This manuscript develops a variational geometric framework for limit cycles in planar dynamical systems based on the symmetric part of the Jacobian matrix. Let \( J = DF \) denote the Jacobian of a vector field and$$S = \frac{1}{2} (J + J^T)$$its symmetric part. A circulation functional is defined on closed curves \( \gamma \) by$$I(\gamma) = \oint_{\gamma} T^T S T \, ds,$$where \( T \) is the unit tangent. The quantity \( I(\gamma) \) measures averaged tangential deformation along \( \gamma \). Vanishing of \( I(\gamma) \) provides a necessary condition for the presence of a limit cycle along a transverse foliation. Under suitable hypotheses, sign changes of \( I \) yield trapping regions and geometric sufficiency criteria for existence. The order of vanishing relates to multiplicity and degeneracy, giving a classification of hyperbolic, semistable, and higher-multiplicity cycles. For polynomial vector fields of degree \( d \), the index inherits algebraic structure. In suitable probing families one obtains$$I(R) = R P(R^2),$$where \( P \) is a polynomial with degree controlled by \( d \). This reduces certain counting questions to the study of real zeros of scalar functions and yields degree-based bounds consistent with known finiteness phenomena. Examples include Hopf-type systems, Liénard systems, and near-Hamiltonian perturbations, where the index reproduces classical bifurcation and stability conditions. The framework relies only on local differential properties and integration along closed curves, without using return maps or time-parametrized Poincaré maps.