Hamlet, Wyndham's, LondonSister Act, Palladium, LondonArcadia, Duke of York's, London

Law's high-octane Hamlet for the new Shakespeare audience refuses to be spooked by the ghost of David Tennant

Can Jude Law succeed? It's a nerve-rackingly hard act to follow: playing the Prince of Denmark so soon after David Tennant's incisive, lionised take on the role. One West End Hamlet doth tread upon another's heel. But I'm inclined to say the more the merrier, if they're all as good as this.

Law's Prince is a human time bomb in Michael Grandage's new production. He stands, at first, in a shadowy corner of Elsinore's towering granite fortress, silently grieving over his father's death and his mother's marriage to his usurper-uncle. However, the moment Penelope Wilton's Gert-rude mildly questions his extended mourning, he snaps back, "I have that within that passes show" – his voice cracking. He has the dangerous intensity of an adolescent still. You can hear his emotional short fuse sizzling.

It does distract somewhat, having the Prince and his pals from Wittenberg looking like scruffy fashion models. Law sports tousled, out-of-bed hair and a deltoid-hugging T-shirt. Yet in another way, this muscularity makes a refreshing change. Here is no wavering wraith, but rather a distinctly physical, frustrated young man – a trained fighter as well as an intellectual – who could easily turn violent.

His hands, never still, dartingly express his every thought. Maybe this is a tad overdone. Given that the show is drawing so many newcomers, you wonder if Law has been asked to illustrate the meaning of Shakespeare's lexicon, like a constant accompanying mime show. Yet his Hamlet is vibrant and icon-like – gilded by crisscrossing shafts of light – as he clasps his forehead in despair, or as he turns his palms into snapping claws when sardonically naming his uncle-ensnaring play "The Mousetrap".

Furthermore, the way he suits the gesture to the word makes his philo-sophising lucid and impassioned. A feverish energy seems to surge from his fingertips as he delivers his soliloquies direct to the audience. These are startlingly intimate lectures, given on the verge of a nervous breakdown.

While verbally fencing, his aggression becomes increasingly concrete, too. He draws a knife more than once, threatening others or jabbing it towards his own wrist. Both Ophelia and Gertrude are manhandled, though his brutality alternates with cheek-cradling tenderness – echoed later in his stroking of Yorick's skull.

He pinions Penelope Wilton's Gertrude on the floor, as if he'd like to throttle her, in the closet scene which – with the hair-raising effect of a horror movie – we watch from the other side of the arras. This is a white gauze (set design by Christopher Oram), like a giant wedding veil-going-on-shroud, in which the stabbed, stumbling Polonius is soon netted.

Wilton is on superb form. Her screaming terror then brooding silence convey an entire dark journey into disillusionment. Ron Cook is outstanding, too, as Polonius: a petty bureaucrat, comically nervous with his superiors and only fitfully loving as a father.

But there are disappointing aspects to this production. Law has less quicksilver wit than Tennant. Gugu Mbatha -Raw isn't convincingly harrowed as the grief-maddened Ophelia. Adam Cork's soundtrack is also hit and miss, souping up Old Hamlet's ghost with hackneyed sci-fi rumblings, as if this production wanted to be a Hollywood movie. And maybe it will be. Grandage's Donmar-at-Wyndham's troupe certainly gives the Royal Shakespeare Company a run for its money, and the RSC has announced that its Tennant Hamlet is to be screened on BBC2.

Meanwhile, it's "Get thee to a nunnery" and then "Get on down" in Sister Act, the musical, newly adapted from the 1992 film which starred Whoopi Goldberg (who now turns co-producer).

This comic yarn – about Deloris, the brassy lounge singer, who's obliged to hole up in a convent and turns the nuns into a red-hot funky choir – is irresistibly enjoyable in parts, though the book, scripted by Cheri and Bill Steinkellner of Cheers, is surprisingly feeble, with a simplified plot and tiresome two-dimensional caricatures.

But, hey, Patina Miller's Deloris has brio and a belting voice. It's hard not to rejoice in this show's impish irreverence and in composer Alan Menken and lyricist Glenn Slater's cheeky conversion of song genres. Deloris's suggestive nightclub number, "Take me to heaven", ends up transmogrified as a gospel hymn with the nuns wowing the Pope, up to their wimples in sequins and disco-dancing against a vast psychedelic rose window. Lord!

Finally, sustaining a delicate balance between love and loss, mortality and the ongoing cycles of life, two couples waltz into the darkness at the end of Tom Stoppard's Arcadia, one of the most exquisite plays of the 20th century.

Arcadia is an extraordinary kind of detective drama, set in an English country house where a trio of researchers – historic, literary and scientific, attracted and repelled, competing and cooperating – try to piece together the past from fragmentary clues and hypotheses.

With comic wit and great intellectual breadth, Stoppard weaves back and forth between 1809 and the modern day, uncovering an elaborate sex farce, mysterious marginalia, a mathematical girl-genius and a haunting romantic tragedy. On the way, he embraces lust, landscape gardening, iterated algorithms and the heat-death of the universe, all with a light touch.

Alas, David Leveaux's production, though contriving to defy Newtonian thermodynamics, takes ages to warm up. Newcomer Jessie Cave, playing the child genius Thomasina, and Dan Stevens, as her young tutor Septimus, never really sparkle. Neil Pearson gains some momentum as the bounding rake, Bernard. But most winningly, it's the actor Ed Stoppard (the playwright's son) who turns the modern-day mathematician Valentine, and his shy wooing of Samantha Bond's prickly Hannah, into the aching heart of his play.

'Hamlet' (0844 482 5120) to 22 Aug; 'Sister Act' (0844 412 2704) to 13 Feb 2010; 'Arcadia' (0870 060 6623) to 12 Sep