'Dancers are the athletes of God" – so said Albert Einstein though no-one, it seems, can confidently pinpoint when or where.

But it's a deliciously quotable quote, and it fits the 2012 Dance GB project like a bespoke glove or possibly a pointe shoe. For even as the UK's athletes are rigorously preparing mind and body for the London Olympics next month, the dancers in all three of our national companies – Scottish Ballet, National Dance Company Wales and English National Ballet – are in the final stages of rehearsing their own contribution to the event: a triple bill of newly commissioned works, all inspired by the Olympics and premiering at Glasgow's Theatre Royal on June 19. The programme will then tour to the Wales Millennium Centre and the Big Top Tent in Greenwich, London.

English National Ballet have recruited Itzik Galili as their dance-maker, while the Welsh have opted for Christopher Bruce, and both these choreographers have substantial, well-proven track records with various companies at home and abroad. Scottish Ballet, meanwhile, have chosen one of the brightest emerging talents on the UK dance scene, Martin Lawrance, and allied his creativity with that of another Martin – 2011 Turner Prize winner Martin Boyce – in what promises to be a visually striking celebration of core Olympic values and the intense, extraordinary physicality of the athletes who compete.

Their sense of historic challenge, of total resolve and aspiration, is threaded through every aspect of this piece, though you might detect a hint of wry humour in the title as well – it's called Run For It.

Ask Lawrance how he felt when Scottish Ballet's artistic director, Ashley Page, first approached him and his humorous side surfaces again as (with apologetic recourse to an expletive or two) he describes the "fight or flight" mix of panic and eager excitement that had him saying "Yes" to the commission even as the spectre of self-doubt was screaming "No! No! No!" inside his head. When it transpired that Run For It would also be a part of the company's touring autumn programme, Lawrance revisited that moment of conflicting emotions and now chuckles at the memory.

"I don't know whether that prospect brought me down to earth or whether it made me even more nervous," he says. "It meant that whatever I came up with wasn't going to disappear after Greenwich; it was going to be seen alongside works by William Forsythe and Hans van Manen in venues across Scotland. No pressure there, then! Because even as you start to take in what an honour it is to be a part of Dance GB, a part of this company's repertoire, you can't help but be aware of the trust that's been placed in you, the responsibility you have to do your best. Yes, the worry is there at the back of your mind that you can't let people down."

All across the globe, countless athletes are probably thinking much the same – and dancers too. When your body is your instrument, trained to meet exceptional standards of finesse and stamina, you want to believe it can – will – do all that you demand of it.

And when it doesn't? Lawrance knows, from personal experience, how that career-ending moment feels. As a dancer with the Richard Alston Dance Company (RADC), he'd reached out to audiences and critics alike with his ability to express nuances of profound humanity within seemingly effortless virtuoso technique. In 2007, those memorable performances stopped. Despite repeated surgeries, Lawrance's knees simply couldn't take the continuing wear and tear of daily class, studio rehearsals and evening shows.

He did, however, have the beginnings of a Plan B that would allow him to stay actively connected with dance: he had a flair for choreography. Alston, recognising Lawrance's potential, did what many folk never would have believed likely – he asked Lawrance to make a piece for RADC (which had previously danced only Alston's work). Since then more commissions – from Alston and others – have reinforced the impression that Lawrance is definitely a talent worth watching.

"It's that whole business of trust again," says Lawrance. "Richard, who has been such a tremendous mentor to me, and now Ashley – you have to believe in yourself if they do. Run For It is the biggest thing I've ever done, but it's also the kind of opportunity you dream about. And now, working with these dancers ... they're just so willing to try anything. They never hold back. I'll say 'Can you indulge me here? Just try this', and they're into it, moving and thinking at full tilt. Really working it, making it spark. Making something to amuse me, surprise me..."

His words melt into a sudden burst of bright laughter that sums up all the pressure-cooker process of creating movement from scratch, that head-of-steam moment when the lid comes off and – oh joy! – ideas that have been sweated through with fearless effort deliver ... well, maybe a whole minute of dance that hits the intended mark.

Helping to drive Lawrance's ideas forward are the rhythmic complexities and cross-currents in John Adams's Son Of Chamber Symphony. He's been determined not to make Run For It into some cavalcade of sporty cliches, even when he's set the dancers tasks that reference pole-vaulting or fencing, karate or marathons in terms of particular body line or dynamic thrust. But he reckons that the Olympic context creeps into everything, including his responses to the music.

"I found myself thinking that the repeats in the first movement – which are, in fact, played by different instruments – are passing on information almost like a baton in a race. The middle movement, we've got a long pas-de-deux, That for me is the marathon: slow at first, because you don't start the marathon at full throttle, but intensifying so by the end, bodies are utterly seized by exhaustion. But then, if you've won the race..."

You can hear the dancer's muscle-memory in his voice. That innate understanding of what it means to push on through the pain and exhaustion to that all-too-transitory pinnacle of achievement.

That "podium moment" is also something that Martin Boyce can identify with, having won the prestigious Turner Prize last year. And it's not all giddy euphoria.

"The weirdest thing was how really low, really down, I was for days afterwards," the artist explains. "Grumping around the house, eating beans on toast, somehow not convinced I'd won the same prize that all those other artists had won. I'd got the Turnip Prize..."

His wife's insistence that they had a proper party to celebrate blew away those glooms, and now he can waggishly evoke them in hugely entertaining anecdotes. Mind you, he never thought that his set design – now a key element in Run For It – would ever translate from small-scale model to full-size actuality.

A couple of years back, Ashley Page had invited Boyce to be part of a creative exchange/choreographic workshop that he had in mind for Scottish Ballet. "Just watching the dancers at such close quarters was inspiring," says Boyce. "Then the music – Steve Reich – struck me as having this pointilliste quality. And such precision. Seeing how the movement came about, I really felt I wanted to show them my response to all this, so I made a model set for something I never thought would happen."

Now, as June 19 draws ever nearer, Boyce is back at Scottish Ballet's Tramway headquarters in Glasgow watching as his concept – a vast geometric canopy of aluminium fins, with a pillar that rises, like a tree trunk, from floor to canopy – takes shape. Like Lawrance's choreography, Boyce's vision wears its Olympic connections lightly.

"I don't like the idea of 'themes', but I think we're all beginning to see these connections," he says. "There is that visual reference to a tree, the canopy that people gather under. That sense of creating a community, which the Olympics does. And you've also got that thing of challenging yourself." There's a chuckle. "The whole set needs to be disassembled within 15 minutes – so we're working on getting that to time."

It sounds as if Scottish Ballet's stage crew, and not just the dancers, will be in full-on athletic mode when Run For It hits its stride.

Dance GB is at the Theatre Royal, Glasgow, June 19-23, with related activities, including a Family Challenge Day, on Saturday, June 16. Full details at www.scottish ballet.co.uk