry or couplings between . This quadratic structure ensures that the leads to linear definitions in the generalized velocities, facilitating the application of variational principles. For systems subject to holonomic constraints, such as f_k(q_1, \dots, q_n, t) = 0 for k = 1, \dots, m, the action is modified using Lagrange multipliers \lambda_k to enforce these restrictions without reducing the coordinate set prematurely. The constrained action becomes S = \int_{t_1}^{t_2} \left[ L(q, \dot{q}, t) + \sum_{k=1}^m \lambda_k(t) f_k(q, t) \right] dt, where the multipliers \lambda_k act as additional dynamical variables that incorporate the constraint forces into the variational framework./05:_Calculus_of_Variations/5.09:_Lagrange_multipliers_for_Holonomic_Constraints) This approach, introduced by Lagrange, preserves the independence of the coordinates while ensuring the stationary path satisfies both the dynamics and constraints. A representative example is the , consisting of two point masses m_1 and m_2 connected by massless rods of lengths l_1 and l_2, with \theta_1 and \theta_2 measuring the angles from the vertical. The is T = \frac{1}{2} (m_1 + m_2) l_1^2 \dot{\theta}_1^2 + m_2 l_1 l_2 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2) + \frac{1}{2} m_2 l_2^2 \dot{\theta}_2^2, and the is V = -(m_1 + m_2) g l_1 \cos \theta_1 - m_2 g l_2 \cos \theta_2. The is then S = \int_{t_1}^{t_2} (T - V) \, dt, capturing the coupled motion without explicit constraint enforcement, as the angular coordinates inherently satisfy the fixed-length constraints.

Relativistic particle action

In special relativity, the action for a free of rest mass m and charge q is formulated to ensure Lorentz invariance, treating the particle's worldline in four-dimensional Minkowski . The fundamental form expresses the action in terms of the \tau, the time measured in the particle's , yielding S = -m c^2 \int d\tau, where c is the . This can equivalently be written using the infinitesimal ds = c d\tau, as S = -m c \int ds, with ds^2 = c^2 dt^2 - d\mathbf{x}^2 in the (+,-,-,-). An alternative parametrization uses coordinate time t as the integration variable, leading to the Lagrangian L = -m c^2 \sqrt{1 - v^2/c^2}, where \mathbf{v} = d\mathbf{x}/dt is the three-velocity, so the action becomes S = \int L \, dt = -m c \int \sqrt{1 - v^2/c^2} \, dt. This form recovers the non-relativistic kinetic energy \frac{1}{2} m v^2 in the low-velocity limit v \ll c, confirming consistency with classical mechanics. The proper time \tau relates to coordinate time via d\tau = dt \sqrt{1 - v^2/c^2}, ensuring both expressions are equivalent. To incorporate interactions with an external electromagnetic field, described by the four-potential A^\mu = (\phi/c, \mathbf{A}), the action extends to S = -m c^2 \int d\tau + \frac{q}{c} \int A_\mu dx^\mu, where dx^\mu = (c dt, d\mathbf{x}) is the four-displacement along the worldline. In three-vector notation (Gaussian units), this corresponds to L = -m c^2 \sqrt{1 - v^2/c^2} + \frac{q}{c} \mathbf{v} \cdot \mathbf{A} - q \phi. The interaction term arises from the minimal coupling principle, replacing the derivative with the covariant derivative to maintain gauge invariance alongside Lorentz symmetry. The entire action is a Lorentz scalar, invariant under transformations mixing space and time coordinates, as it depends solely on the invariant spacetime interval ds. This invariance guarantees that the equations of motion, derived via the principle of stationary action, transform covariantly between inertial frames, underpinning the relativistic dynamics of the particle.

Foundational principles

Principle of stationary action

The principle of stationary action, also known as , asserts that the true of a between specified initial and final configurations at times t_1 and t_2 renders the action functional S stationary with respect to small variations in the path, such that the first variation vanishes: \delta S = 0. This principle provides a variational foundation for , unifying diverse phenomena under a single extremal condition on the action S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt, where L is the . To derive this condition, consider a varied path q(t) + \delta q(t) with fixed endpoints, so \delta q(t_1) = \delta q(t_2) = 0. The first variation of is then \delta S = \left[ p \delta q \right]_{t_1}^{t_2} + \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} \right) \delta q \, dt = 0, where p = \partial L / \partial \dot{q} denotes the generalized momentum. The boundary term vanishes due to the fixed endpoints, leaving the integral term, which must hold for arbitrary \delta q(t). This implies that the integrand vanishes pointwise, yielding the without requiring the action to achieve a global minimum or maximum—stationarity suffices, as the true path may correspond to a in the space of trajectories. Historically, introduced this principle in his 1834 paper, framing it through the "principal function," which represents the stationary value of as a function of initial and final coordinates and times, thereby linking to a satisfying partial differential equations. This formulation elevated from an earlier least-action concept to a more general stationary principle, foundational to subsequent developments in .

Maupertuis's principle

Maupertuis's principle, formulated in 1744, posits that for a physical system with fixed total energy E, the true trajectory between two configuration points q_1 and q_2 is the one that minimizes the abbreviated action W = \int_{q_1}^{q_2} \mathbf{p} \cdot d\mathbf{q}, where \mathbf{p} is the momentum. This variational condition selects paths in configuration space at constant energy, reducing the dynamics to a geodesic problem on the energy surface. The principle emerges from the broader principle of stationary action when holds, allowing the time-dependent action S = \int (T - V) dt to be reparameterized such that the fixed-energy constraint transforms it into the integral over , independent of time. In this form, the stationary path satisfies the for conservative systems, bridging and through analogous minimization. A key application lies in , where Maupertuis adapted of least time to a , deriving of : for light rays crossing media with different speeds, the path minimizes \int n ds, equivalent to \int p dq with n proportional to . In , it applies to problems like the brachistochrone, framing the curve of fastest descent under as minimizing an action-like quantity along fixed-energy contours, though typically solved via least time. This formulation assumes a time-independent potential and conservative forces to enforce energy conservation, restricting its use to equilibrium-like paths and excluding dissipative or explicitly time-varying systems where the full action principle is required.

Hamilton's principal function

In classical mechanics, Hamilton's principal function, often denoted as S(\mathbf{q}, \mathbf{Q}; t), represents the value of the action functional evaluated along the extremal trajectory that connects an initial configuration \mathbf{Q} at time t = 0 to a final configuration \mathbf{q} at time t. This function encapsulates the integral of the Lagrangian over the true path of motion, distinguishing it from the action considered over arbitrary paths in the principle of stationary action. Introduced by William Rowan Hamilton in his foundational work on dynamics, S provides a complete solution to the equations of motion for a given system, linking initial and final states through the variational principle. A key property of Hamilton's principal function is its role as a of the first kind for transformations in . Specifically, it relates the old coordinates and momenta (\mathbf{q}, \mathbf{p}) to the new ones (\mathbf{Q}, \mathbf{P}) via the partial derivatives: p_i = \frac{\partial S}{\partial q_i}, \quad P_i = -\frac{\partial S}{\partial Q_i}, where the indices i label the . These relations ensure that the transformation preserves the structure of Hamilton's equations, allowing S to map the time-dependent trajectory to a set of constant new coordinates and momenta. By specifying initial conditions through \mathbf{Q} and \mathbf{P}, S thus generates the full of the system without solving the differential equations directly. The principal function facilitates the of solutions over time by providing an explicit means to compute positions and momenta at any instant from the initial data. For a given set of constants \mathbf{P}, the extremal is determined such that S minimizes , and the resulting \mathbf{q}(t) and \mathbf{p}(t) follow Hamilton's equations. This is particularly useful in systems where direct is challenging, as S transforms the problem into finding constants of motion. For systems with time-independent Hamiltonians, Hamilton's principal function separates into a time-dependent linear term and a time-independent part: S(\mathbf{q}, \mathbf{Q}; t) = W(\mathbf{q}, \mathbf{Q}) - E t, where W is known as Hamilton's characteristic function and E is the total energy, a constant of the motion. This decomposition highlights the role of W in the time-independent case, analogous to S but focused on fixed-energy surfaces rather than explicit time intervals.

Hamilton–Jacobi equation

The is a first-order (PDE) in that governs the evolution of the principal function S, also known as the action, in the formulation of dynamics. It was originally formulated by in 1834 as a central tool for transforming and solving the . This equation shifts the focus from solving Hamilton's ordinary differential equations directly to finding a suitable that simplifies the system. The standard form of the time-dependent Hamilton–Jacobi equation for a system with q_i and H(q, p, t) is \frac{\partial S}{\partial t} + H\left(q, \frac{\partial S}{\partial q}, t\right) = 0, where the momenta are given by p_i = \frac{\partial S}{\partial q_i}. This PDE arises naturally in the context of canonical transformations generated by S, allowing the original dynamics to be reformulated. The relation p = \frac{\partial S}{\partial q} identifies S as the generator of the transformation, linking the action directly to the variables. Solutions to the Hamilton–Jacobi equation are sought in the form of a complete integral, S(q, \alpha_i, t; a_i), which depends on the coordinates q, time t, and n arbitrary constants \alpha_i (for an n-degree-of-freedom system), with the additional parameters a_i denoting explicit dependence if present. Carl Gustav Jacob Jacobi emphasized in 1842 that such a complete integral, satisfying \det\left(\frac{\partial^2 S}{\partial q_i \partial \alpha_j}\right) \neq 0, fully determines the general solution to the equations of motion through differentiation. The constants \alpha_i serve as the new momenta in the transformed coordinates, facilitating the integration of the system. A key advantage of the lies in its ability to generate canonical transformations to new coordinates (Q_i, P_i) where the transformed vanishes (K = 0), rendering the new momenta P_i constant and solving the dynamics trivially. This transformation often exploits separability of the PDE, allowing additive decomposition of S into terms dependent on individual coordinates, which is particularly useful for integrable systems. For example, consider a in one dimension with H = \frac{p^2}{2m}. A complete solution is S(q, \alpha, t) = \alpha q - \frac{\alpha^2}{2m} t, where \alpha is the constant , yielding p = \alpha and the trivial motion q(t) = q_0 + \frac{\alpha}{m} t. In multiple s, this generalizes to S = \sum_i \alpha_i q_i - \frac{\sum_i \alpha_i^2}{2m} t.

Key derivations

Euler–Lagrange equations

The Euler–Lagrange equations arise as the necessary condition for functional to be stationary under admissible variations of the path in configuration space, as established in the principle of stationary action. These equations provide the explicit differential in for systems described by . Consider a system with a single , where the is L(q, \dot{q}, t). The action is given by S = \int_{t_1}^{t_2} L(q(t), \dot{q}(t), t) \, dt. For the action to be stationary, the first variation \delta S = 0 for variations \delta q(t) that vanish at the endpoints t_1 and t_2. The variation of is \delta S = \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q} \right) dt. Integrating the second term by parts yields \int_{t_1}^{t_2} \frac{\partial L}{\partial \dot{q}} \delta \dot{q} \, dt = \left[ \frac{\partial L}{\partial \dot{q}} \delta q \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) \delta q \, dt. The boundary term vanishes due to the endpoint conditions on \delta q, leaving \delta S = \int_{t_1}^{t_2} \left[ \frac{\partial L}{\partial q} - \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) \right] \delta q \, dt = 0. Since this holds for arbitrary \delta q, the integrand must vanish, yielding the Euler–Lagrange equation \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0. For a system with n generalized coordinates \mathbf{q} = (q_1, \dots, q_n), the L(\mathbf{q}, \dot{\mathbf{q}}, t) leads to a set of n coupled Euler–Lagrange equations, one for each coordinate: \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0, \quad i = 1, \dots, n. If the Lagrangian depends explicitly on time t, the derivation proceeds identically, as the total time derivative in the accounts for any such dependence without altering the form of the equations. Non-holonomic constraints, which are linear in the velocities and not integrable to constraints (e.g., \sum_k a_{jk}(\mathbf{q}, t) \dot{q}_k + a_{t j}( \mathbf{q}, t) = 0 for j = 1, \dots, [m](/page/M)), are incorporated using Lagrange multipliers \lambda_j(t). The modified Euler–Lagrange equations become \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = \sum_{j=1}^m \lambda_j \frac{\partial a_j}{\partial \dot{q}_i}, along with the constraint equations themselves, where the multipliers \lambda_j represent the constraint forces. As an illustrative example, consider a one-dimensional harmonic oscillator with Lagrangian L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} m \omega^2 x^2, where m is the mass and \omega is the angular frequency. Applying the Euler–Lagrange equation gives \frac{d}{dt} \left( m \dot{x} \right) - \left( - m \omega^2 x \right) = 0 \implies m \ddot{x} + m \omega^2 x = 0 \implies \ddot{x} + \omega^2 x = 0, which is the standard equation of simple harmonic motion.

Field-theoretic extensions

The action principle extends to classical field theories by treating fields as continuous functions over , generalizing the variational approach from finite-dimensional mechanical systems to infinite . In this context, the action functional S is defined as a of a density \mathcal{L}, which depends on the field variables, their first derivatives, and possibly the coordinates themselves. This formulation maintains the core idea of extremizing the action to derive , but now over a four-dimensional manifold, typically . The general expression for in field theory is S[\phi] = \int \mathcal{L}(\phi, \partial_\mu \phi, x) \, d^4 x, where \phi(x) represents the field (which could be scalar, vector, or tensorial), \partial_\mu denotes the with respect to spacetime coordinates x^\mu, and the integral spans a suitable in four-dimensional . Stationarity of S under infinitesimal variations \delta \phi that vanish at the boundaries yields the field-theoretic Euler-Lagrange equations: \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \right) - \frac{\partial \mathcal{L}}{\partial \phi} = 0. These equations govern the dynamics of the field, analogous to the point-particle case but accounting for propagation and interactions across space and time. For a real scalar field, a fundamental example is the Klein-Gordon action, with Lagrangian density \mathcal{L} = \frac{1}{2} (\partial_\mu \phi)(\partial^\mu \phi) - \frac{1}{2} m^2 \phi^2, where m is the field's mass. Substituting into the Euler-Lagrange equation produces the Klein-Gordon equation (\partial_\mu \partial^\mu + m^2) \phi = 0, describing a relativistic spin-0 particle and serving as the basis for many field-theoretic models. This form emerged from early attempts to relativize quantum mechanics, as independently derived by Klein and Gordon in 1926. In , the free-field action takes the covariant form S = -\frac{1}{4} \int F_{\mu\nu} F^{\mu\nu} \, d^4 x, where F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu is the antisymmetric field-strength tensor constructed from the four-potential A_\mu. The resulting Euler-Lagrange equations recover in their Lorentz-covariant representation: \partial_\mu F^{\mu\nu} = 0 (homogeneous equations follow identically from the definition of F_{\mu\nu}). This structure inherently respects the Poincaré symmetries of , ensuring Lorentz invariance without additional constraints.

Symmetry and conservation laws

In 1918, established a profound between symmetries and laws in variational principles, demonstrating that every of functional implies the existence of a or quantity. Specifically, Noether's first states that if is invariant under a of the coordinates and fields, then there exists a divergence-free current whose integral over space yields a conserved charge. This provides a systematic framework for deriving laws directly from the symmetries of the physical system's , revolutionizing the understanding of fundamental principles in classical and field theories. A canonical example arises from time-translation invariance, where remains unchanged under a shift \delta t = \epsilon (with \epsilon ). This symmetry, reflecting the homogeneity of time, leads to the , as the H is along the system's trajectory. Similarly, spatial-translation invariance under \delta x^i = \epsilon^i (for each spatial i) implies the conservation of linear , with the \mathbf{p} remaining , underscoring the of . Rotational invariance, corresponding to infinitesimal rotations \delta \theta^k = \epsilon^k, yields conservation of \mathbf{L}, which is preserved when is independent of the orientation of the . In field theories, extends to , generating the canonical stress-energy tensor T^{\mu\nu} as the associated with translations in four-dimensional . The vanishing divergence \partial_\mu T^{\mu\nu} = 0 ensures local and momentum, providing the foundational Noether current for relativistic field actions invariant under Poincaré transformations. This tensor encapsulates the distribution of energy, momentum, and stress within the field configuration, with its integral over space giving the total .

Advanced formulations

Path integral approach

The of provides an alternative to the operator-based approach by expressing the transition from an initial |q_i\rangle at time t=0 to a final \langle q_f| at time t=T as a sum over all possible paths q(t) between these points, each weighted by a determined by the classical S. This is given by \langle q_f | q_i \rangle = \int \mathcal{D}q(t) \, \exp\left( \frac{i}{\hbar} S \right), where the functional \int \mathcal{D}q(t) represents the measure over all paths satisfying the boundary conditions q(0) = q_i and q(T) = q_f, and S = \int_0^T L(q, \dot{q}, t) \, dt with L the . This formulation generalizes the classical principle of stationary to , where between paths with nearby actions leads to constructive or destructive contributions depending on the differences. In the classical limit as \hbar \to 0, the rapid oscillations of the exponential phase cause destructive for paths deviating significantly from the one where the action is stationary, i.e., the classical path satisfying \delta S = 0. The method of phase then approximates the by the contribution from this dominant path, recovering the classical trajectory and the principle of least action. For paths near the classical one, higher-order expansions of the action around the yield semiclassical corrections, such as those in the , highlighting the smooth transition from quantum to classical behavior. To operationalize the functional integral, Feynman introduced a discretization of time into N small intervals of length \epsilon = T/N, representing the amplitude as a multiple integral over intermediate positions q_1, q_2, \dots, q_{N-1}: \langle q_f | q_i \rangle = \lim_{N \to \infty} \left( \frac{m}{2\pi i \hbar \epsilon} \right)^{N/2} \int dq_1 \cdots dq_{N-1} \, \exp\left( \frac{i}{\hbar} \sum_{k=1}^N L(q_{k-1}, \frac{q_k - q_{k-1}}{\epsilon}, t_{k-1}) \epsilon \right), where the short-time propagator for each slice approximates the evolution under the Hamiltonian H = T + V, with kinetic energy T handled exactly and potential V treated as constant over \epsilon. This time-sliced product of propagators converges to the full path integral in the continuum limit \epsilon \to 0. The derivation connects directly to the through the Trotter product formula, which justifies the approximation e^{-i H \epsilon / \hbar} \approx e^{-i T \epsilon / \hbar} e^{-i V \epsilon / \hbar} for small \epsilon, with the error vanishing as O(\epsilon^2). Applying this iteratively over the full time T yields the time evolution operator e^{-i H T / \hbar}, and differentiating with respect to T recovers the i \hbar \partial_t \psi = H \psi. This equivalence demonstrates that the provides a complete, equivalent description of non-relativistic .

Quantum field theory applications

In quantum field theory (QFT), the action S is defined as the spacetime integral of the Lagrangian density, S = \int d^4 x \, \mathcal{L}, where the integral is over Minkowski and \mathcal{L} encodes the dynamics of the fields. This formulation extends the classical action principle to relativistic quantum systems, providing the foundation for quantization. For a in \phi^4 theory, a prototypical model for self-interacting bosons, the density is \mathcal{L} = \frac{1}{2} \partial^\mu \phi \partial_\mu \phi - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!} \phi^4, where m is the mass and \lambda the . In (QCD), describing strong interactions, the includes the non-Abelian gauge sector for gluons and Dirac fields for quarks: \mathcal{L}_\text{QCD} = -\frac{1}{4} G^a_{\mu\nu} G^{a\mu\nu} + \sum_f \bar{q}_f (i \gamma^\mu D_\mu - m_f) q_f, with G^a_{\mu\nu} the field strength tensor, D_\mu the covariant derivative, and summation over quark flavors f. The in QFT, building on the non-relativistic version for , expresses the generating functional Z[J] as Z[J] = \int \mathcal{D}\phi \, \exp\left( \frac{i}{\hbar} \left( S[\phi] + \int d^4 x \, J(x) \phi(x) \right) \right), where J is an external source linearly coupled to the field \phi, and the measure \mathcal{D}\phi functional-integrates over all field configurations. This functional, introduced by Feynman in his approach to , serves as the vacuum-to-vacuum transition amplitude in the presence of sources and generates all correlation functions via functional derivatives: the n-point function is \langle \phi(x_1) \cdots \phi(x_n) \rangle = \frac{1}{Z{{grok:render&&&type=render_inline_citation&&&citation_id=0&&&citation_type=wikipedia}}} \left. \frac{\delta^n Z[J]}{\delta J(x_1) \cdots \delta J(x_n)} \right|_{J=0}. Schwinger's complements this by varying the directly in the quantum theory to derive and symmetries. Perturbative expansions in QFT rely on the terms in the action, visualized through Feynman diagrams where correspond to local interaction vertices in \mathcal{L}. For \phi^4 , the quartic term yields a four-point vertex with factor -i[\lambda](/page/Lambda), while propagators arise from the kinetic term, enabling diagrammatic computation of amplitudes as series in powers of \lambda. In QCD, the three- and four-gluon from the non-linear lead to complex gluon self-interactions, facilitating at high energies. These diagrams systematically organize the expansion of Z[J] around free-field solutions, with integrals capturing quantum corrections. Renormalization addresses ultraviolet divergences in loop diagrams by redefining parameters in the bare , yielding a finite \Gamma[\phi], the Legendre transform of W[J] = -i \hbar \log Z[J], which incorporates all quantum effects as a non-local functional of the mean field. In \phi^4 theory, one-loop renormalization adjusts and coupling to absorb infinities, preserving below the scale. For QCD, the at low energies integrates out high-momentum modes, leading to descriptions of dynamics. This framework, rooted in the , ensures the action's central role in unifying classical principles with quantum phenomenology.

Contemporary developments

In general relativity, the dynamics of spacetime is governed by the Einstein-Hilbert action, which takes the form S = \frac{1}{16\pi G} \int R \sqrt{-g} \, d^4x + S_m, where R is the Ricci scalar, g is the determinant of the metric tensor, G is Newton's gravitational constant, and S_m incorporates the action for matter fields. This formulation, proposed by in 1915, provides a for deriving the , extending the classical action principle to curved geometries. Effective field theories represent another modern application of , treating it as a low-energy effective description of underlying high-energy physics, organized by powers of a small expansion such as over a heavy mass scale. In , exemplifies this approach, using an based on the spontaneously broken chiral symmetry of to describe low-energy interactions. Seminal work by in 1979 established the phenomenological framework, enabling systematic calculations of amplitudes and form factors with controlled uncertainties from higher-order terms. The further extends the role of actions in relating gravitational theories to quantum field theories on boundaries. In the AdS/CFT correspondence, proposed by in 1997, the action of a gravitational theory in computes partition functions equivalent to those of a on the boundary, facilitating non-perturbative insights into strongly coupled systems. This duality has implications for understanding black hole entropy and through boundary actions. Post-2000 developments have explored actions in candidate frameworks. In , the Holst action modifies the Einstein-Hilbert form by including a term proportional to the Immirzi parameter, enabling a that discretizes into spin networks while preserving invariance. This formulation, reviewed in recent analyses, supports background-independent quantization but remains challenged by defining a full dynamics. Similarly, Erik Verlinde's 2010 entropic gravity proposal reinterprets the Einstein-Hilbert action as emerging from holographic entropic forces, deriving gravitational laws from principles without fundamental curvature. Despite these advances, the action principle's role in quantum gravity remains incomplete, as no unified theory reconciling with has been achieved, with ongoing research highlighting tensions in and unification.