Single-phase to 3-phase conversion
I'm understanding this question to mean that you're trying to run a three-phase motor off a single-phase line. If you're trying to run the motor directly off the AC line, the phase angles involved will make it difficult to get the motor started, which is part of the reason three-phase exists in the first place. Single-phase motors usually have motor start caps for just that reason. That sounds like what you're describing.
The simple answer to your question is that to get three-phase AC from single-phase AC, you need to rectify the single-phase AC line into DC, then run the DC back through an inverter to get controlled three-phase AC. There are other electronic approaches, but they're less common in my (limited) experience. There are also mechanical approaches, which may be more convenient if you have the parts.
I'd suggest using a drive to operate your three-phase motor. Typical variable-frequency three-phase drives are exactly what I described above: a rectifier, followed by an inverter. I can't speak as to what's on the market in a given power class, but larger three-phase drives typically have terminals for the three-phase AC line input, the DC bus, and the three-phase motor output. If you have those terminals, you have two options.
One is to run single-phase AC through the three-phase input of the drive. If the voltages are correct, the drive should operate fine. The caveat is that you'll have to derate the drive somewhat. The input diodes are spec'd assuming that the drive's constant-power load will be distributed among three legs of the rectifier. If you distribute that same load over just two legs, those diodes will get hotter. The internal bus capacitors will also get hotter, because they'll see more ripple current without the third phase. Check with the drive manufacturer for the derating info.
If your drive has DC bus terminals, your other option is to skip it's internal rectifier and use an external one. Rectify the single-phase AC, then use that DC as the input to the drive. This will let you avoid derating the drive. My company makes something exactly for that purpose, though its power range may be larger than is cost-effective for your application. You'd have to price both options out to find out for sure. Read this for more details.
This is actually a very common and legitimate desire, for example to run quasi-industrial machine tools equipped with 3-phase motors in a home workshop.
Three approaches are generally available:
use capacitors to accomplish a phase shift to manufacture at least enough sense of a third phase to tell the motor in which direction you wish it to start. This is closely related to how single phase motors are often started with a shift capacitor which is then cut out by a centrifugal clutch. Some capacitors may remain in the circuit while running. Although low cost and simple, this is the least advantageous solution, and the uneven usage of the phases won't let you operate at designed efficiency. You can sometimes see these solutions listed in machinery catalogs as "static phase converters"
use a "rotary converter" which is sort of a rotational-inertia auto-transformer. Basically, you get another 3-phase motor of comparable or larger size, and spin it up with a start cap (or even, though not recommended given what can go wrong, a pull rope). You connect both motors in parallel, with the idler generating a third phase for the load motor from its rotational inertia (any 3-phase machine will readily operate as a motor or a generator, depending only on if it is mechanically rotating ahead or behind the electrical phase rotation). Some will further fine tune these with capacitors. Although this is a complicated setup it tends to work fairly well, and with a well sized idler gives advantages of 3-phase power such as quickly reversing during lathe threading operations.
use modern power semiconductors to rectify the available power source to DC, then PWM synthesize 3 perfectly phased sine waves. These are typically called variable (or sometimes vector) frequency drives, and have the additional advantage of enabling variation in the applied voltage and frequency, which is to say the speed of synchronous rotation, allowing the user to fill in the gaps between manual belt/gearing options or potentially eliminate the need for a transmission altogether. Prices for this technology are closing in on that of rotary converters, especially when you consider the cost of copper in a rotary converter idler motor.