I couldn't tell you about Alexander Hamilton specifically, but the Sparta that supposedly resulted from the laws of Lycurgus has been one of the great political foils since the 4th century BC. The modern standard work on this is still Elizabeth Rawson's The Spartan Tradition in European Thought (1969), but there has been a recent surge in scholarship on the afterlife of Sparta as a socio-political ideal. One of the key works is the volume edited by Hodkinson and Morris, Sparta in Modern Thought (2012).

As Rawson already pointed out, Sparta has been used to support an endless parade of political ideas, ranging across the political spectrum and across cultures and historical periods. In most of these appropriations only the life of full Spartan citizens is considered, and the fact that Spartan society was supported by a large serf underclass is conveniently forgotten.

In its essence, Lycurgan Sparta provides an example of the extent to which the ideal citizen can be created by the state through direct interference in the selection, upbringing, environment, dress, speech, diet, activities and opportunities of children. The Greeks themselves, at least as early as the final years of the 5th century BC, were already discussing the merits and drawbacks of the Lycurgan system. Proponents emphasised the health and moderate lifestyle of Spartan citizens, and the stability and power of their state. Detractors noted the rampant inequality even within the citizen body, and the state's tendency toward conservatism and inertia. Aristotle also really didn't like the fact that women had more rights in Sparta than they did elsewhere in Greece, and attributed much of their loss of power to this fact.

The key question here is which produces better citizens - freedom or control. This question remains forever relevant, and Sparta lingers in the background of any modern discussion on nature vs nurture, state educational curricula and the desirability of a prescribed set of moral/intellectual/physical training for young citizens.

Another branch of the Spartan tradition is about its nature as a state. Political thinkers have often stereotyped Sparta as the ideal of the stable, authoritarian, self-sustaining land power, in contrast with the aggressive, democratic, unstable sea power of which Athens is the primal example. The opposition between Early Modern France and England has been cast in this mould - and so has the Cold War. I am told that the association of the US with Athens and Russia with Sparta is part of the reason why Thucydides still features heavily in the education of American military officers.

But the association did not always go that way around. In earlier times, it was the United States that preferred to see itself as the Spartan type - an agricultural land of proud, hardy men, with a government system resistant to mob rule internally, and disinclined to involve itself with adventures overseas. They feared the example of Athens, with its fickle and costly policies and its constant ambition for more. Even in current politics, it is possible to caricature Republicans as "Spartan" and Democrats as "Athenian".

More radical political movements have made their own use of the idea of Sparta. Marxists took it to represent a sort of proto-communism - an ideal rural autarky in which the means of production were in the hands of a government that theoretically consisted of all adult male citizens. The Nazis, meanwhile, focused on the idea of a "pure" class of citizens exercising their right to rule over their enslaved inferiors, and liked to portray themselves as the descendants of the Spartiate Herrenvolk.

So, while I can't say in what ways Lycurgus and Sparta influenced particular figures, it is safe to say that they would have been aware of the stereotype of Sparta, and that they would have had strong opinions about whether that was an ideal to strive for or a dystopia to avoid. Most people since Classical times have done so.

Lycurgus comes off as more of a legendary figure than a historical one. Spartans in antiquity remembered Lycurgus as the ultimate lawgiver who created the Spartan system as we remember it seemly on his own.While many scholars tend to believe there really was an important man named Lycurgus in the Spartan past, they are skeptical he single handedly created the classical Sparta. Instead they tend to think that the Spartan system evolved over time, and specifically took a turn after the victory over Messenia

The tendency to attribute long term change to a single man is a reoccurring theme in history. For instance, I have heard professors compare the remembrance of Lycurgus to that of Romulus of Rome. They probably didn't do as much as they we remembered for , but it was easier for the following generations to explain their state as a somewhat divine act as opposed to gradual change.