In analytical mechanics, generalized coordinates are a minimal set of independent parameters, such as lengths, angles, or other variables, that completely and unambiguously specify the configuration of a mechanical system at any given time, allowing for a more efficient description than the full set of Cartesian coordinates, particularly in systems with constraints.^[1][2][3] This concept was introduced by Joseph-Louis Lagrange in his seminal 1788 work Mécanique Analytique, where he reformulated classical mechanics in a purely analytical framework, emphasizing energy principles over Newtonian forces to derive equations of motion.^[4][5] The number of generalized coordinates equals the system's degrees of freedom, which is typically 3N for N unconstrained particles in three-dimensional space but reduced by the number of holonomic constraints.^[1][4] In Lagrangian mechanics, these coordinates $ q_i $ and their time derivatives $ \dot{q}_i $ are used to express the kinetic energy $ T $ and potential energy $ V $, forming the Lagrangian $ L = T - V $, from which Lagrange's equations $ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = Q_i $ (where $ Q_i $ are generalized forces) yield the system's dynamics without explicitly resolving constraint forces.^[1][2][4] This approach offers significant advantages, including coordinate invariance, simplified handling of complex geometries like pendulums or multi-body systems, and applicability to non-inertial frames or continuous systems in field theory.^[2][3]

Fundamentals

Definition and motivation

In classical mechanics, generalized coordinates refer to a minimal set of independent parameters $ q_1, q_2, \dots, q_n $ that fully describe the configuration of a system possessing $ n $ degrees of freedom, eliminating redundancy in the representation of the system's state.^[6] These coordinates serve as a flexible framework, distinct from the fixed Cartesian system, by parameterizing positions in a way that aligns directly with the system's intrinsic geometry and constraints.^[2] The concept originated in the late 18th century through the pioneering work of Joseph-Louis Lagrange, who introduced it in his seminal 1788 treatise Mécanique Analytique to advance the field of analytical mechanics.^[7] Lagrange's formulation shifted the focus from force-based Newtonian methods to a variational principle, using these coordinates to derive equations of motion systematically without relying on geometric intuitions.^[8] The primary motivation for generalized coordinates lies in their ability to streamline the analysis of constrained mechanical systems by reducing the number of variables from the 3N Cartesian coordinates (where N is the number of particles) to just n, incorporating constraints implicitly and avoiding the need to solve extraneous equations.^[6] This simplification is particularly advantageous in Lagrangian and Hamiltonian mechanics, where it facilitates the derivation of equations of motion by excluding non-contributory forces, such as those perpendicular to the motion.^[8] Furthermore, they promote computational efficiency for intricate systems, such as those involving rigid bodies, and enable the adoption of curvilinear systems like polar or spherical coordinates, which better capture rotational or symmetric dynamics.^[2]

Relation to Cartesian coordinates

In classical mechanics, the positions of the particles in a system of $N$ particles are typically described in Cartesian coordinates as $\mathbf{r}_i = (x_i, y_i, z_i)$ for $i = 1, \dots, N$. To employ generalized coordinates $q_1, q_2, \dots, q_s$, where $s \leq 3N$ is the number of degrees of freedom, each Cartesian component is expressed as a function of these generalized coordinates and possibly time: $x_i = x_i(q_1, \dots, q_s, t)$, $y_i = y_i(q_1, \dots, q_s, t)$, $z_i = z_i(q_1, \dots, q_s, t)$.^[9][10] This transformation maps the full 3N-dimensional Cartesian space to a reduced s-dimensional space, incorporating any constraints through the choice of the $q_k$. The nature of this transformation depends on whether the underlying constraints are time-independent or time-dependent. In scleronomic systems, the constraints do not explicitly involve time, so the position functions simplify to $x_i = x_i(q_1, \dots, q_s)$, without the $t$ argument.^[2][9] In contrast, rheonomic systems have time-dependent constraints, leading to explicit time reliance in the transformation: $x_i = x_i(q_1, \dots, q_s, t)$.^[2][10] This distinction affects the form of the dynamical equations, as time dependence in rheonomic cases introduces additional terms. The set of generalized coordinates $q_1, \dots, q_s$ parameterizes the configuration space, which is an s-dimensional manifold representing all possible configurations of the system accessible under the given constraints.^[9][10] Each point in this space corresponds to a unique specification of the system's position, with the manifold's geometry determined by the transformation to Cartesian coordinates; for scleronomic systems, the configuration space is time-independent, while for rheonomic systems, it evolves with time. The velocities in Cartesian coordinates are obtained by differentiating the position functions with respect to time, yielding the transformation

$$\mathbf{v}_i = \dot{\mathbf{r}}_i = \sum_{k=1}^s \frac{\partial \mathbf{r}_i}{\partial q_k} \dot{q}_k + \frac{\partial \mathbf{r}_i}{\partial t},$$

where the partial derivatives form the columns of the Jacobian matrix $\mathbf{J}$ with elements $J_{\alpha k} = \partial x_{\alpha i}/\partial q_k$ for the $\alpha$-th Cartesian component.^[9][10] This Jacobian relates the generalized velocities $\dot{q}_k$ to the Cartesian velocities $\mathbf{v}_i$, and its nonzero determinant ensures the transformation is locally invertible, preserving the structure of the configuration space.^[10]

Constraints and Degrees of Freedom

Holonomic constraints

Holonomic constraints are those that can be expressed in the form $ f_j(q_1, \dots, q_n, t) = 0 $ for $ j = 1, \dots, m $, where the $ q_i $ are generalized coordinates and $ t $ is time, allowing the system's configuration to be described on a lower-dimensional manifold parameterized by the remaining independent coordinates./13%3A_Lagrangian_Mechanics/13.03%3A_Holonomic_Constraints) These constraints arise from integrable relations, such as geometrical conditions like a fixed distance between particles in a rigid body, which restrict motion without depending on velocities.^[11] For a system of $ N $ particles initially with $ 3N $ Cartesian degrees of freedom, $ m $ independent holonomic constraints reduce the number of generalized coordinates to $ n = 3N - m $.^[2] The integrability of holonomic constraints stems from their representation as Pfaffian differential forms that are exact differentials, enabling the elimination of dependent variables to yield explicit functions of coordinates and time.^[12] Specifically, a constraint in the differential form $ \sum_i a_i , dq_i + a_t , dt = 0 $ is holonomic if there exists an integrating factor making it the total differential of some function $ f(q, t) $, thus $ df = 0 $ implies $ f = \text{constant} $.^[13] Constraints without explicit time dependence, known as scleronomic, simplify to $ f_j(q_1, \dots, q_n) = 0 $, further restricting the system to a fixed configuration space.^[11] In the Lagrangian formulation, holonomic constraints can be enforced either by directly substituting the relations to reduce variables or, more generally, by introducing Lagrange multipliers $ \lambda_j $ to augment the Lagrangian as $ \mathcal{L}' = \mathcal{L} - \sum_j \lambda_j f_j $, preserving all coordinates while accounting for the constraints through the equations of motion.^[14] This approach ensures the generalized coordinates effectively parameterize the admissible configurations on the constraint manifold.^[2]

Non-holonomic constraints

Non-holonomic constraints are velocity-dependent relations that cannot be expressed as functions of the generalized coordinates $q_i$ and time $t$ alone, but instead take the form of linear differential equations known as Pfaffian constraints:

$$\sum_{k=1}^n a_{jk}(q, t) \, dq_k + a_{j0}(q, t) \, dt = 0,$$

where $j = 1, \dots, m$ labels the constraints, and the coefficients $a_{jk}$ and $a_{j0}$ depend on the coordinates and time.^[15] These are not exact differentials, meaning they cannot be integrated to yield a constraint solely on the configuration variables, distinguishing them from holonomic constraints that reduce the system's configuration space.^[2] The non-integrability of such constraints is determined by the Frobenius theorem, which provides necessary and sufficient conditions for a distribution of vector fields (or equivalently, a Pfaffian system) to be integrable. Specifically, for the constraint forms to be integrable, the exterior derivatives of the Pfaffian equations must lie within the ideal generated by the constraints themselves; if this involutivity condition fails, the constraints are non-holonomic, restricting allowable velocity fields without confining the accessible configuration space.^[16][17] A classic example is the rolling without slipping of a wheel or disk on a plane, where the no-slip condition imposes $\dot{x} = r \dot{\theta} \cos \phi$ and $\dot{y} = r \dot{\theta} \sin \phi$, with $x, y$ as the contact point coordinates, $\theta$ as the rotation angle, $\phi$ as the heading angle, and $r$ the radius; these link velocities but do not integrate to a position constraint, allowing the disk to reach any $(x, y)$ while following curved paths. Another instance is a sphere rolling on a plane, where similar velocity relations prevent sliding but permit full positional freedom in the plane.^[18] In systems with $N$ particles, non-holonomic constraints do not reduce the dimension of the configuration space, which remains $3N$ (or $n$ generalized coordinates), but they limit the admissible trajectories in that space, constraining the dynamics to a sub-bundle of the tangent space.^[19] The number of degrees of freedom for configuration is thus $n = 3N - 0$ (no reduction), but the effective dynamical freedom is $n - m$, where $m$ is the number of independent constraints.^[20] Addressing non-holonomic systems poses challenges because the constraints cannot be used to eliminate coordinates directly, as in the holonomic case; instead, formulations like quasi-coordinates—parameters proportional to velocities rather than positions—or Appell's equations, which incorporate the constraints into higher-order terms involving accelerations, are required to derive the equations of motion. These methods, originally developed by Appell in 1899, allow for the systematic inclusion of non-integrable constraints without reducing the coordinate set.^[21]

Physical Quantities

Kinetic energy

In classical mechanics, the kinetic energy $T$ of a system consisting of $N$ particles, each with mass $m_i$ and position vector $\mathbf{r}_i$, is fundamentally expressed in Cartesian coordinates as

$$T = \frac{1}{2} \sum_{i=1}^N m_i \left\| \frac{d\mathbf{r}_i}{dt} \right\|^2,$$

where $\left\| \frac{d\mathbf{r}_i}{dt} \right\|^2 = \dot{\mathbf{r}}_i \cdot \dot{\mathbf{r}}_i$ represents the squared speed of the $i$-th particle.^[22] To express this in generalized coordinates $q_1, \dots, q_n$, the position vectors are written as $\mathbf{r}_i = \mathbf{r}_i(q_1, \dots, q_n, t)$, reflecting the system's configuration space parametrized by the $q_k$. The velocity of each particle then follows from the chain rule:

$$\dot{\mathbf{r}}_i = \sum_{k=1}^n \frac{\partial \mathbf{r}_i}{\partial q_k} \dot{q}_k + \frac{\partial \mathbf{r}_i}{\partial t}.$$

For scleronomic systems, where the transformation between generalized and Cartesian coordinates is time-independent (i.e., $\frac{\partial \mathbf{r}_i}{\partial t} = 0$), the velocity simplifies to

$$\dot{\mathbf{r}}_i = \sum_{k=1}^n \frac{\partial \mathbf{r}_i}{\partial q_k} \dot{q}_k.$$

Substituting this into the kinetic energy expression and expanding the dot product yields

$$\left\| \dot{\mathbf{r}}_i \right\|^2 = \sum_{k=1}^n \sum_{l=1}^n \dot{q}_k \dot{q}_l \left( \frac{\partial \mathbf{r}_i}{\partial q_k} \cdot \frac{\partial \mathbf{r}_i}{\partial q_l} \right).$$

Thus, the total kinetic energy becomes

$$T = \frac{1}{2} \sum_{k=1}^n \sum_{l=1}^n g_{kl}(q, t) \dot{q}_k \dot{q}_l,$$

where the coefficients $g_{kl}$ form the elements of the metric tensor (or mass matrix) on the configuration space, defined by

$$g_{kl} = \sum_{i=1}^N m_i \left( \frac{\partial \mathbf{r}_i}{\partial q_k} \cdot \frac{\partial \mathbf{r}_i}{\partial q_l} \right).$$

This metric tensor $g_{kl}$ is symmetric ($g_{kl} = g_{lk}$) and, for physically realizable systems, positive definite, ensuring $T > 0$ for any nonzero $\dot{q}$.^[22][2] The resulting form of $T$ is a homogeneous quadratic function in the generalized velocities $\dot{q}_k$, which endows the configuration space with a Riemannian metric structure induced by the system's inertia; this geometric interpretation underscores how $g_{kl}$ quantifies the "inertial coupling" between different directions in coordinate space.^[23] In rheonomic systems (time-dependent transformations), cross terms involving $\dot{q}_k$ and $\frac{\partial \mathbf{r}_i}{\partial t}$ may appear, but the quadratic structure in $\dot{q}$ persists as the dominant feature.^[2] For systems subject to constraints, the expression for $T$ adapts to the choice of coordinates. In the presence of holonomic constraints, which can be integrated to reduce the number of independent coordinates to the degrees of freedom $n$, the generalized coordinates $q_k$ are selected to automatically satisfy the constraints, allowing $T$ to be directly formulated in the reduced set as above, with the metric $g_{kl}$ reflecting the constrained configuration manifold.^[2] For non-holonomic constraints, which impose velocity-dependent restrictions without integrable position constraints, a larger set of coordinates is typically used; the kinetic energy is expressed in these coordinates, but the allowable velocities $\dot{q}$ are projected onto the subspace orthogonal to the constraint gradients (e.g., via the constraint Jacobian), ensuring compatibility while preserving the quadratic form of $T$.^[2]

Generalized momentum

In Lagrangian mechanics, the generalized momentum $ p_k $ conjugate to the generalized coordinate $ q_k $ is defined as the partial derivative of the Lagrangian $ L $ with respect to the corresponding generalized velocity $ \dot{q}_k $, given by

$$p_k = \frac{\partial L}{\partial \dot{q}_k},$$

where $ L = T - V $ is the Lagrangian, with $ T $ denoting the kinetic energy and $ V $ the potential energy of the system.^[8] Since the potential energy $ V $ generally does not depend on the velocities, this simplifies to $ p_k = \frac{\partial T}{\partial \dot{q}k} $.^[24] For systems where the kinetic energy takes a quadratic form in the generalized velocities, as expressed through the metric tensor $ g{kl} $ from the kinetic energy section, the generalized momentum assumes the linear form $ p_k = \sum_l g_{kl} \dot{q}_l $.^[25] A significant property of the generalized momentum arises when the Lagrangian is independent of the coordinate $ q_k $, rendering $ q_k $ a cyclic or ignorable coordinate. In such cases, the Euler-Lagrange equation yields $ \frac{d p_k}{dt} = -\frac{\partial L}{\partial q_k} = 0 $, implying that $ p_k $ is conserved throughout the motion.^[26] This conservation corresponds to symmetries in the system; for instance, in polar coordinates describing motion under a central force, the angular coordinate $ \theta $ is cyclic, and its conjugate momentum $ p_\theta = m r^2 \dot{\theta} $ represents the angular momentum, which remains constant.^[27] Such conserved generalized momenta provide integrals of motion that simplify the analysis of dynamical systems.^[28] The generalized momentum relates to the linear momenta in Cartesian coordinates through the coordinate transformation. Specifically, if the position vector in Cartesian coordinates is expressed as a function of the generalized coordinates, the conjugate momentum is $ p_k = \sum_i \frac{\partial x_i}{\partial q_k} p_{x_i} + \sum_i \frac{\partial y_i}{\partial q_k} p_{y_i} + \sum_i \frac{\partial z_i}{\partial q_k} p_{z_i} $, where $ p_{x_i}, p_{y_i}, p_{z_i} $ are the components of the linear momenta.^[28] This expression demonstrates that each generalized momentum is a weighted sum of the Cartesian linear momentum components, with weights given by the partial derivatives of the Cartesian positions with respect to $ q_k $; in the special case where the generalized coordinates are Cartesian, $ p_k $ directly coincides with the linear momentum.^[8] In the framework of Hamiltonian mechanics, the generalized momenta serve as independent variables alongside the coordinates, facilitating a phase space description of the dynamics. The Hamiltonian function $ H $ is obtained from the Lagrangian via the Legendre transform,

$$H = \sum_k p_k \dot{q}_k - L,$$

typically resulting in $ H = T + V $ when expressed in terms of $ q_k $ and $ p_k $.^[29] The time evolution of the system is then dictated by Hamilton's canonical equations:

$$\dot{q}_k = \frac{\partial H}{\partial p_k}, \quad \dot{p}_k = -\frac{\partial H}{\partial q_k}.$$

This formulation highlights the symmetry between coordinates and momenta, enabling powerful techniques for solving mechanical problems and extending to quantum mechanics.^[30]

Examples

Bead on a wire

A classic example of using generalized coordinates involves a bead of mass $ m $ constrained to slide without friction along a curved wire in three-dimensional space. The wire's shape is arbitrary but fixed, and the bead's position can be parameterized by the arc length $ s $ measured along the wire from a reference point, or, in the special case of a circular wire of radius $ R $, by the angle $ \theta $ from a reference direction. This setup enforces a holonomic constraint, reducing the system's degrees of freedom from three (in Cartesian coordinates) to one.^[6][31] The single generalized coordinate is $ s $, the distance traveled along the wire, which fully specifies the bead's position given the wire's geometry. For a circular wire, $ s = R \theta $, so $ \theta $ serves equivalently as the generalized coordinate. The kinetic energy $ T $ of the bead is then $ T = \frac{1}{2} m \dot{s}^2 $, reflecting the bead's speed solely along the wire's tangent, as the constraint eliminates motion perpendicular to it. The potential energy $ V $ due to gravity (assuming the wire lies in a vertical plane with height increasing along $ s $) is $ V = m g h(s) $, where $ h(s) $ is the vertical height as a function of $ s $.^[6][31] The Lagrangian for the system is formed as $ L = T - V = \frac{1}{2} m \dot{s}^2 - m g h(s) $. Applying the Euler-Lagrange equation for the conservative system (with no non-conservative generalized forces), $ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{s}} \right) - \frac{\partial L}{\partial s} = 0 $, yields the equation of motion. This simplifies to $ m \ddot{s} = -\frac{\partial V}{\partial s} $, or equivalently $ m \ddot{s} = -m g \frac{d h}{d s} $, where the constraint forces (normal to the wire) are handled implicitly without explicit computation.^[6][31] This approach demonstrates a key advantage of generalized coordinates: the three-dimensional problem is reduced to a single ordinary differential equation in one degree of freedom, avoiding the need for Lagrange multipliers to enforce the constraints explicitly.^[6]

Simple pendulum

The simple pendulum consists of a point mass $ m $ attached to a massless rod of fixed length $ l $, constrained to swing freely in a vertical plane under the influence of gravity, with the pivot point fixed in space.^[32] This system has one degree of freedom due to the holonomic constraint enforcing the fixed rod length.^[32] The natural choice for the generalized coordinate is the angle $ \theta $ measured from the downward vertical.^[32] In terms of Cartesian coordinates with the pivot at the origin and the positive $ y $-axis upward, the position of the mass is given by

$$x = l \sin \theta, \quad y = -l \cos \theta.$$

The corresponding velocity components are

$$\dot{x} = l \dot{\theta} \cos \theta, \quad \dot{y} = l \dot{\theta} \sin \theta.$$

These transformations express the constrained motion solely in terms of $ \theta $ and $ \dot{\theta} $.^[32] The kinetic energy $ T $ is the radial form derived from the speed squared $ \dot{x}^2 + \dot{y}^2 = l^2 \dot{\theta}^2 $, yielding

$$T = \frac{1}{2} m l^2 \dot{\theta}^2.$$

The potential energy $ V $, taking the reference at the lowest point, is

$$V = -m g l \cos \theta,$$

where $ g $ is the acceleration due to gravity.^[32] The Lagrangian $ L = T - V $ then becomes

$$L = \frac{1}{2} m l^2 \dot{\theta}^2 + m g l \cos \theta.$$

^[32] Applying Lagrange's equation $ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) - \frac{\partial L}{\partial \theta} = 0 $ to this system produces the nonlinear differential equation of motion

$$\ddot{\theta} + \frac{g}{l} \sin \theta = 0.$$

This equation captures the pendulum's oscillatory dynamics, reducible to simple harmonic motion for small $ \theta $.^[32]

Double pendulum

The double pendulum consists of two point masses, $m_1$ and $m_2$, attached to massless rods of lengths $l_1$ and $l_2$, respectively, with the second rod connected to the end of the first, allowing motion in a plane under gravity.^[4] This system has two degrees of freedom, making it a classic example of coupled oscillators where the motion of one pendulum influences the other.^[4] The generalized coordinates are the angles $\theta_1$ and $\theta_2$, both measured from the downward vertical.^[4] The position of the first mass is given by Cartesian coordinates $x_1 = l_1 \sin \theta_1$, $y_1 = -l_1 \cos \theta_1$, assuming the pivot is at the origin and the positive y-axis points upward.^[33] For the second mass, the position is $x_2 = x_1 + l_2 \sin \theta_2 = l_1 \sin \theta_1 + l_2 \sin \theta_2$, $y_2 = y_1 - l_2 \cos \theta_2 = -l_1 \cos \theta_1 - l_2 \cos \theta_2$.^[33] The kinetic energy $T$ of the system is derived from the velocities of both masses:

$$T = \frac{1}{2} (m_1 + m_2) l_1^2 \dot{\theta}_1^2 + \frac{1}{2} m_2 l_2^2 \dot{\theta}_2^2 + m_2 l_1 l_2 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2).$$

This expression captures the coupling term $m_2 l_1 l_2 \dot{\theta}_1 \dot{\theta}_2 \cos(\theta_1 - \theta_2)$, which arises from the relative motion between the pendulums.^[4] The potential energy $V$ is $V = -(m_1 + m_2) g l_1 \cos \theta_1 - m_2 g l_2 \cos \theta_2$, leading to the Lagrangian $L = T - V$.^[4] Applying the Euler-Lagrange equations to the Lagrangian yields two coupled nonlinear ordinary differential equations for $\ddot{\theta}_1$ and $\ddot{\theta}_2$:

$$(m_1 + m_2) l_1 \ddot{\theta}_1 + m_2 l_2 \cos(\theta_1 - \theta_2) \ddot{\theta}_2 + m_2 l_2 \sin(\theta_1 - \theta_2) \dot{\theta}_2^2 + (m_1 + m_2) g \sin \theta_1 = 0,$$

$$m_2 l_2 \ddot{\theta}_2 + m_2 l_1 \cos(\theta_1 - \theta_2) \ddot{\theta}_1 - m_2 l_1 \sin(\theta_1 - \theta_2) \dot{\theta}_1^2 + m_2 g \sin \theta_2 = 0.$$

These equations highlight the nonlinear coupling, which can lead to chaotic behavior for certain initial conditions and energies.^[33][34]

Spherical pendulum

A spherical pendulum consists of a point mass $ m $ attached to a light, inextensible rod of fixed length $ l $, suspended from a fixed point and allowed to swing freely in three-dimensional space under the influence of gravity, with the constraint that the mass moves on the surface of a sphere of radius $ l $.^[35] This system has two degrees of freedom, making it an ideal example for illustrating generalized coordinates in Lagrangian mechanics.^[36] The motion is conveniently described using spherical coordinates, where the generalized coordinates are the polar angle $ \theta $ (measured from the downward vertical axis) and the azimuthal angle $ \phi $ (measuring longitude around the vertical).^[35] The position of the mass in Cartesian coordinates is given by the transformations:

$$\begin{align*} x &= l \sin \theta \cos \phi, \\ y &= l \sin \theta \sin \phi, \\ z &= -l \cos \theta, \end{align*}$$

assuming the suspension point is at the origin and the $ z $-axis points upward.^[35] The kinetic energy $ T $ of the mass is derived from its velocity components in spherical coordinates:

$$T = \frac{1}{2} m l^2 \left( \dot{\theta}^2 + \sin^2 \theta \, \dot{\phi}^2 \right),$$

while the potential energy $ V $, due to gravity, is

$$V = - m g l \cos \theta,$$

where $ g $ is the acceleration due to gravity.^[35] The Lagrangian $ \mathcal{L} = T - V $ thus becomes

$$\mathcal{L} = \frac{1}{2} m l^2 \left( \dot{\theta}^2 + \sin^2 \theta \, \dot{\phi}^2 \right) + m g l \cos \theta.[](https://farside.ph.utexas.edu/teaching/336k/Newton/node82.html)$$

Applying the Euler-Lagrange equation to the $ \theta $-coordinate yields the equation of motion:

$$m l^2 \ddot{\theta} = m l^2 \sin \theta \cos \theta \, \dot{\phi}^2 - m g l \sin \theta,$$

which simplifies to

$$\ddot{\theta} = \sin \theta \cos \theta \, \dot{\phi}^2 - \frac{g}{l} \sin \theta.[](https://farside.ph.utexas.edu/teaching/336k/Newton/node82.html)$$

The $ \phi $-coordinate is cyclic in the Lagrangian, implying that the conjugate generalized momentum $ p_\phi = \frac{\partial \mathcal{L}}{\partial \dot{\phi}} = m l^2 \sin^2 \theta , \dot{\phi} $ is conserved, representing the constant angular momentum about the vertical axis.^[35] This conservation allows reduction of the system to a single effective equation in $ \theta $, highlighting the utility of generalized coordinates for capturing symmetries in constrained systems.^[36]

Virtual Work in Generalized Coordinates

Principle of virtual work

The principle of virtual work states that for a system in equilibrium, the total virtual work performed by all applied forces through any virtual displacement consistent with the system's constraints is zero. This is mathematically expressed as $\delta W = \sum_i \mathbf{F}_i \cdot \delta \mathbf{r}_i = 0$, where $\mathbf{F}_i$ are the forces acting on each particle and $\delta \mathbf{r}_i$ are the corresponding infinitesimal virtual displacements that satisfy the constraints without violating them.^[37][38] In the framework of generalized coordinates $q_k$, which parameterize the configuration space while incorporating constraints such as holonomic ones, the virtual work is reformulated as $\delta W = \sum_k Q_k \delta q_k$, where $Q_k$ are the generalized forces conjugate to the coordinates $q_k$. The generalized forces are defined by $Q_k = \sum_i \mathbf{F}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q_k}$, capturing the component of the applied forces that contributes to changes in the generalized coordinates.^[2][38][37] For systems subject to conservative forces derivable from a potential energy function $V$, the generalized forces take the form $Q_k = -\frac{\partial V}{\partial q_k}$, allowing the principle to connect directly with energy-based formulations. Extending this to dynamic systems, the principle links to d'Alembert's principle, which incorporates inertial effects by requiring $\sum_i (\mathbf{F}_i - m_i \mathbf{a}_i) \cdot \delta \mathbf{r}_i = 0$, where $m_i \mathbf{a}_i$ are the inertial forces, thus treating dynamics as a form of constrained equilibrium.^[2][37][38] This approach was developed by Joseph-Louis Lagrange in his seminal work Mécanique Analytique (1788), where he employed the principle of virtual work to derive equations of motion in generalized coordinates, bypassing direct resolution of Newtonian forces and constraint reactions.^[39][40] When combined with the expression for kinetic energy in generalized coordinates, the principle yields the Lagrange equations of motion, providing a powerful tool for analyzing complex mechanical systems.^[37][38]

Application to generalized forces

In systems with generalized coordinates, the principle of virtual work is extended to define generalized forces $ Q_k $ that account for non-conservative forces acting on the particles of the system. Specifically, for a system of $ N $ particles with positions $ \mathbf{r}_i $, the generalized force corresponding to the $ k $-th coordinate $ q_k $ is given by

$$Q_k = \sum_{i=1}^N \mathbf{F}_i^{\mathrm{nc}} \cdot \frac{\partial \mathbf{r}_i}{\partial q_k},$$

where $ \mathbf{F}i^{\mathrm{nc}} $ are the non-conservative forces on the $ i $-th particle, such as friction or external drives. This expression arises because the virtual work $ \delta W = \sum{i=1}^N \mathbf{F}_i^{\mathrm{nc}} \cdot \delta \mathbf{r}_i $ can be rewritten using $ \delta \mathbf{r}_i = \sum_k \frac{\partial \mathbf{r}_i}{\partial q_k} \delta q_k $, yielding $ \delta W = \sum_k Q_k \delta q_k $. For ideal constraints, the reaction forces associated with holonomic constraints perform no virtual work, as the virtual displacements $ \delta \mathbf{r}_i $ are chosen to lie in the admissible subspace tangent to the constraint manifold, ensuring their contribution to $ Q_k $ vanishes.^[41][2] This formulation extends naturally to non-holonomic systems, where constraints are velocity-dependent and non-integrable, typically of the form $ \sum_j a_{jk}(q, t) \dot{q}j = 0 $. In such cases, the virtual displacements $ \delta q_k $ must satisfy $ \sum_k a{jk} \delta q_k = 0 $ for each constraint, ensuring they are perpendicular to the constraint velocity directions in the tangent space. The generalized forces $ Q_k $ are then computed similarly, but now only over the non-conservative forces, with constraint reactions again contributing zero virtual work under ideal (frictionless) conditions. This approach preserves the structure of the virtual work principle while accommodating systems like rolling without slipping or sliding with velocity constraints.^[2][42] Representative examples illustrate the computation of $ Q_k $. In rotational systems, such as a pendulum driven by an external torque $ \tau $ about the pivot, the generalized force for the angle coordinate $ \theta $ is $ Q_\theta = \tau $, directly reflecting the work done through a virtual angular displacement $ \delta \theta $. For sliding systems involving friction, like a block on a rough surface with frictional force $ \mathbf{f} = -\mu N \hat{v} $, the generalized force for the position coordinate $ x $ becomes $ Q_x = \mathbf{f} \cdot \frac{\partial \mathbf{r}}{\partial x} = -\mu N $ if motion aligns with the displacement direction. These cases highlight how $ Q_k $ captures dissipative or applied effects not derivable from a potential.^[43][41] The generalized forces enter the equations of motion via the Lagrange-d'Alembert form, which generalizes Lagrange's equations for systems with non-conservative forces or non-holonomic constraints:

$$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_k} \right) - \frac{\partial L}{\partial q_k} = Q_k,$$

where $ L = T - V $ is the Lagrangian, with $ T $ the kinetic energy and $ V $ the potential for conservative forces. For non-holonomic cases, additional terms involving Lagrange multipliers $ \lambda_j $ may enforce the constraints, modifying the right-hand side to $ Q_k - \sum_j \lambda_j a_{jk} $. This yields a complete dynamical description without explicitly resolving constraint forces. However, the method assumes ideal constraints where reactions do no work, limiting its direct application to frictional constraints; in pure statics, it neglects inertial terms, reducing to equilibrium conditions $ \sum_k Q_k \delta q_k = 0 $.^[2][41][43]