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We derive conditions for a nonholonomic system subject to nonlinear constraints (obeying Chetaev's rule) to preserve a smooth volume form. When applied to affine constraints, these conditions dictate that a basic invariant density exists if and only if a certain 1-form is exact and a certain function vanishes (this function automatically vanishes for linear constraints). Moreover, this result can be extended to geodesic flows for arbitrary metric connections and the sufficient condition manifests as integrability of the torsion. As a consequence, volume-preservation of a nonholonomic system is closely related to the torsion of the nonholonomic connection. Examples of nonlinear/affine/linear constraints are considered. 

Abstract Herein, we demonstrate “direct”¹³C hyperpolarization of¹³C‐acetate via signal amplification by reversible exchange (SABRE). The standard SABRE homogeneous catalyst [Ir‐IMes; [IrCl(COD)(IMes)], (IMes=1,3‐bis(2,4,6‐trimethylphenyl), imidazole‐2‐ylidene; COD=cyclooctadiene)] was first activated in the presence of an auxiliary substrate (pyridine) in alcohol. Following addition of sodium 1‐¹³C‐acetate, parahydrogen bubbling within a microtesla magnetic field (i.e. under conditions of SABRE in shield enables alignment transfer to heteronuclei, SABRE‐SHEATH) resulted in positive enhancements of up to ≈100‐fold in the¹³C NMR signal compared to thermal equilibrium at 9.4 T. The present results are consistent with a mechanism of “direct” transfer of spin order from parahydrogen to¹³C spins of acetate weakly bound to the catalyst, under conditions of fast exchange with respect to the¹³C acetate resonance, but we find that relaxation dynamics at microtesla fields alter the optimal matching from the traditional SABRE‐SHEATH picture. Further development of this approach could lead to new ways to rapidly, cheaply, and simply hyperpolarize a broad range of substrates (e.g. metabolites with carboxyl groups) for various applications, including biomedical NMR and MRI of cellular and in vivo metabolism.