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Conjugate variables

In physics, conjugate variables are pairs of dynamical quantities that are mathematically and physically linked, typically through Legendre transformations or Poisson brackets, allowing the description of systems in terms of alternative but equivalent representations of energy or action. These pairs consist of one extensive variable (like or ) and its intensive counterpart (like or ), whose product often corresponds to work, , or other forms of energy transfer. Common examples include q and p in , T and S in , and P and V. The concept originates in , where conjugate variables form the basis of and Hamilton's canonical equations, \dot{q}_i = \frac{\partial H}{\partial p_i} and \dot{p}_i = -\frac{\partial H}{\partial q_i}, with H being the total energy expressed as a function of these pairs. This framework generalizes Newton's laws for complex systems, enabling transformations that preserve the form of the equations and facilitating quantization in . In , conjugate pairs appear in the differential forms of potentials like U, where dU = T dS - P dV + \mu dn, linking intensive "forces" (e.g., T, P, \mu) to extensive "displacements" (e.g., S, V, n) and underpinning Legendre transforms between potentials such as U(S, V, n) and G(T, P, n). Beyond these core areas, conjugate variables extend to , where they describe fluctuations and response functions, and to field theories, pairing fields with their momenta (e.g., \phi and conjugate \Pi). Their defining property—that small changes in one induce responses in the conjugate via partial derivatives—ensures laws and in physical laws, making them essential for understanding , , and dynamical evolution across scales.

General Principles

Definition and Mathematical Foundations

Conjugate variables in physics are pairs of dynamical quantities, typically denoted as (q, p) in , that are ly conjugate, meaning they satisfy the relation {q, p} = 1 in or appear as dual variables in the differential forms of thermodynamic potentials via Legendre transformations. This canonical structure allows for equivalent descriptions of physical systems, such as expressing in terms of different sets of variables while preserving the underlying dynamics. In , for example, pairs like T and S, or P and V, appear in the internal differential dU = T \, dS - P \, dV + \mu \, dn, where intensive variables (e.g., T, P) conjugate to extensive ones (e.g., S, V). The mathematical foundations rest on symplectic geometry and variational principles. In phase space, conjugate pairs form coordinates of a symplectic manifold with the 2-form \omega = \sum_i dq_i \wedge dp_i, ensuring volume preservation under Hamiltonian flows as per Liouville's theorem. The Legendre transform provides a key tool for switching between conjugate representations: for instance, the Hamiltonian H(q, p) is the Legendre transform of the Lagrangian L(q, \dot{q}), defined as H = p \dot{q} - L with p = \partial L / \partial \dot{q}. This duality generalizes across fields, linking "displacements" in one variable to "forces" in its conjugate, with the product q p often corresponding to energy terms. In specific contexts, such as quantum mechanics, position q and momentum p exhibit Fourier duality, where wavefunctions in position and momentum spaces are related by Fourier transforms, leading to the uncertainty principle. However, this is a consequence of the canonical structure rather than its foundation.

Properties of Conjugate Pairs

Conjugate variables in classical mechanics are defined such that their Poisson bracket equals unity, providing a fundamental measure of their relational structure. The Poisson bracket for a pair of canonical coordinates q and p is given by \{q, p\} = \frac{\partial q}{\partial q} \frac{\partial p}{\partial p} - \frac{\partial q}{\partial p} \frac{\partial p}{\partial q} = 1, which generalizes to multiple dimensions as \{q_i, p_j\} = \delta_{ij} for the set of conjugate pairs (q_i, p_i). This property ensures that the variables generate transformations preserving the symplectic structure, contrasting with quantum mechanics where the commutator [q, p] = i\hbar introduces inherent uncertainty, though the classical Poisson bracket serves as its structural analog for deterministic evolution. A key property of conjugate pairs is their invariance under canonical transformations, which are coordinate changes that preserve the form of Hamilton's equations. Specifically, the Poisson bracket remains unchanged: if (Q, P) are new variables related to (q, p) by a canonical transformation, then \{Q, P\} = \{q, p\} = 1. This invariance underscores the symplectic nature of phase space, where conjugate pairs form the coordinates of a symplectic manifold equipped with a non-degenerate, closed 2-form \omega = \sum dq_i \wedge dp_i. The symplectic structure guarantees that transformations preserve the geometric properties of phase space, ensuring the volume-preserving flow of dynamical systems. In this framework, emerges as a direct consequence: the volume occupied by an ensemble of trajectories remains constant under evolution, reflecting the incompressibility of the flow generated by the form. Conjugate pairs thus underpin the conservation of information in classical , as the preservation of volume implies no dissipation or creation of elements. The uniqueness of conjugate variables is established through criteria like the Legendre transform, which relates functions of one variable to their duals via the conjugate pair. For instance, in transitioning from to , the p is defined as p = \partial L / \partial \dot{q}, and the H is the Legendre transform H = p \dot{q} - L, ensuring the pair (q, p) satisfies the condition and variational principles. This transform criterion confirms conjugacy by linking extremal paths in configuration space to those in , maintaining the structure across dual descriptions.

Classical Mechanics

Hamiltonian Formulation

In Hamiltonian mechanics, the formulation of classical dynamics shifts from the Lagrangian's dependence on generalized coordinates q_i and velocities \dot{q}_i to a description using coordinates q_i and their conjugate momenta p_i, defined as p_i = \frac{\partial L}{\partial \dot{q}_i}, where L(q, \dot{q}, t) is the of the system. The function H(q, p, t), which generates the , is obtained via the : H(q, p, t) = \sum_i p_i \dot{q}_i - L(q, \dot{q}, t), with the velocities \dot{q}_i solved in terms of the momenta p_i. For systems with scleronomic constraints and kinetic energy quadratic in velocities, H equals the total energy T + V, where T is kinetic and V is potential energy. The time evolution of the system is governed by Hamilton's equations: \dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}, which form a symmetric set of first-order differential equations derived from the that the action S = \int (p_i \dot{q}_i - H) \, dt is stationary with respect to variations in q_i and p_i. This approach highlights the role of conjugate pairs in preserving the structure of the dynamics through the of . Generalized coordinates allow the Hamiltonian formulation to adapt to non-Cartesian systems, where q_i are position-like parameters and p_i are momentum-like quantities tailored to the symmetry. For example, in polar coordinates for a particle of mass m in a central potential V(r), the coordinates are q_r = r (radial distance) and q_\theta = \theta (angle), with Lagrangian L = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) - V(r); the conjugate momenta are then p_r = m \dot{r} and p_\theta = m r^2 \dot{\theta}, yielding Hamiltonian H = \frac{p_r^2}{2m} + \frac{p_\theta^2}{2 m r^2} + V(r). Here, p_\theta conserves as angular momentum due to rotational invariance. The of the system is the $2n-dimensional manifold coordinatized by the conjugate pairs (q_i, p_i), where n is the number of ; these pairs define the structure over the configuration space, enabling the representation of trajectories as flows under the . This framework underscores how conjugate variables encapsulate both positional and momentum information symmetrically, facilitating analysis in systems with constraints or symmetries.

Derivatives of the Action

In , the action principle posits that the trajectory of a minimizes or renders stationary functional S, defined as the integral of the L over time: S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt, where q denotes and \dot{q} their velocities. This principle, formulated by , underlies the variational approach to , with the physical path satisfying \delta S = 0 for variations \delta q vanishing at the endpoints t_1 and t_2. The variation of the action yields \delta S = \left[ \frac{\partial L}{\partial \dot{q}} \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. For \delta S = 0, the integrand must vanish, leading to the Euler-Lagrange equations: \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) = \frac{\partial L}{\partial q_i}. These equations govern the motion, and the canonical conjugate momentum p_i emerges naturally as p_i = \frac{\partial L}{\partial \dot{q}_i}, or equivalently in functional terms, p = \frac{\delta S}{\delta \dot{q}}, pairing position-like q_i with momentum-like p_i. The boundary term in the variation of identifies the conjugate relation through \delta S = p \delta q - H \delta t when varying the endpoints. A deeper connection arises in the Hamilton-Jacobi theory, where the principal function S(q, P, t) serves as a for a to new coordinates (Q, P). The old momenta are given by p_i = \frac{\partial S}{\partial q_i}, while the new coordinates satisfy Q_j = \frac{\partial S}{\partial P_j}, thus defining conjugate pairs via partial derivatives of S. This function obeys the Hamilton-Jacobi equation: \frac{\partial S}{\partial t} + H\left( q, \frac{\partial S}{\partial q}, t \right) = 0, where H is the Hamiltonian, obtained from the Lagrangian via Legendre transform. Solving this partial differential equation provides a complete integral, yielding the trajectories and conserving the new momenta P as constants. As an illustrative example, consider the simple harmonic oscillator with Lagrangian L = \frac{1}{2} m \dot{q}^2 - \frac{1}{2} k q^2. The conjugate momentum follows directly as p = \frac{\partial L}{\partial \dot{q}} = m \dot{q}, pairing position q and momentum p. The principal function S can be constructed by integrating along the classical path, S = \int L \, dt, explicitly yielding p = \frac{\partial S}{\partial q} and confirming the conjugate relation through the Hamilton-Jacobi equation, where solutions can be expressed using elementary functions parameterized by the energy.

Quantum Mechanics

Commutation Relations

In quantum mechanics, the classical conjugate variables, such as position q and momentum p, are promoted to Hermitian operators \hat{q} and \hat{p} acting on a Hilbert space, satisfying the canonical commutation relation [\hat{q}, \hat{p}] = i \hbar, where \hbar = h / 2\pi is the reduced Planck's constant. This promotion, known as canonical quantization, replaces classical c-numbers with non-commuting operators to capture the wave-like and particle-like duality of quantum systems. The canonical commutation relations (CCR) arise from a correspondence principle linking classical and quantum mechanics, where the classical Poisson bracket \{q, p\}_{\mathrm{PB}} = 1 is mapped to the quantum commutator via \{A, B\}_{\mathrm{PB}} \to \frac{1}{i\hbar} [\hat{A}, \hat{B}]. This yields the explicit form [\hat{q}, \hat{p}] = i \hbar for the position-momentum pair, ensuring that quantum dynamics preserve the structure of classical Hamiltonian equations through the correspondence \dot{q} = \{q, H\}_{\mathrm{PB}} \to \frac{1}{i\hbar} [\hat{q}, \hat{H}]. The CCR form the algebraic foundation of quantum theory, dictating how observables interfere in measurements. These relations are realized in different representations of the . In the , states evolve as wavefunctions \psi(q) in position space, with \hat{q} \psi(q) = q \psi(q) and \hat{p} \psi(q) = -i \hbar \frac{d}{dq} \psi(q), satisfying the CCR upon substitution. The momentum-space wavefunction \tilde{\psi}(p) is the of \psi(q), \tilde{\psi}(p) = \frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty} e^{-i p q / \hbar} \psi(q) \, dq, interchanging the roles of \hat{q} and \hat{p} via i \hbar \frac{d}{dp}. In the , operators evolve in time while states remain fixed, with the CCR ensuring unitarity of the . For systems with multiple , the generalize to [\hat{q}_i, \hat{p}_j] = i \hbar \delta_{ij}, [\hat{q}_i, \hat{q}_j] = [\hat{p}_i, \hat{p}_j] = 0 for i, j = 1, \dots, n, where \delta_{ij} is the , extending the single-pair algebra to the for n dimensions. This multi-variable form underpins the quantization of complex systems, such as multi-particle quantum fields.

Uncertainty Principle Implications

In quantum mechanics, the commutation relations of conjugate variables, such as position q and momentum p satisfying [q, p] = i\hbar, lead to the through the Robertson-Schrödinger inequality. This inequality states that for any two observables A and B, (\Delta A)^2 (\Delta B)^2 \geq \left( \frac{1}{2} \langle \{A, B\} \rangle - \langle A \rangle \langle B \rangle \right)^2 + \left( \frac{1}{2i} \langle [A, B] \rangle \right)^2, where \Delta A and \Delta B are standard deviations. For conjugate pairs, the anticommutator term vanishes in the symmetric case, yielding \Delta q \Delta p \geq \frac{\hbar}{2}. This establishes a fundamental limit on the precision of simultaneous measurements of conjugate variables. The uncertainty principle generalizes to other conjugate pairs beyond position and momentum. For energy E and time t, the Mandelstam-Tamm formulation gives \Delta E \Delta t \geq \frac{\hbar}{2}, where \Delta t represents the time scale over which the system's expectation value changes appreciably, though time acts as a classical parameter rather than a quantum operator, introducing interpretive caveats such as distinguishing between lifetime and evolution time. Similarly, for angular momentum components L_x, L_y, and L_z obeying [L_x, L_y] = i\hbar L_z (and cyclic permutations), the pairwise relations are \Delta L_x \Delta L_y \geq \frac{\hbar}{2} |\langle L_z \rangle|, with extensions to triple-component inequalities like \Delta L_x \Delta L_y \Delta L_z \geq \frac{\hbar^3}{8} |\langle \mathbf{L} \rangle| for tighter bounds. These generalizations highlight how non-commutativity imposes trade-offs across various conjugate pairs in quantum systems. Physically, these uncertainties limit simultaneous knowledge of conjugate variables, manifesting in wave-particle duality through the relationship between and representations of . A localized in space requires a broad distribution to satisfy the bound, leading to and spreading over time. This trade-off is illustrated conceptually in the , where determining an electron's path through a specific slit localizes its , increasing uncertainty and washing out the pattern that reveals wave-like behavior. Such implications underscore the intrinsic quantum indeterminacy in measuring conjugate quantities.

Thermodynamics

Thermodynamic Conjugate Pairs

In thermodynamics, conjugate variables are pairs of intensive and extensive properties that appear in the differential expression of thermodynamic potentials, where one variable is the partial derivative of the potential with respect to the other. These pairs, often denoted as (X_i, Y_i), satisfy the form dΦ = ∑ Y_i dX_i for a potential Φ, with Y_i typically intensive (independent of system size) and X_i extensive (scaling with system size). This structure reflects a duality where conjugate variables maximize informational content in thermodynamic relations, analogous to broader mathematical dualities but specialized to equilibrium systems. The fundamental example arises in the internal energy U, expressed as a function of its natural variables entropy S, volume V, and particle number N: dU = T \, dS - P \, dV + \mu \, dN where T is (conjugate to S), -P is (conjugate to V), and μ is (conjugate to N). Here, the coefficients Y_i (T, -P, μ) are intensive variables obtained as partial derivatives: T = (∂U/∂S){V,N}, P = -(∂U/∂V){S,N}, μ = (∂U/∂N)_{S,V}. These pairs link (T dS), mechanical work (-P dV), and matter exchange (μ dN) in reversible processes. Legendre transforms generate other thermodynamic potentials by replacing a natural extensive X_i with its conjugate Y_i, creating a new potential whose excludes the term Y_i dX_i. The transform for a pair is Φ' = Φ - Y_i X_i, yielding natural variables that include Y_i instead of X_i, which is useful for constraints like constant or . This process preserves the conjugate structure, as the new follows from the exactness of dΦ. For the pressure-volume pair, the Legendre transform defines enthalpy H = U + P V, with natural variables S, P, and N: dH = T \, dS + V \, dP + \mu \, dN where V (now extensive, conjugate to P) replaces -P dV. Transforming further for the temperature-entropy pair gives the G = H - T S = U + P V - T S, with natural variables T, P, and N: dG = -S \, dT + V \, dP + \mu \, dN Here, -S (extensive, conjugate to T) and V (conjugate to P) appear, making G suitable for constant-temperature, constant-pressure conditions common in . In these conjugate pairs, extensive variables like S, V, and N scale linearly with system size (e.g., doubling the system doubles them), while intensive conjugates like T, P, and μ remain unchanged, ensuring thermodynamic potentials are homogeneous functions of degree one in extensive variables. This scaling property underpins for integrating the differentials, yielding relations like U = T - P + μ for closed systems.

Role in Potentials and Work

In thermodynamics, conjugate pairs play a central role in expressing the infinitesimal work and heat transfers during reversible processes. For a hydrostatic system, the mechanical work done on the system is given by \delta W = -P \, dV, where P is the pressure (intensive) and V is the volume (extensive), while the heat transfer is \delta Q = T \, dS, with T the temperature and S the entropy. These expressions arise from the differential form of the internal energy, dU = T \, dS - P \, dV, highlighting how conjugate pairs quantify energy exchanges in equilibrium processes. This framework generalizes to other conjugate pairs beyond hydrostatic systems. In magnetic systems, for instance, the work term becomes \delta W = \mu_0 H \, dM, where H is the magnetic field strength and M is the total magnetization, analogous to the pressure-volume pair but for magnetic interactions. Such generalizations allow conjugate variables to describe work in diverse thermodynamic contexts, from mechanical compression to field-induced changes. Conjugate pairs also underpin stability criteria through second-order derivatives of thermodynamic potentials. For thermal stability, the specific heat at constant volume must satisfy C_V = T \left( \frac{\partial S}{\partial T} \right)_V > 0, ensuring positive entropy response to fluctuations. Similarly, mechanical stability requires the isothermal compressibility \kappa_T = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T > 0, derived from the - conjugate pair, which confirms that decreases under increased , preventing . These conditions, expressed via conjugate relations, guarantee that small perturbations restore equilibrium. In phase transitions, discontinuities in conjugate variables or their derivatives signal changes in system order according to the Ehrenfest classification. transitions exhibit jumps in first derivatives of the , such as (related to the temperature-entropy pair) or volume change (pressure-volume pair), indicating phase coexistence. Higher-order transitions involve discontinuities in higher derivatives, like specific heat divergences in second-order cases, where conjugate pairs reveal the nature of the transition without . Maxwell relations, derived from the exactness of thermodynamic differentials, further illustrate the interconnectedness of conjugate pairs in potentials. For the , the differential is dG = -S \, dT + V \, dP, leading to the relation \left( \frac{\partial S}{\partial P} \right)_T = -\left( \frac{\partial V}{\partial T} \right)_P, which links cross-derivatives of and across and . This equality ensures thermodynamic consistency and enables prediction of response functions from measurable quantities.

Other Applications

Fluid Mechanics

In fluid mechanics, conjugate variables appear prominently in the description of irrotational, incompressible flows, particularly in two dimensions. The velocity potential \phi and the stream function \psi serve as a pair of conjugate variables, satisfying the Cauchy-Riemann equations \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} and \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}, which ensure that the velocity components are u = \frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y} and v = \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}. These functions are harmonic conjugates, both obeying Laplace's equation \nabla^2 \phi = 0 and \nabla^2 \psi = 0, reflecting the irrotational nature of the flow where vorticity \omega = \nabla \times \mathbf{v} = 0. This duality allows the complex potential f(z) = \phi + i\psi to analytically represent the flow, providing a powerful tool for solving boundary value problems in potential flow theory./10:_Inviscid_Flow_or_Potential_Flow/10.4_Conforming_Mapping/10.4.1:_Complex_Potential_and_Complex_Velocity) In the context of action principles for fluid motion, conjugate variables emerge from the distinction between Lagrangian and Eulerian descriptions. The Lagrangian approach uses material coordinates \mathbf{a} (labeling fluid particles) as the configuration variables, with their conjugate momenta being the momentum per unit mass \mathbf{m} = \rho_0 \dot{\mathbf{q}}, where \mathbf{q}(\mathbf{a}, t) is the position map and \rho_0 is the reference density. This formulation derives from the action integral over Lagrangian paths, leading to Euler-Lagrange equations that recover the fluid equations upon transformation to Eulerian coordinates \mathbf{r}. The conjugate pair (\mathbf{q}, \mathbf{m}) captures the conservation of mass and momentum along particle trajectories, contrasting with the Eulerian view where fields like velocity \mathbf{v}(\mathbf{r}, t) are defined at fixed points in space. For ideal fluids, the Hamiltonian structure highlights these conjugates, with the total Hamiltonian given by H = \int \left( \frac{1}{2} \rho v^2 + \rho \Phi \right) dV, where \rho is density, \mathbf{v} is velocity, and \Phi is the gravitational potential. In irrotational flows, \mathbf{v} = \nabla \phi, and perturbations in density \rho are conjugate to variations in \phi, as seen in the linearized Euler equations where acoustic modes couple potential fluctuations to density waves. This pairing facilitates the stability analysis of flows and the derivation of wave equations from the Hamiltonian. Applications of these concepts extend to vortex dynamics, where in certain Hamiltonian formulations for two-dimensional flows, the circulation \Gamma acts as a conjugate variable to the enclosed area of vortex structures. For instance, in point vortex models, the conserved circulation of individual vortices pairs with the geometric area influenced by their interactions, enabling variational principles to describe merging and evolution of coherent structures. This framework underscores the role of conjugate pairs in preserving topological invariants like during adiabatic processes.

Electromagnetism

In the Hamiltonian formulation of classical electrodynamics, the electromagnetic field is treated as a system of dynamical variables analogous to position and momentum in particle mechanics. The primary canonical conjugate pair consists of the vector potential \mathbf{A}(\mathbf{r}, t) and its conjugate momentum density \boldsymbol{\pi}(\mathbf{r}, t), where \boldsymbol{\pi} = -\mathbf{E} in units where the permittivity of free space \epsilon_0 = 1. This pairing arises from the Lagrangian density for the free electromagnetic field, \mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu}, with the field strength tensor F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu. The conjugate momentum is obtained as \pi^i = \frac{\partial \mathcal{L}}{\partial (\partial_0 A_i)} = F^{0i} = -E^i, confirming the identification in the spatial components. The scalar potential A_0 (or \phi) does not have a time derivative in the Lagrangian and thus possesses no conjugate momentum; it acts as a Lagrange multiplier enforcing the Gauss constraint \nabla \cdot \mathbf{E} = \rho. To eliminate redundancy due to gauge freedom, the Coulomb gauge \nabla \cdot \mathbf{A} = 0 is often imposed, reducing the independent degrees of freedom to the two transverse components of \mathbf{A} and \mathbf{E}. The magnetic field is then expressed as \mathbf{B} = \nabla \times \mathbf{A}, ensuring \nabla \cdot \mathbf{B} = 0. The total Hamiltonian for the field in vacuum is H = \int d^3\mathbf{r} \left[ \frac{1}{2} \mathbf{E}^2 + \frac{1}{2} \mathbf{B}^2 \right], representing the total electromagnetic energy. Hamilton's equations of motion follow from the Poisson bracket structure, defined by \{A_i(\mathbf{r}), \pi_j(\mathbf{r}')\} = \delta_{ij} \delta^3(\mathbf{r} - \mathbf{r}'), with vanishing brackets for other pairs. This yields \dot{A}_i = \{A_i, H\} = \pi_i = -E_i and \dot{\pi}_i = \{\pi_i, H\} = -(\nabla \times \mathbf{B})_i, reproducing Faraday's law \dot{\mathbf{B}} = \nabla \times \dot{\mathbf{A}} = \nabla \times \boldsymbol{\pi} = -\nabla \times \mathbf{E} and Ampère's law \dot{\mathbf{E}} = -\dot{\boldsymbol{\pi}} = \nabla \times \mathbf{B} - \mathbf{J} (in vacuum, \mathbf{J} = 0) upon substitution. The longitudinal components are constrained, while the transverse parts propagate as waves. This formulation unifies the field dynamics and facilitates quantization by promoting the Poisson brackets to commutators. For charged particles coupled to the field, the total includes the interaction term \int \mathbf{J} \cdot \mathbf{A} \, d^3\mathbf{r}, where the current \mathbf{J} arises from particle modified by the : the canonical \mathbf{p} = m\mathbf{v} + q\mathbf{A}. This extends the conjugate structure to encompass matter-field interactions, preserving the overall . Seminal treatments emphasize how this approach reveals the as an infinite collection of oscillators in modes, each with conjugate coordinates Q_\lambda and P_\lambda.