There were 10 cities that were honored with a commemorative coin during the classic era (
Quick Bits #57 - City Commemorative Coins) - three of them are the capital of their respective State: Albany, NY / Columbia, SC / Providence, RI.
In 1936, a half dollar was struck to commemorate the 250th anniversary of the city of Albany, New York receiving its charter (July 22, 1686). Albany became the permanent capital of New York in 1797. Before being formally designated as New York's capital and the site of State Government, the State Legislature had met in Kingston (1777), Poughkeepsie (1778-1785) and then alternated between New York City and Albany.
Albany's more central location within the State was a significant factor in the decision to make it the permanent State Capital.
1936 Albany, NY Charter 250th Anniversay

The 1936 Columbia, SC Sesquicentennial half dollar was minted in commemoration of the sesquicentennial (150th) anniversary of the founding of the city of Columbia, South Carolina. Columbia was created to host the State Capital and Government - it was designed city. It was created by the South Carolina State Legislature in 1786 while it met in Charleston - South Carolina's first capital.
As with Albany, NY, Columbia's site is more centrally-located within SC - the primary reason for its selection.
1936 Columbia, SC Sesquicentennial

In January 1936, a bill calling for the minting of 50-cent coins "in commemoration of the three hundredth anniversary of the founding of the city of Providence, Rhode Island." was introduced in the US Senate. Though the resultant coin is commonly referred to as the "Rhode Island," its authorizing legislation clearly calls out the City of Providence which became Rhode Island's permanent capital in 1900. Prior to this official designation, the Rhode Island State Government met in the State's County Seats on a rotating basis. In 1854, Newport and Providence were designated as co-Capitals - this continued until 1900 when Providence was named the State's sole Capital.
1936 Providence, RI Tercentenary

So, there they are, three State Capitals honored on three US commemorative half dollars. The "Capital-ness" of the Albany, NY and Providence, RI coins may not have been the primary reason behind their issue - unlike the Columbia, SC coin - but the fact that each was its State Capital at the time the commemorative coin was issued can't be denied and hence the "connection"!
For other of my posts about commemorative coins and medals, including other stories about each of the half dollars presented here, see:
Commems Collection.