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We study the lower central series filtration L_k for a symplectic quotientA=A_{2n}/<w> of the free algebra A_{2n} on 2n generators, where w=\sum[x_i,x_{i+n}]. We construct an action of the Lie algebra H_{2n} of Hamiltonianvector fields on the associated graded components of the filtration, and usethis action to give a complete description of the reduced first component\bar{B}_1(A)= A/(L_2 + AL_3) and the second component B_2=L_2/L_3, and weconjecture a description for the third component B_3=L_3/L_4.

The lower central series of the symplectic quotient of a free associative algebra