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Conserved quantity

In physics, a conserved quantity is a measurable property of an isolated that remains constant over time, even as the system's internal state evolves through interactions or processes. These quantities, also known as constants of the motion, underpin fundamental conservation laws that govern the behavior of and in classical and . The most prominent conserved quantities include total energy, which remains invariant in isolated systems due to the time-independence of physical laws; linear momentum, the vector sum of individual momenta that stays constant in the absence of external forces; and , which is preserved under rotational symmetries. Other examples encompass , which is conserved in electromagnetic interactions, and , a quantity maintained in strong and weak nuclear processes. These conservation principles provide powerful constraints for predicting system evolution and are exact for closed systems, with no known violations in fundamental physics. The deep connection between conserved quantities and symmetries is formalized by , which asserts that every of (or ) of a implies a corresponding conserved quantity. For instance, translational invariance in space leads to momentum conservation, while rotational invariance yields angular momentum conservation, and time-translation invariance ensures . This theorem, developed in 1918, extends across classical field theories, , and , highlighting symmetries as the origin of these enduring laws.

Definition and Examples

Definition

In dynamical systems, a conserved quantity is defined as a function of the system's state variables that remains constant along the trajectories of the system, thereby invariant under its time evolution. This invariance implies that the value of the quantity does not change as the system progresses from one state to another, distinguishing it from other observables that may vary. For instance, in physical contexts, quantities such as total energy or linear momentum often serve as conserved quantities in isolated systems. Mathematically, if the state of the system is represented by \mathbf{x}(t), where t denotes time, a conserved quantity C(\mathbf{x}) satisfies the condition \frac{dC}{dt} = 0. This differential equation ensures exact conservation, meaning the quantity holds a strict constant value throughout the system's dynamics in ideal, closed scenarios. However, in real-world systems subject to external influences, dissipation, or modeling approximations, conservation may only be approximate, with the quantity varying slowly or within bounded fluctuations over relevant timescales. The presence of conserved quantities plays a crucial role in analyzing dynamical systems by effectively reducing the number of independent , allowing for a lower-dimensional description of the system's behavior. This reduction simplifies the integration of and aids in understanding long-term stability and qualitative features, without requiring exhaustive computation of all variables.

Physical Examples

In , total is conserved in isolated systems, where no energy enters or leaves, manifesting as the interconversion between kinetic and potential forms during motion. Linear momentum remains conserved in the absence of external forces, as seen in the uniform motion of objects or the balanced outcomes of interactions like billiard ball collisions. Similarly, is conserved without external torques, evident in phenomena such as a spinning top maintaining its rate until intervenes or a altering body position to control speed during flight. The recognition of these conserved quantities traces back to foundational developments in and . In , Isaac Newton's laws of motion, published in 1687, implied the conservation of through the third law's action-reaction principle, providing a basis for analyzing interactions without changes. In , emerged as in the mid-19th century, with Julius Robert Mayer's 1842 observation that and work are interchangeable forms of in isolated processes, formalizing the principle for broader physical systems. Beyond classical mechanics, specific conserved quantities appear in other domains. Electric charge is conserved in all electromagnetic interactions, ensuring that the net charge in a —such as during electron transfers in circuits or particle decays—remains unchanged, as verified through countless experiments in and particle accelerators. In particle physics, is conserved, where protons and neutrons (baryons) carry a value of +1, maintaining balance in reactions like neutron or high-energy collisions, a principle upheld in observations from facilities like . These principles are observable in simple experiments. A demonstrates as it swings, converting at the highest points to at the bottom, with the total remaining constant in the absence of dissipative losses like air resistance. Collisions, such as those between gliders on an air track, illustrate , where the total before impact equals the total after, whether the collision is or inelastic, as measured by changes. Such examples underscore how conserved quantities govern predictable behaviors in physical systems, often linked to underlying symmetries like time-invariance for .

Symmetries and Noether's Theorem

Noether's Theorem

Noether's first theorem establishes a profound connection between symmetries and conservation laws in physical theories. In 1918, developed this result in her seminal paper "Invariante Variationsprobleme," addressing fundamental questions posed by and about the nature of conservation laws in Albert Einstein's general , where appeared to manifest as an identity rather than a . presented work to the Royal Society of Sciences in on July 26, 1918, highlighting its resolution of these issues through the invariance of variational problems. The theorem's insight—that symmetries dictate conserved quantities—has since become a cornerstone of , applicable across classical and quantum regimes. The theorem states that for every differentiable of functional, defined as the integral of the over time, there corresponds a conserved quantity. In the framework of for a system with generalized coordinates q_i and L(q, \dot{q}, t), consider an infinitesimal \delta q_i = \epsilon \xi_i(q, t), where \epsilon is an infinitesimal parameter and \xi_i are the generators of the symmetry. The S = \int_{t_1}^{t_2} L \, dt is invariant under this if the variation of the satisfies \delta L = \frac{d F}{dt} for some function F(q, t), known as the gauge function (often F = 0 for strict symmetries). Along the solutions to the Euler-Lagrange equations, this invariance implies the existence of a conserved Noether charge. The general derivation proceeds from the principle of least action. The total variation of the action under the transformation is \delta S = \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial q_i} \delta q_i + \frac{\partial L}{\partial \dot{q}_i} \delta \dot{q}_i \right) dt + \left[ \frac{\partial L}{\partial \dot{q}_i} \delta q_i \right]_{t_1}^{t_2}, assuming fixed endpoints where boundary terms vanish. Substituting \delta \dot{q}_i = \frac{d}{dt} (\delta q_i) and integrating by parts yields \delta S = \int_{t_1}^{t_2} \left( \left( \frac{\partial L}{\partial q_i} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_i} \right) \delta q_i + \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \delta q_i \right) \right) dt. For \delta S = 0 (or up to boundary terms), and using the Euler-Lagrange equations \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial L}{\partial q_i} which hold on physical trajectories, the remaining term must satisfy \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \delta q_i \right) = \delta L. Given the symmetry condition \delta L = \frac{d F}{dt}, it follows that \frac{d}{dt} \left( \sum_i \frac{\partial L}{\partial \dot{q}_i} \xi_i - F \right) = 0, so the quantity Q = \sum_i \frac{\partial L}{\partial \dot{q}_i} \xi_i - F is conserved, \frac{dQ}{dt} = 0. This conserved charge Q generates the symmetry transformation in . For instance, time-translation invariance yields as the associated charge. The theorem relies on several key assumptions to ensure its validity in classical mechanics. The symmetries must be continuous and differentiable, forming a Lie group whose infinitesimal generators \xi_i are smooth functions, allowing the expansion in \epsilon. The Lagrangian is assumed to be a first-order differential function, typically at most quadratic in velocities, leading to second-order Euler-Lagrange equations, and the action is varied with fixed endpoints. Invariance holds "on-shell," meaning conservation is guaranteed only for trajectories satisfying the equations of motion. The theorem extends naturally to quantum mechanics, where the conserved charge becomes a Hermitian operator commuting with the Hamiltonian, preserving the symmetry in the quantum Hilbert space. These conditions ensure the theorem's broad applicability while highlighting its foundational role in linking geometric symmetries to dynamical invariants.

Applications to Symmetries

Noether's theorem links continuous symmetries of a physical system's to corresponding conserved quantities, providing a foundational explanation for many conservation laws in classical and field theories. A prime example is , which asserts that the laws of physics are invariant under shifts in time. This symmetry yields the , where the total energy—typically the for closed systems—remains constant throughout the system's evolution. Spatial translation symmetry, meaning the laws of physics are the same everywhere in space, leads to the conservation of linear momentum. In systems exhibiting this , the total linear momentum vector is preserved, as demonstrated in the motion of isolated particles or collections of particles without external forces. Rotational symmetry, or the invariance of physical laws under arbitrary rotations in space, corresponds to the conservation of . For instance, in central force problems like planetary orbits, the total angular momentum about the center remains fixed, explaining phenomena such as the stable elliptical paths predicted by Kepler's laws. Internal symmetries extend these principles beyond . Gauge symmetries, such as those in under U(1) transformations, give rise to the conservation of through , where the associated is covariantly conserved on-shell. In non-Abelian gauge theories like , similar symmetries conserve , though it is confined within hadrons. , present in certain massless field theories, leads to the conservation of the , relating to the of the energy-momentum tensor vanishing. When symmetries are broken, conservation laws become approximate. In the weak nuclear interaction, parity symmetry—invariance under spatial inversion—is violated, as established experimentally in beta decay processes, leading to slight deviations from perfect conservation in parity-related quantities, while stronger conservations hold in electromagnetic and strong interactions.

Mathematical Formulation in Dynamical Systems

Ordinary Differential Equations

In dynamical systems described by ordinary differential equations (ODEs) of the form \dot{\mathbf{r}} = \mathbf{f}(\mathbf{r}, t), where \mathbf{r} is the and \mathbf{f} is the , a scalar function H(\mathbf{r}, t) serves as a conserved quantity if its total time derivative vanishes along trajectories. This condition is given by \nabla H \cdot \mathbf{f} + \frac{\partial H}{\partial t} = 0, ensuring that H remains constant for solutions of the system. Geometrically, this implies that the level sets of H are invariant under the flow generated by \mathbf{f}, providing a of the that constrains the motion. Algebraically, it equates the of H along \mathbf{f} to the negative of any explicit time dependence, allowing identification of conserved quantities through solvability of this , known as the transport equation for H. Conserved quantities, often termed first integrals, act as constants of motion that reduce the dimensionality of system. For an n-th order ODE or an equivalent first-order system in n dimensions, each independent first integral restricts the dynamics to an (n-1)-dimensional manifold, effectively lowering the required. This reduction facilitates solving the system by successive quadratures, as the problem decouples into lower-dimensional subsystems on the surfaces defined by the constants. In non-autonomous cases, time-dependent first integrals similarly preserve this utility, though explicit t-dependence complicates the reduction. For non-Hamiltonian systems, where standard structures do not apply, Darboux provides a framework for identifying conserved quantities using integrating factors. This approach, originating from classical methods for polynomial s, involves constructing cofactors and Darboux polynomials—functions whose directional derivatives along \mathbf{f} yield multiples of themselves—to generate first integrals via products or ratios. Integrating factors in this context multiply the vector field to yield an exact form, whose primitives are conserved quantities, extending applicability to dissipative or irregular systems without relying on variational principles. A representative example is the simple harmonic oscillator, governed by the second-order ODE \ddot{x} + \omega^2 x = 0, or equivalently the first-order system \dot{x} = v, \dot{v} = -\omega^2 x. Here, the total energy H(x, v) = \frac{1}{2} v^2 + \frac{1}{2} \omega^2 x^2 satisfies the conservation condition, as \frac{dH}{dt} = v \dot{v} + \omega^2 x \dot{x} = v(-\omega^2 x) + \omega^2 x v = 0, reducing the system to motion on elliptical level curves in the (x, v)-plane. This first integral halves the effective order, yielding the solution x(t) = A \cos(\omega t + \phi) directly from quadrature on the energy manifold.

Integrable Systems

In dynamical systems, particularly , integrability refers to the existence of a complete set of conserved quantities that allow for exact solutions. For a with n , it is defined as integrable if there are n independent conserved quantities I_1, \dots, I_n (including the H = I_1) that are in , meaning their brackets vanish: \{I_i, I_j\} = 0 for all i, j. This condition ensures the system can be reduced to quadratures, enabling analytical integration of the . The Liouville-Arnold theorem provides a geometric characterization of such systems. It states that for a compact, of the defined by the conserved quantities, the foliates into invariant n-dimensional tori, on which the motion is quasi-periodic with frequencies determined by the gradients of the integrals. This structure implies that trajectories are confined to these tori, avoiding ergodic filling of the surface and enabling action-angle coordinates where the depends only on the actions. A classic example is the , describing the motion of a particle in a $1/r potential, which has three degrees of freedom. Its conserved quantities include the total energy E, the vector \mathbf{L} (with three components, though one is redundant due to rotational invariance), and the Laplace-Runge-Lenz vector \mathbf{A} = \mathbf{p} \times \mathbf{L} - m k \hat{\mathbf{r}}, where \mathbf{p} is , m , and k the coupling constant. These quantities are in , confirming integrability and explaining the closed elliptical orbits. Another example is the finite Toda lattice, a one-dimensional chain of n particles with nearest-neighbor exponential interactions V(q_{i+1} - q_i) = e^{-(q_{i+1} - q_i)}. This system possesses n independent conserved quantities in involution, derivable from its Lax pair formulation, allowing exact solution via spectral invariants of the associated . The Toda lattice illustrates how integrability persists in nonlinear chains, with applications to dynamics. While integrable systems are solvable, small perturbations often lead to non-integrability. The Kolmogorov-Arnold-Moser (KAM) theorem addresses this by showing that, for sufficiently small non-integrable perturbations of an integrable , a set of positive measure of the invariant tori persists, with slightly deformed frequencies, provided the unperturbed frequencies are non-degenerate. However, the complementary set of tori is destroyed, giving rise to chaotic regions and highlighting the fragility of full integrability. This partial persistence underscores the distinction between integrable and generic dynamical systems.

In Classical Mechanics

Lagrangian Mechanics

In Lagrangian mechanics, conserved quantities emerge naturally from the structure of the function L(q, \dot{q}, t), where q represents and \dot{q} their time derivatives. When the Lagrangian does not depend explicitly on time, L = L(q, \dot{q}), the system exhibits time-translation invariance, leading to the conservation of the function. This energy E is defined as E = \sum_i \dot{q}_i \frac{\partial L}{\partial \dot{q}_i} - L, which represents the total energy in many systems, such as the sum of kinetic and potential energies for scleronomic constraints. To demonstrate conservation, consider the Euler-Lagrange equations, \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) = \frac{\partial L}{\partial q_i}, for each coordinate i. The time derivative of E is \frac{dE}{dt} = \sum_i \left[ \ddot{q}_i \frac{\partial L}{\partial \dot{q}_i} + \dot{q}_i \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial \dot{q}_i} \ddot{q}_i - \frac{\partial L}{\partial q_i} \dot{q}_i \right] - \frac{\partial L}{\partial t}. The terms involving \ddot{q}_i cancel, and substituting the Euler-Lagrange equation yields \frac{dE}{dt} = -\frac{\partial L}{\partial t}. For time-independent L, \frac{\partial L}{\partial t} = 0, so \frac{dE}{dt} = 0, proving E is conserved along the system's trajectory. Similarly, spatial translation invariance manifests through cyclic coordinates, where the is independent of a particular q_j, so \frac{\partial L}{\partial q_j} = 0. The Euler-Lagrange equation then simplifies to \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_j} \right) = 0, implying the conjugate p_j = \frac{\partial L}{\partial \dot{q}_j} is constant. This conserved corresponds to linear in Cartesian coordinates for translationally invariant systems. A simple example is the free particle in one dimension, with Lagrangian L = \frac{1}{2} m \dot{x}^2, where x is the position. Here, L is independent of x, making x cyclic; the conjugate momentum is p = \frac{\partial L}{\partial \dot{x}} = m \dot{x}, which is conserved, reflecting constant velocity in the absence of forces. More generally, for infinitesimal symmetry transformations \delta q = \epsilon K(q, \dot{q}, t) and \delta t = \epsilon \Lambda(q, \dot{q}, t), where \epsilon is small, the Lagrangian transforms such that the action integral remains invariant up to a total derivative. This leads to a conserved Noether current in the Lagrangian framework: the quantity Q = \frac{\partial L}{\partial \dot{q}} \delta q - L \delta t satisfies \frac{dQ}{dt} = 0 on solutions to the Euler-Lagrange equations, yielding a conserved charge associated with the symmetry. These currents generalize the earlier cases, with time independence (\delta t = \epsilon, \delta q = 0) recovering energy conservation and coordinate independence (\delta q_j = \epsilon, others zero) recovering momentum. This approach connects directly to Noether's theorem by linking symmetries to these conserved quantities without requiring Hamiltonian reformulation.

Hamiltonian Mechanics

In , conserved quantities are identified within the framework of , where the dynamics are governed by the function H(q, p), with q representing and p generalized momenta. This formulation leverages the structure of , endowed with a non-degenerate, closed 2-form \omega = \sum_i dq_i \wedge dp_i, which ensures the preservation of phase space volume under . The , a bilinear operation derived from this geometry, provides the algebraic tool for detecting conservation: for two smooth functions f and g on phase space, the is defined as \{f, g\} = \sum_i \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right). This bracket satisfies antisymmetry, bilinearity, and the , endowing the space of functions with a Poisson structure that underlies the of Hamiltonian vector fields. A function f(q, p) is a conserved quantity along the trajectories of the system if its Poisson bracket with the Hamiltonian vanishes, \{f, H\} = 0, assuming H is time-independent. This condition implies that the total time derivative \dot{f} = \{f, H\} is zero, so f remains constant under the flow generated by Hamilton's equations \dot{q}_i = \frac{\partial H}{\partial p_i} and \dot{p}_i = -\frac{\partial H}{\partial q_i}. In the broader Poisson geometric context, this extends to Poisson manifolds, where the symplectic leaves foliate the phase space, but in standard Hamiltonian mechanics, the invertible Poisson tensor corresponds directly to the symplectic form. Conserved quantities thus label invariant tori or level sets in phase space, facilitating the analysis of integrable systems./15%3A_Advanced_Hamiltonian_Mechanics/15.02%3A_Poisson_bracket_Representation_of_Hamiltonian_Mechanics) Symplectic invariance arises because conserved quantities generate canonical transformations that preserve the Hamiltonian and the symplectic form. Specifically, if f is conserved, the time-\tau flow \phi_\tau generated by the X_f (defined via \omega(X_f, \cdot) = -df) leaves H invariant, as \mathcal{L}_{X_f} H = \{f, H\} = 0. Such transformations are maps, maintaining the structure and thus the form of Hamilton's equations. This generation property links conservation directly to the symmetries of the geometry. A representative example is the of in central force problems, where the potential depends only on the radial distance r = |\mathbf{q}|. The is H = \frac{|\mathbf{p}|^2}{2m} + V(r), and the \mathbf{L} = \mathbf{q} \times \mathbf{p} satisfies \{\mathbf{L}, H\} = 0 due to the rotational invariance of H in . Each component L_i generates infinitesimal rotations that preserve H, confirming and enabling reduction to effective one-dimensional radial motion.

Extensions to Other Theories

Quantum Mechanics

In quantum mechanics, conserved quantities are represented by Hermitian operators \hat{C} that correspond to observable physical properties. Such an operator is conserved if it commutes with the system's Hamiltonian \hat{H}, satisfying [\hat{C}, \hat{H}] = 0. This commutation relation ensures that the operator remains time-independent in the , where the time evolution of operators is governed by the equation \frac{d\hat{C}}{dt} = \frac{i}{\hbar} [\hat{H}, \hat{C}] (assuming no explicit time dependence in \hat{C}). As a result, the expectation value \langle \hat{C} \rangle in any state does not change over time, preserving the associated physical quantity during the system's dynamics. Noether's theorem extends to quantum mechanics by linking continuous symmetries of the Hamiltonian to conserved operators via unitary transformations. A symmetry generated by an infinitesimal unitary operator \hat{U} = e^{-i \epsilon \hat{G}/\hbar} (where \hat{G} is the generator and \epsilon is a small parameter) implies that [\hat{H}, \hat{U}] = 0, making \hat{G} a conserved charge that also commutes with \hat{H}. For instance, translational invariance of the Hamiltonian under spatial shifts generates the total momentum operator \hat{p} as the conserved quantity, while rotational invariance yields the angular momentum operator \hat{\mathbf{L}}. This quantum adaptation of Noether's theorem underscores how symmetries dictate the existence of conserved operators, facilitating the classification of quantum states and the prediction of measurement outcomes. Certain conserved quantities impose superselection rules, which restrict the allowable quantum superpositions by decomposing the into orthogonal sectors. These rules arise when the conserved operator, such as the \hat{Q}, commutes not only with \hat{H} but also with all local observables, forbidding coherent superpositions between states in different eigenspaces of \hat{Q}. For example, a system cannot be placed in a superposition of states with differing total charge, as such states would be unobservable due to the inability to couple them via physical operations. Superselection rules thus enforce classical-like behavior for these quantities, limiting quantum coherence while preserving . A prominent example occurs in the quantum , where the central potential preserves , rendering the angular momentum operators \hat{L}^2 and \hat{L}_z conserved since they commute with the \hat{H} = -\frac{\hbar^2}{2m} \nabla^2 - \frac{e^2}{4\pi \epsilon_0 r}. This conservation leads to degeneracy in the energy eigenstates: for a given n, states with the same n but different azimuthal quantum numbers m_l (from -l to +l) share identical energies, as \hat{H} is independent of the orientation specified by m_l. The $2l + 1-fold degeneracy in m_l for fixed orbital angular momentum quantum number l directly stems from this , enabling the atom's lines to exhibit multiplet structures upon further perturbations.

Field Theory and Relativity

In field theories, Noether's theorem extends the concept of conserved quantities to continuous systems by associating symmetries of the action with conserved currents in spacetime. For a Lagrangian density \mathcal{L}(\phi, \partial_\mu \phi) invariant under infinitesimal transformations \delta \phi = \epsilon K(\phi) and \delta x^\mu = \epsilon \Xi^\mu, the theorem yields a conserved Noether current J^\mu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \delta \phi - \mathcal{L} \delta x^\mu, which satisfies the continuity equation \partial_\mu J^\mu = 0 on solutions to the equations of motion. This framework applies to relativistic field theories, where the integral of the time component over a spatial slice gives a conserved charge, reflecting the flow of the quantity through spacetime. Quantum field theory builds on this by promoting currents to operator-valued distributions, ensuring conservation in the sense of equal-time commutators. Relativistic examples prominently feature the energy-momentum tensor arising from translation symmetries. Under s \delta x^\mu = \epsilon^\mu, the Noether current is the energy-momentum tensor T^\mu{}_\nu = \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\nu \phi - \delta^\mu_\nu \mathcal{L}, conserved via \partial_\mu T^\mu{}_\nu = 0. In , the symmetric stress-energy tensor T_{\mu\nu} sources the curvature through the , and its conservation \nabla_\mu T^{\mu\nu} = 0 follows from the contracted second Bianchi identity \nabla_\mu G^{\mu\nu} = 0, where G^{\mu\nu} is the , independent of due to the invariance of the theory. In gauge theories, local symmetries lead to conserved charges via the global subgroup, as captured by Noether's first theorem. For , the U(1) gauge symmetry under phase transformations \psi \to e^{i\alpha} \psi (with \alpha constant globally) yields the conserved electromagnetic current j^\mu = \bar{\psi} \gamma^\mu \psi, satisfying \partial_\mu j^\mu = 0 and integrating to the . Local gauge invariance enforces this through the , ensuring anomaly-free conservation at the classical level. The of incorporates numerous conserved quantities from accidental global symmetries, such as B and L, preserved perturbatively but violated non-perturbatively via processes that change both B and L by three units while conserving B - L. Beyond the , extensions like mechanisms for masses introduce L-violating terms, such as Majorana mass operators, potentially observable in processes like , while maintaining approximate conservation for low-energy phenomenology.