She unveiled a statue marking her visit, met jockeys and others involved in racing, and was shown stud horses at the centre in Kildare, west of Dublin.
Irish agriculture minister Simon Coveney said the Queen's trip had now entered a "more relaxing" phase.
She has expressed "deep sympathy" for victims of past British-Irish conflict.
The Queen is making the first trip to the country by a reigning UK monarch for 100 years and the first since it gained independence.
At a state dinner at Dublin Castle on Wednesday, she offered "sincere thoughts and deep sympathy" to people who had suffered during the two countries' "troubled past".
An apology had not been expected, BBC royal correspondent Nicholas Witchell said, but the Queen came "pretty close".
Continue reading the main storyThe Queen's among horses. She'll be happy. They don't bow, curtsey or fawn”End Quote Peter Hunt Royal correspondent, BBC News Earlier on Wednesday, her engagements had included a visit to Croke Park sports stadium in the capital, where 14 people were killed by British forces 91 years ago, and the earlier laying of a wreath at a service honouring Ireland's World War I dead.
BBC Ireland correspondent Mark Simpson said that having reached "half-time" in the Queen's four-day trip, there was now "no more politics - just horses and showbiz today".
The Queen is due to attend a variety show at Dublin's National Convention Centre on Thursday evening.
Mr Coveney said: "The first two days were very much about the emotive history of the two islands, reconciliation and moving forward and respecting the history. I think the Queen did that in an extraordinarily generous way.
"I think the second half of this visit is, I hope, more relaxing - and she's very interested in racing and horse breeding."
He said: "Horse racing is part of what it means to be Irish and an important industry for us here."
'Reconciliation'Addressing guests at the state dinner on Wednesday, the Queen opened her only public speech of the trip in Irish.
"A hUachtarain agus a chairde (President and friends)," she said.
The Queen opened her speech in IrishShe went on: "It is a sad and regrettable reality that through history our islands have experienced more than their fair share of heartache, turbulence and loss.
"These events have touched us all, many of us personally, and are a painful legacy. We can never forget those who have died or been injured, and their families.
"To all those who have suffered as a consequence of our troubled past I extend my sincere thoughts and deep sympathy."
She added: "With the benefit of historical hindsight we can all see things which we would wish had been done differently or not at all."
Continue reading the main storyI think people were very taken by the extent of her words and the clarity and ringing tone in which she said them”End Quote Enda Kenny Irish prime minister The Queen praised her host, President Mary McAleese, who had "done a great deal to promote this understanding and reconciliation" and "set out to build bridges".
Mrs McAleese, in her address to guests, said a new chapter had been opened between Ireland and Britain.
"The harsh facts cannot be altered nor loss nor grief erased but with time and generosity, interpretations and perspectives can soften and open up space for new accommodations," she said.
'Remarkable contribution'UK Prime Minister David Cameron was present at the state dinner, after earlier flying in and holding talks with Ireland's prime minister, Taoiseach Enda Kenny.
"As so often, her majesty has been able to put things and say things and do things in a way that massively strengthens the relationship," Mr Cameron said.
Mr Kenny said: "I think it was a remarkable contribution in what is a remarkable state visit.
"I think people were very taken by the extent of her words and the clarity and ringing tone in which she said them."
Sinn Fein president Gerry Adams, whose party did not attend the dinner, said he believed that the Queen's expression of sympathy was genuine.
But he added: "Queen Elizabeth's acknowledgement that the relationship between Britain and Ireland has not been entirely benign is a gross understatement.
"This will be forgiven if the future policy of her government is about building an entirely new future based on genuine equality, and mutual respect."
The Queen and Prince Philip's signatures on the visitors' book at the official residence of the Irish president
Wreath-laying at the Garden of Remembrance, the Queen (left) with President Mary McAleese (right)
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The Queen inspects the Book of Kells at Trinity College
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The Queen signs the visitors' book at Government Buildings, while meeting the Irish prime minister, on right
Irish veterans meet the Queen and the Irish president at the Irish War Memorial Gardens
The Queen is offered a pint at the Guinness Storehouse
The Queen and Prince Philip receive a hurley stick, at Croke Park stadium
The Queen wore a diamond harp brooch for her speech at Dublin Castle
The Queen views the stallions at the Irish National Stud
Thursday 19 May: Visit to National Stud at Kildare.
Friday 20 May: Tour of St. Patrick's Rock, Cashel and the English Market and Tyndall Institute, Cork.
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Magazine Home UK Africa Asia-Pac Europe Latin America Mid-East South Asia US & Canada Business Health Sci/Environment Tech Entertainment Video England Northern Ireland Scotland Wales UK Politics Education Magazine 26 May 2011Last updated at 12:33 GMT Share this page Delicious Digg Facebook reddit StumbleUpon Twitter Email Print Obama v Bush: A tale of two state visits By Vanessa BarfordBBC News 
Barack Obama has completed his three-day state visit to the UK. But how did it compare with the last state visit by a US president, when George Bush came to the country eight years ago in 2003? Obama v Bush Entourage Obama's entourage was believed to consist of about 500 people - including 200 secret service agents. The Obamas, whose US security codenames are Renegade and Renaissance, were also reportedly joined by a team of medics, chefs and White House staff. The Bushes, whose alternate US identities were Trailblazer and Tempo, were thought to have had an entourage of about 700. As well as secret service agents, national security advisers and government officials, there were also 15 sniffer dogs and five cooks on hand. Wheels 
Obama's bomb-proof Cadillac is known as The Beast. 
Bush's Cadillac de Ville was sometimes called the Stagecoach. Police security The Metropolitan Police Service said the policing operation for the entire duration of the state visit involved 5,000 officer shifts. The full cost of policing Obama's visit won't be confirmed until later this year, but it's expected to fall well short of the cost of the state visit by Pope Benedict - ?6.9m. Bush's state visit to London cost police more than ?4.1m, according to the former Metropolitan Police Commissioner Sir John Stevens. All police leave was cancelled during the visit and the salaries for officers' shifts cost about ?2.5m, while overtime, transport and catering came to more than ?1.6m. Protests 
A small group of radical Muslims gathered outside Downing Street to protest against the visit. Next to the anti-Obama protest, a few members of the right-wing English Defence League staged a counter-demonstration, with police separating the two sides when they threatened to come to blows. 
Tens of thousands of people marched against the Iraq war, which climaxed with the toppling of an 25ft effigy of Mr Bush, in a symbolic echo of the destruction of Saddam's statue in Baghdad. One protester also delivered a parting shot by throwing an egg at the presidential cavalcade, but the egg missed. Casual dining Obama and Cameron ditched their jackets and dished out burgers at a barbecue in the garden of No 10, in an event hosted by their wives for families of UK and US servicemen. British bangers, Kentish lamb chops, corn on the cob, Jersey Royal potatoes and caesar salad were also on the menu. Dessert was summer berries and ice cream. Much was made of Blair and Bush and their wives eating fish, chips and mushy peas at the Dun Cow Inn in Tony Blair's Sedgefield constituency. The starter was cream of potato and leek soup and dessert was a lemon creme brulee. Mr Bush washed it down with a non-alcoholic lager. Mrs Bush and the Blairs sipped soft drinks. Moment of rapport 
Playing table tennis in a visit to a south London academy 
Having a beer (albeit non-alcoholic) in a pub in Sedgefield Formal dining On the state banquet menu for the Obamas was sole with watercress, lamb in basil, roasted radish and courgettes, green beans and roast potatoes. It was followed by vanilla charlotte with morello cherries. The Bushes' Buckingham Palace menu was consomme with sorrel, roast halibut with herbs, breast of chicken with basil, roast potatoes and Savoy cabbage, followed by vanilla praline and coffee ice-cream. Banquet VIPs 
Famous names included actors Tom Hanks and Kevin Spacey and actress Helena Bonham Carter. 
Celebrity guests included actor Sir Michael Caine and entrepreneur Sir Richard Branson. Gifts The Queen gave Michelle Obama an antique gold and red coral brooch in the form of roses, while Obama received a book containing a selection of letters, from the royal archives, between past US presidents and British monarchs. The Queen received a leather-bound album containing photographs of her parents' trip to America in 1939. The Queen gave Laura Bush a jewellery box made by her nephew Viscount Linley and Bush a solid silver rule engraved with the royal cypher and the US presidential seal, and a specially-bound copy of The Royal Treasures, A Gold Jubilee Celebration. Meanwhile, the Bushes gave the royal couple a specially commissioned set of china. Talking points Libya and the developments across the Middle East, in Syria and Yemen and the fight against terrorism were discussed during 90 minutes of talks between Obama and Cameron. Other issues included the global economy, climate change and international security. Iraq featured heavily in the Bush-Blair talks but it did not totally dominate. International terrorism, the terror attacks against British targets in Turkey, Guantanamo internees and free trade were among the other key issues discussed. First Lady 
As well as co-hosting a barbecue with Samantha Cameron, Michelle Obama spoke to schoolgirls at Oxford University. The trip - arranged at her suggestion - aimed to encourage the girls to think about studying for a degree. All eyes were also on Michelle when she met the Queen but there was no repeat of her "faux pas" hug two years ago. 
Former First Lady Laura Bush kept a relatively low profile during the 2003 three-day state visit, seemingly happy to let her husband have the limelight. Laura and Cherie Blair watched a children's Shakespeare performance together, and the pair were also treated to a menu prepared by British celebrity chef Nigella Lawson. Slip up Obama suffered an awkward moment during the palace banquet when he proposed a toast "to the Queen" - which prompted the orchestra to start playing the national anthem. Obama carried on talking and raised his glass to the monarch before the music had finished. There was no widely reported mishap by Bush in 2003, however during the Queen's state visit to the US in 2007, the former president slipped up in a speech on the lawn outside the White House. He suggested the Queen had been on the throne since the 18th century. Then when he realised his mistake, he turned to the monarch and winked at her. Newspaper headline The Daily Mirror's headline on 25 May 2011 was a picture of the Queen and Obama, saying: "The Best of Buddies". The Daily Telegraph's headline on 20 November 2003 was a picture of Bush and the Queen, saying: "Bush: You're our closest friend". Trips abroad 
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