LOUISVILLE, Ky. — Senator Rand Paul of Kentucky officially declared himself a candidate for the Republican presidential nomination on Tuesday, promising a crowd of cheering supporters that he is ready to shake up Washington and disprove those in his own party who doubt that a fiercely libertarian conservative can be a serious contender.

“The Washington machine that gobbles up our freedoms and invades every nook and cranny of our lives must be stopped,” Mr. Paul told a jubilant audience at the Galt House hotel. “I want to be part of a return to prosperity.”

In a speech outlining his small-government vision for the country, Mr. Paul leaned heavily on his biography, describing his experience as an eye surgeon, a career that inspired him after his grandmother’s vision failed. Recalling his own story of living the American dream, Mr. Paul scolded both Republicans and Democrats for failing Americans.

“What kind of America will our grandchildren see?” he asked. “It seems to me that both parties and the entire political system are to blame.”