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Mean field games equations are examined for conservation laws. The system ofmean field games equations consists of two partial differential equations: theHamilton-Jacobi-Bellman equation for the value function and the forwardKolmogorov equation for the probability density. For separable Hamiltonians,this system has a variational structure, i.e., the equations of the system areEuler-Lagrange equations for some Lagrangian functions. Therefore, one can usethe Noether theorem to derive the conservation laws using variational anddivergence symmetries. In order to find such symmetries, we find symmetries ofthe PDE system and select variational and divergence ones. The paper considersseparable, state-independent Hamiltonians in one-dimensional state space. Itexamines the most general form of the mean field games system for symmetriesand conservation laws and identifies particular cases of the system which leadto additional symmetries and conservation laws.

Conservation laws of mean field games equations