Within Known Physics
There are three vaguely plausible methods for changing a planet's orbit that come to mind without breaking the accepted laws of physics:
- direct application of thrust
- Consistent close passes of a series of massive objects
- targeted solar flares
Direct application of thrust is going to be incredibly disruptive - it's either by impact or by insolation changes.
Impact as a direct application of force is practical on a societal time scale, but not on an individual time scale. It would require minor alterations to many cometary/asteroidal bodies in crossing orbits, with lead times of one to two decades per each. Remember, a small perturbation can result in a significant change in position over a long time. The important elements of the project require ensuring that the final impact is on the correct position - the trailing (retrograde) face of venus to speed it up, or the spinward (prograde) face of mars. (Remembering the WWIVnet reduction of Niven's ditty: Spin out, out back, back in, in spin. Accelerate to spinward, you move out. To back, move in.) The issue is that this would take thousands of impacts, all of which would need to change the orbital velocity in the same direction, over a scale of a couple centuries, in order to have any appreciable effect. We are just now getting to the technological point where this is feasible - but it will likely never be practical.
The use of altered insolation could produce a change in atmospheric loss in such a manner as to create a very slow, but steady, acceleration. Light pressure is already a known factor in orbital mechanics by itself; the mirror array would be accelerated away, so would need some form of acceleration of its own. Perhaps, a larger sail array. Timescale is, again, centuries. Mass is tremendous - comparable to entire navies. In the case of Venus, the desired thinning effect might be useful, as well.
Moving massive objects by at the correct path is doable, but without a non-N-space drive, requires insane amounts of power to alter them. To accelerate, you need to route them close enough to the correct side. For venus, this means parading a bunch of bodies past the spinward side of Venus, and preferably, most of them doing so on the outbound leg of their own path. For mars, you want them on the trailing side and inbound leg. Ideally, for venus, they also should be moving prograde, while for mars, retrograde. Minor errors? Accumulate new moons. The drawback is that, to get sufficient mass, you're looking at moving large asteroids.
It is theoretically possible to trigger solar flares; doing so could be used to apply recurrent pressure over several centuries. The side effects, however, are that doing so is likely to be seen as a threat of wiping out all life in the system.
According to Wikipedia, solar wind generates between 1e-9 and 6e-9 nPa. Venus has a cross sectional area of about 1.15e14 m^2, a mass of around 4.87e24 kg, for a total net of about (3.5x1.15/4.8)e(-9+14-24) = 0.8e-19 nm/s^2 or 8e-26 m/s^2. Pressure at venus should be about 2.7x as strong... It would take 1,000,000,000,000,000,000 years to hit 8m/s. A CME can be 1000x more energetic and can add additional from atmospheric heating... It's theoretically useful, but in practice, worthless. If one's engineering them, one can possibly also get deeper upshot, more energy, and more particle density.
Just outside current physics
- A giant Alcubierre-White Warp Drive
- Gravity Manipulation - if the math is to be believed, gravity is a manipulable force.
The Alcubierre-White Warp Drive, in theory, could be used to place a body into a "fall into orbit" position. It doesn't matter if it generates superluminal pseudovelocities - only that it moves the object by changing the topology of space. Without Dr. White's transformations, the math would indicate insanely high energy requirements; even with, moving a bubble that large is likely to require a stupidly high amount of energy.
Gravity manipulation is believed theoretically possible by a number of physicists. If one can either enhance or impede gravity's pull artificially, one can then make use of that to alter orbits. But the process may be surprisingly convoluted. You move out with it reducing, then turn it to enhancing to accelerate inbound, then turn it to reduce again as one passes perihelion, to increase overall orbital speed. Unfortunately, this also increases eccentricity. Careful manipulation of the process can reround the orbit, but during the transfer process, it's actually going to get closer. In practice, however, almost no one is willing to admit to thinking it within easy reach. Many physicists think it impossible, citing that gravity is a fundamental property of matter, not a force generated by matter; if it's a fundamental principle, it isn't alterable.