Are tendons and ligaments classified under the muscular system or the bone system?
Basically they are not bones or muscles and that's why I have doubt.
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Tendons are intrinsically part of muscles. Ligaments are not. So for your question, we need to create a classification system which is a bit artificial, based on the tissue structure of the various components of the musculoskeletal system.
The musculoskeletal system is made up of hard and soft tissues. The hard tissue includes bones and cartilages (articular cartilages), while the soft tissues are the muscles, tendons, synovial membranes, joints capsule and ligaments. (emphasis mine)
Here is an illustration of the biceps brachii muscle (arm) from Kenhub:
It looks a little different in real life, but this clearly illustrates how tendons (the white things at the ends of the muscle) are part of the muscle. So, to answer your question, tendons are part of the muscular system (a term that I have never used as a physician. I would just say "muscles".)
Ligaments are supporting structures of fibrous tissue of nearly the same composition as tendons, but do not arise from muscle per se. (For a reason I cannot comprehend, little research is devoted to the embryonic development of ligaments.)
These are the ligaments of the elbow (and more fibrous connective tissue):
Clearly they are not bone, and they are not muscle.
If you need to artificially divide muscle vs. skeletal systems, then:
Muscle: muscle and tendons
Skeletal: bone and cartilage
"Other": ligaments, bursae, synovial fluid and membranes, etc.
The easiest thing to define is what is bone and what is not bone. Then bone may well stand alone, even without cartilage. But no one would say cartilage is part of the "muscular system".
So I hope you see that this way of separately classifying tissues which are meant to work together is artificial and perhaps misleading.
Some ontologies such as the FMA and Uberon draw a distinction between the skeleton and the skeletal system, with the former consisting only of bones/cartilage, and the latter including ligaments and joints. A broader concept musculoskeletal system includes muscles/tendons with skeletal system elements.
This link shows the uberon concept of 'skeletal ligament' and its placement within the skeletal system: