I have a knee injury, so I cannot do things like running, cycling, jumping, or virtually anything else that puts pressure on the legs.
What are good upper-body exercises that I could use to burn fat?
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The late Dr. Leonard Schwartz's Heavyhands program dealt specifically with how the upper body and most specifically the arms were able to boost aerobic performance by taking some of the strain off the legs, ankles, and feet. As a matter of fact he developed because he was a runner and incurred a leg injury. It was huge in the 80's and still has plenty of devoted followers to this day. His book Heavyhands as well as some of the associated DVD's are available from used book sellers. There is also plenty of information about Heavyhands that is available via free search via your choice of search engine.
Unfortunately, it is harder to burn more calories and gain aerobic fitness just from using your upper body simply because you are missing out on using the legs in the activity. But here are some suggestions that may or may not depending on your condition:
Rowing machine without moving your knees
Elliptical with no incline
FYI, any activity burns fat
Make a pair or set of upper-body exercises you can do, and do those with no rest between them and little or no rest between rounds. Use bodyweight exercises as well as resistance tools like dumbbells, barbells, and kettlebells. For example:
This approach of pairing or mixing can be extended to include any exercise you are capable of doing. For instance, I recently did 4 rounds of 10 one-arm dumbbell bench press and 10 one-arm dumbbell row. One can mix weights with bodyweight, as well: push-ups and bent-over rows, for example.
What makes this so cardio-intensive? Short rests, using a moderate weight (or moderately difficult exercise), using the time spent doing one exercise to rest the muscles used in the other exercise, and doing several rounds. If you have those elements, you can get quite sweaty and out of breath with simple pairings.