Partial Projection Operator

• Partial projection operator is a mathematical tool that decomposes tangent spaces using an almost product structure and complementary projectors.
• It is applied in Riemannian, Poisson, and symplectic geometries to analyze constrained systems like nonholonomic dynamics.
• Its implementation facilitates the derivation of equations of motion without Lagrange multipliers in systems such as the Chaplygin–Carathéodory sleigh and contact sub-Riemannian geometry.

A partial projection operator is a geometric or algebraic tool designed to project elements (often vectors, vector fields, or operators) onto subspaces or subbundles that are associated with constraints, foliations, or nonholonomic structures on manifolds. In the context of constrained systems, these operators enable the precise decomposition of tangent vectors or more general geometric objects into components tangent to constraint distributions and their complements, facilitating the formulation and analysis of equations of motion under nonholonomic and sub-Riemannian constraints.

1. Almost Product Structures and Projector Definition

The foundation of partial projection operators is the introduction of an almost product structure on a finite-dimensional manifold $W$. This is formalized by a $(1,1)$ tensor field $F$ satisfying $F^2 = \mathrm{id}$, which induces a direct sum decomposition at each point $z \in W$: $T_z W = D_z \oplus D_z^\perp$ Here, $D_z$ denotes a distribution of constant rank, often interpreted as the subspace of allowed directions (e.g., constrained, nonholonomic, or “horizontal” directions), while $D_z^\perp$ (sometimes denoted $D^c$) is its complementary subspace (“vertical” or unconstrained directions).

With such a splitting, two complementary projection operators are defined: $$P: T W \rightarrow D^c \qquad Q = \mathrm{id} - P: T W \rightarrow D$$ These satisfy the idempotency and complementarity relations: $P^2 = P, \quad Q^2 = Q, \quad P + Q = \mathrm{id}$ When the splitting is orthogonal with respect to a Riemannian metric $g$, they can be written symmetrically as: $$Q = \frac{1}{2}(\mathrm{id} + F), \qquad P = \frac{1}{2}(\mathrm{id} - F)$$ However, oblique decompositions (non-orthogonal cases) are also treated within the same framework.

2. Construction in Riemannian, Poisson, and Symplectic Settings

Riemannian Geometry

Given a Riemannian metric $g$, a polarization by a constant-rank distribution $D$ allows the metric to split as $g = g^D + g^{D^c}$, with $g^D$ the restriction to $D$ and $g^{D^c}$ its complement. In a local coordinate system with adapted coframe $\{dz^\alpha, \omega^a\}$, the projectors are concretely expressed: $$P(Z) = \sum_a Z^a X_a,\qquad Q(Z) = \sum_\alpha \left(Z^\alpha + r^\alpha_a Z^a\right) \frac{\partial}{\partial z^\alpha}$$ and in matrix form,

$$P = \begin{bmatrix} r^\alpha_a & \mathrm{id} \end{bmatrix},\qquad Q = \mathrm{id} - P$$

The musical morphisms induced by the metric (${}^\sharp_g$ and $\flat_g$) are exploited to clarify the intrinsic projective properties, yielding: $$g = g_P + g_Q,\qquad g_Q(\cdot,\cdot) = g(Q^*(\cdot), Q^*(\cdot)),\quad g_P(\cdot,\cdot) = g(P^*(\cdot), P^*(\cdot))$$ This splitting is crucial for D'Alembertian reduction: in constrained mechanical systems, constraint forces reside in $D^c$ and physical evolution is projected into $D$.

Poisson and Symplectic Geometry

On a Poisson manifold $(W, \Pi)$, where $\Pi$ is a bi-vector with $[\Pi,\Pi]=0$, the corresponding bundle morphism ${}^\sharp: T^*W \to TW$ is constructed via $\sharp(\alpha) = \Pi(\alpha, \cdot)$. For degenerate $\Pi$, the flow is decomposed into symplectic leaves and their transverse complements: $T_z W = T_z S \oplus S^c$ Projections onto a leaf ($T_S$) yield an induced Poisson structure on transverse submanifolds $M$: $\Pi_M = P_{T_M}(\Pi_W)$ Explicitly, this recovers the Dirac bracket: $$\{f, g\}_M = \{f, g\}_W - \{f, x^\alpha\}_W N_{\alpha\beta} \{x^\beta, g\}_W$$ where $N_{\alpha\beta}$ is the inverse of the symplectic matrix restricted to the leaf.

An operator of the form

$$q = g^*(\cdot, \omega^a)\,G_{ab}\,g(\omega^b, \cdot)$$

with $G_{ab} = (g(\omega^a, \omega^b))^{-1}$, also realizes a projector from the metric and constraint one-forms, matching $Q$ when the distribution is orthogonal.

3. Application to Nonholonomic and Sub-Riemannian Systems

In nonholonomic mechanics, admissible motions are confined to a nonintegrable distribution $D$. The partial projection operator directly decomposes variational displacements $\delta z$ into $D$ (“horizontal”) and $D^c$ (“vertical”) components, allowing derivation of equations of motion without introducing Lagrange multipliers. The Euler-Lagrange vector $E_L$ is split: $E_L - \text{(constraint force)} = Q(E_L)$ Constraint forces are inherently accommodated by the geometry of $Q$.

Example: Chaplygin–Carathéodory Sleigh

A classical nonholonomic system with coordinates $(x, y, \theta)$ and constraint $$v = -r \dot{\theta} + \cos\theta\, \dot{y} - \sin\theta\, \dot{x} = 0$$. The constraint covector is assembled into a matrix $A$, and the projection is constructed as

$Q = A^* G^{-1} A,\qquad G = AA^{*T}$

Projecting the Euler-Lagrange equations yields the nonholonomic dynamical system consistent with D’Alembert’s principle.

Example: Contact Sub-Riemannian Geometry

For a free particle in $\mathbb{R}^3$ under $\omega = dz - ydx = 0$, projections with respect to the adapted metric and basis give rise to the Heisenberg Lie algebra among projected vector fields, and induce a pseudo-Poisson structure in phase space.

4. Summary of Key Formulas

• Projectors for almost product structure:

$$Q = \frac{1}{2}(\operatorname{id} + F),\quad P = \frac{1}{2}(\operatorname{id} - F)$$

• Compatibility with metric:

$$g = g_P + g_Q,\quad g_P(v,w) = g(P(v), P(w)),\quad g_Q(v,w) = g(Q(v), Q(w))$$

• Local decomposition in a foliated chart:

$$P(Z) = \sum_a Z^a X_a,\quad Q(Z) = \sum_\alpha \left(Z^\alpha + r^\alpha_a Z^a\right) \frac{\partial}{\partial z^\alpha}$$

• Poisson reduction (Dirac bracket formula):

$$\{f, g\}_M = \{f, g\}_W - \{f, x^\alpha\}_W N_{\alpha\beta} \{x^\beta, g\}_W$$

5. Theoretical and Practical Impact

Partial projection operators—rooted in the geometric splitting of tangent or cotangent spaces and extended through metric, Poisson, or symplectic structures—enable the intrinsic decomposition of dynamics and forces in constrained mechanical systems. This method provides:

• A principled alternative to coordinate-based constraint elimination or Lagrange multipliers, seamlessly accommodating nonholonomic and sub-Riemannian structures.
• A local and global framework for the analysis of both holonomic and nonholonomic constraints across diverse geometric settings.
• A direct mechanism for deriving geometrically meaningful equations of motion, encoding constraints at the operator level rather than through separate algebraic conditions.

The versatility is exemplified by the treatment of both the Chaplygin–Carathéodory sleigh (finite-dimensional, planar motion with direction constraint) and the particle with a contact constraint (embedding of sub-Riemannian geometry and Lie algebraic structure).

In summary, partial projection operators offer a unified and geometrically transparent approach for modeling, analyzing, and deriving dynamics in constrained systems across a broad array of settings, from classical mechanics to geometric control and sub-Riemannian analysis (Pitanga et al., 2011).