Think circular for a smoother action
We deal with back and forth motion all the time when we play the violin, whether it be down bow and up bow, or retaking the bow. This motion becomes jerky When we think that we are traveling two points back and forth. Instead, let's think about traveling in circles to make the motion smoother.
The biggest advantage of moving in circles is that you can keep moving without ever stopping when you are following a circle. On the contrary, if you are following a straight line back and forth, you must stop before you can change the direction of the movement. This is an important concept string players need to understand, and apply to their playing.
The key phrase here is that, in circles you can keep moving "without ever stopping". A movement of an object will be smoother if it can keep moving. A circular motion enables you to realize this, and a number of things in violin playing can become smoother by employing the concept of a circular motion.
Circular motion in string crossings
For instance, take a look at No.11 from Sevcik's School of Violin Techniques, Op.1, Book 1.
This example involves string crossings between two strings, which should be done from your wrist. You can move your hand up and down from your wrist to change strings, however, when you are moving your hand up and down, it needs to stop when it can change the direction. Instead, if you move your hand in circular motion, it can keep moving without stopping, which results in a smoother motion. In this example, you can move your hand clockwise when it starts with a down bow, and counter-clockwise when it starts with an up bow.
Circular motion in changing the direction of the bow
We can apply the concept of a circular motion when you want a smooth bow change from a down bow to an up bow, or vice versa. The easiest way to visualize this is to think about a figure 8 in sideways. You push your arm out when you play a down bow. Then you start to pull your arm in at the bow change on the up bow. You will draw a small clockwise circle at the tip, and a counter-clockwise circle at the frog.
This will keep you from stopping the movement of your bow completely at the bow change.
Circular motion in retaking the bow
Drawing a circle upward will make retaking of the bow smoother and more efficient.
Make a quick and small circle for occasions as you see here. You draw a counter-clockwise circle for a down bow retake, and a clockwise retake for an up bow retake.
Down bow retake
Up bow retake
Below are some examples of when we should make a big and quick circle.
Down bow retake
Circular motion in vibrato
The same concept can be applied to the vibrato. Instead of merely moving your finger up and down, think in circular motion and move your fingers counter-clockwise. In reality, your finger will not move in circles that much, but it will help smooth out your vibrato just by thinking and trying to move in circles.
When you think your playing is jerky and abrupt, see if you can apply the idea of a circular motion to what you are doing. It might help you resolve the problem you are experiencing.