Two types of mimetic gravity models with higher derivatives of the mimetic field are analyzed in the Hamiltonian formalism. For the first type of mimetic gravity, the Ricci scalar only couples to the mimetic field and we demonstrate the number of degrees of freedom (DOFs) is three. Then in both Einstein frame and Jordan frame, we perform the Hamiltonian analysis for the extended mimetic gravity with higher derivatives directly coupled to the Ricci scalar. We show that different from previous studies working at the cosmological perturbation level, where only three propagating DOFs show up, this generalized mimetic model, in general, has four DOFs. To understand this discrepancy, we consider the unitary gauge and find out that the number of DOFs reduces to three. We conclude that the reason why this system looks peculiar is that the Dirac matrix of all secondary constraints becomes singular in the unitary gauge, resulting in extra secondary constraints and thus reducing the number of DOFs. Furthermore, we give a simple example of a dynamic system to illustrate how gauge choice can affect the number of secondary constraints as well as the DOFs when the rank of the Dirac matrix is gauge dependent.
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