Unified Lagrangian-Hamiltonian Formalism

• Unified Lagrangian-Hamiltonian formalism is a comprehensive framework that integrates Lagrangian and Hamiltonian descriptions by including higher-stage constraints in an extended configuration space.
• The approach systematically extends the configuration space with auxiliary variables, enabling explicit treatment of gauge symmetries and the recovery of the original dynamics when these variables vanish.
• By simplifying the constraint hierarchy and validating the Dirac conjecture, the formalism facilitates efficient symmetry analysis, constraint reduction, and quantization of singular systems.

The unified Lagrangian-Hamiltonian formalism is a comprehensive framework that encodes both the Lagrangian and Hamiltonian descriptions—especially for singular systems with higher-stage constraints—within a single geometric and algebraic structure. By systematically extending the configuration space, constructing an extended Lagrangian that manifests all algebraic constraints, and providing closed-form gauge symmetry generators, this formalism clarifies and simplifies the relationship between constraints, gauge symmetries, and the Dirac conjecture in constrained dynamics.

1. Construction of the Extended Lagrangian

Starting from a singular Lagrangian $L(q, \dot{q})$ whose constraint structure (primary, secondary, etc.) is determined by the Dirac-Bergmann algorithm, the formalism introduces new configuration space variables $s^a$—one for each higher-stage constraint $T_a$. These auxiliary variables are included in a closed-form extended Lagrangian: $$\tilde{L}(q, \dot{q}, s) = L(q, v(q, \omega(q, \dot{q}, s)), \dot{q}) + \omega_i(q, \dot{q}, s) \left[ \dot{q}^i - v^i(q, \omega(q, \dot{q}, s)) \right] - s^a T_a(q, \omega(q, \dot{q}, s)),$$ as in Eq. (21).

Here, $v^i$ is an auxiliary function relating velocities to momenta, and $\omega_i = \omega_i(q, \dot{q}, s)$ are new functions defined by inverting the relation

$$\dot{q}^i - v^i(q, \tilde{p}, \dot{q}) - s^a \frac{\partial T_a(q, \tilde{p})}{\partial \tilde{p}_i} = 0,$$

locally near $s = 0$. The original Lagrangian is recovered for $s = 0$: $\tilde{L}(q, \dot{q}, 0) = L(q, \dot{q}).$

The key property is that all higher-stage constraints $T_a$ enter $\tilde{L}$ directly, rather than appearing only as late-generation constraints in the Dirac-Bergmann sequence. In the extended Hamiltonian analysis, the momenta conjugate to $s^a$ are primary constraints $\pi_a = 0$, and their conservation enforces $T_a = 0$ at the next stage. Thus the entire tower of constraints "collapses," with all original constraints appearing as primary or secondary constraints of the extended system.

2. Extension of Configuration Space and Geometric Interpretation

The configuration space is augmented as $(q, s)$, where the number of extra variables $s^a$ matches the number of higher-stage constraints. These variables acquire physical meaning: those associated with first-class constraints emerge as gauge fields. Their transformation laws under infinitesimal gauge symmetries generally involve derivatives of local parameters, exemplified by

$$\dot{q}^i \rightarrow D_\tau q^i = \dot{q}^i - s^a \frac{\partial T_a}{\partial \dot{q}^i},$$

mirroring the structure of covariant derivatives.

In practical models, these auxiliary variables $s^a$ play a nontrivial role in both the equations of motion and the realization of local symmetries.

3. Explicit Construction of Gauge Symmetry Generators

The gauge symmetries of $\tilde{L}$ admit a closed-form expression directly in terms of the original first-class constraints $G_{I_1}$: $$\delta_{I_1} q = \epsilon^{I_1} \left. \{ q, G_{I_1}(q, p) \} \right|_{\partial \tilde{L}},$$ with $\epsilon^{I_1}(\tau)$ arbitrary local (possibly time-dependent) parameters and where the bracket is evaluated consistently with the derivatives taken in the Lagrangian context.

The variations of the auxiliary fields $s^a$ are: $$\delta s = \dot{\epsilon}^{I_1} K^a_{I_1} + \epsilon^{I_1} \left( b^a_{I_1} + s^b c^a_{b I_1} + \dot{q}^i c^a_{i I_1} \right),$$ where the tensors $K, b, c$ are structure functions determined by the closure of the constraint algebra—see Eq. (47).

An important operational property is that all first-class constraints of the original Lagrangian become explicit gauge symmetry generators for $\tilde{L}$, and, crucially, the transformations of $s^a$ naturally involve time derivatives of the gauge parameters (as is typical for gauge fields).

4. Clarification of the Dirac Conjecture

The explicit realization of all first-class constraints as gauge generators in the extended formalism constitutes a constructive proof of the Dirac conjecture in the Lagrangian context. For any singular system, the passage to the extended system (with Lagrangian $\tilde{L}$) ensures that the infinitesimal symmetry transformations of all configuration variables—regulated by Eq. (46) and Eq. (47)—are exhaustively generated by the first-class constraints.

This process decomposes general symmetries (possibly involving higher derivatives of the local parameters in $L$) into a sum of simpler symmetries (at most linear in derivatives of the gauge parameters) in the extended formulation, streamlining the analysis of local invariances.

5. Central Formulas and Their Roles

The following table summarizes the central formulas and their significance in the unified Lagrangian-Hamiltonian formalism:

Key Equation	Purpose	Reference
$\tilde{L}(q, \dot{q}, s)$ (Eq. 21)	Defines the extended Lagrangian with explicit constraints	Eq. (21)
$\tilde{L}(q, \dot{q}, 0) = L(q, \dot{q})$	Ensures recovery of original Lagrangian when auxiliary variables vanish	Eq. (22)
$$\delta_{I_1} q = \epsilon^{I_1} \{ q, G_{I_1} \}|_{\partial \tilde{L}}$$	Gives gauge symmetry transformation of configuration variables	Eq. (46)
$\delta s = \dot{\epsilon}^{I_1} K^a_{I_1} + \cdots$	Gives gauge symmetry transformation of gauge (auxiliary) fields	Eq. (47)
$$\dot{q}^i \rightarrow D_\tau q^i = \dot{q}^i - s^a \frac{\partial T_a}{\partial \dot{q}^i}$$	Covariantization of velocities in the gauge-invariant kinetic structure	Eq. (43)

All symmetry structures, including the full constraint algebra and all local invariances, become explicit and tractable within this construction.

6. Practical and Conceptual Impact

The unified Lagrangian-Hamiltonian formalism resolves ambiguities in constraint implementation by encoding higher-stage constraints directly in the modified Lagrangian and configuration space. All local symmetries are rendered manifest; their algebraic structure is simplified to gauge-type symmetries, and the full set of constraints is produced in a controlled manner at a lower generation within the constraint hierarchy. The method applies equally to systems with both first- and second-class constraints and does not require classification of constraints at intermediate steps.

This framework enables more efficient identification and handling of gauge symmetries and provides constructive tools for quantization, systematic gauge fixing, and reduction of dynamics to true physical degrees of freedom.

7. Summary

The extended (unified) Lagrangian–Hamiltonian formalism, as constructed for general singular systems, employs auxiliary variables to absorb higher-stage constraints into an extended Lagrangian. The approach yields a closed, explicit symmetry structure whose gauge transformations are generated by all first-class constraints of the original system, confirming the Dirac conjecture in the Lagrangian setting. This not only simplifies symmetry analysis but also provides a manifestly unified structure bridging Lagrangian and Hamiltonian formulations, essential for the systematic paper of singular systems, constraint reduction, and quantization (0901.3893).