By sea.
We call it the Peloponnesian War today because it was between the Athenian Empire and the Peloponnesian League, of which Sparta was a member and leader. At the time, they referred to it the war between Sparta and its allies and Athens and its allies.
The advantage that Athens had during the Peloponnesian wars was that they had a bigger army. They also had a larger Navy.
During the Peloponnesian war.
Sparta had the largest area after it conquered the southern Peloponnese, while Athens had limited land in Attica, until it created an empire for 50 years of the GREEK cities in the eastern Aegean Sea littoral.
Sparta .
Sparta and its allies and Athens and its allies fought for domiance. The Spartan side won with Persian assistance.
Sicily.
plato
Sparta and its allies.
The Peloponnesian League was formed by Sparta in the 6th century BCE as a coalition of city-states in the Peloponnese, initially to counter the influence of Athens and its expanding power. Members of the league resented Athens due to its imperialistic policies, particularly during the Delian League's dominance, which fostered a sense of inequality and exploitation among its allies. Athens's naval superiority and aggressive expansion often threatened the autonomy of other city-states, leading to rising tensions and ultimately the outbreak of the Peloponnesian War.
The city-states of Athens and Sparta as well as their allies were involved in the war that started at 431 and ended at 404 BCE. The Spartans and their allies were the winners of the war that later was named "the Peloponnesian war"
Phase 1 - the Peloponnesian League invested Athens, which sent out amphibious expeditions targeting the Peloponnesian League cities. Phase 2 - a phony peace during which Athens tried to capture Syracuse in Sicily. Phase 3 - more direct hostilities during which Persia supplied the funds for the Peloponnesian League to mount a fleet which could match the Athenian one. Athens defeated at sea, invested and surrendered.