Would it be feasible to land large (Boeing 747 and Airbus A380) and all other aircraft on a sled on the runway, the kinetic energy of the aircraft is then transferred to the sled, then the sled uses regenerative braking to produce electricity for use by other electric sleds that taxi the aircraft to the runway for taxi, given that a lot of fuel is used during taxi and that fuel is expensive, causes green house gas pollution when burnt. This would save costs on tire wear brake pad wear and if designed properly improve braking safety
What you propose might be technically possible, but that doesn't make it a good idea.
However, any benefits would by tiny compared to the immense cost, the extraordinary engineering efforts required, and the horrendous risk of something going wrong.
Airports have long, wide runways for good reasons. Airliners don't touch down on a single spot. Your "sled" would need to be a very wide and extremely long platform.
The sled would need to be able to bear the weight of a large airliner as it touches down. A large plane can damage a concrete runway. The sled is going to need to be extraordinarily strong.
Your platform will need to accelerate from zero to the speed of a landing airliner in very short order; if it causes the plane to decelerate too rapidly instead, it will tear bits off the plane, or limbs and heads off the passengers.
Good luck designing and manufacturing this huge, strong and and extremely lightweight sled.
(Maybe you can design the sled to match the speed of the incoming plane. That would solve some problems, but only at the expense of introducing some even more exciting new ones, like having just constructed the world's largest and most lethal railgun.)
What's going to happen when something goes wrong? You now have a wheel-less airliner, in difficulties, at landing speed, plus a huge sled also possibly travelling at 300+ km/h. I don't even want to be in the same town as that, never mind at the same airport.
What will you gain? Not much in the way of saved energy from regenerative braking; the energy cost of maintaining the machinery that can cope with such savage braking effects will greatly outweigh it anyway.
And actually, taxiing is good for aircraft engines, because it allows them to warm up before being used at full power (it may even be good for them after landing, to allow them to cool down more slowly).
So, nice idea, but in this case, the wheel, all of 5000+ years old, is an invention better-suited to airliner landings.
To put some numbers around the answer by @Harper. According to this page, a 737-800 has a max landing weight of 144,00 pounds (65,000 kg) and landing speed of 155 knots (~80 m/s). That's E=0.5 m v^2 = 0.5 *65000 * 80^2 = 208 MJ of energy. 1 kW-hour is about 3.6 MJ, so we are talking 57 kW-hours. A kW-hour of electricity sells for about 10 cents. So there is about \$5.7 worth of energy in a jet liner landing, assuming that your system was 100% efficient. Once you take into account efficiency losses, you are probably talking \$2.80. That is so small compared to how much the system would cost, it is not worth it.
(edit: dropped the factor of 1/2 in the energy calculation originally, so it's even less money than I originally said).
On landing - There's no energy in it - airplanes aren't that heavy. If you're thinking there's \$1000 of energy there, forget it... there might be \$1 worth on a 737. That's figuring accelerating the platform to meet the plane, aero drag on both platform and plane, rolling resistance, and the sheer difficulty of storing a 50MW surge of energy you'd get catching a loaded 747. The battery pack alone - it'd be much cheaper to pay for an extra windmill at a wind farm, paint "Danzig airport" on the blades, and call it done. Given the shabby returns, the extreme safety risk of catching an airplane on a sliding platform is not worth it. Arrestor wires would help on all fronts but there's still only a couple bucks of energy there.
Actually the real win for such a platform system would be ground ops. Plane lands on its own, takes a high-speed exit to clear the runway for traffic, taxis onto a platform, and shuts engines down. The platform then shuttles to the apron or gate. This solves one of the biggest safety problems in aviation: the ground game, giving railroad-style positive movement control. It will also save a ton of fuel because aircraft won't spin up engines until near takeoff, and would shut them down immediately on landing - cutting hours off engine runtime and overhaul intervals.
I don't know how I'd feel about using a platform for takeoff, the safety win would be positive movement control (never this) and being catapulted to airborne before V1 decision point. But, many things to go wrong.
The platform would be powered via induction, slot rails (like DC trolleys) or surface current collection and would have a "startcart" onboard which could provide standby power, HVAC and bleed for engine start.
As others have mentioned, there is an issue in terms of what you get back, and the complexity and cost of building a sled to catch the airplane. But there are other issues as well which will be affected by such a change. You'd have an issue with moving the plane afterwards; once landed, do you keep the plane on a sled and move that around (costing lots of energy, changes to infrastructure, capacity issues if you run out of sleds), or do you move the plane off the sled and if so, how? Ramps may work, but takes a lot of area to avoid having to steep inclines, and there might also be a time penalty compared to the more conventional land-taxi-park.
It is unlikely that the sleds can reuse the runways as-is, so you might have to build separate runways for aircraft landing on sleds and those landing conventionally or taking off. Will it be possible for planes to cross the sled runway? If not, some airports will be handicapped (to say the least) as there might not be enough room to build taxiways around. And in any case, taxi times will increase if you can't cross, unless you build the sled-based runways away from everything else (increasing cost as you need more area for the airport).
Some parts of the world experience weather ranging from -30C with lots of snow and wind, to +30C with lots of water. How will the sleds handle that without breaking down? What if the sled breaks down during a landing roll; what will the effect be on the aircraft sitting on top?
This is just a few thoughts I had as a result of reading what others have posted. Everything boils down to cost. Complexity is not an issue by itself, but a more complex system will also have a higher maintenance requirement and more failure modes, driving the cost up. And while todays system isn't perfect, the combination of cost, complexity and flexibility is better than anything else that has been thought of or tested so far.