Basic Kinematics: Constant Velocity w/o Rotation

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In a scenario where a body's velocity remains constant in magnitude but changes direction, a net force must act perpendicular to its motion. This results in parabolic motion, like projectile motion, if the force direction remains constant. If the force direction changes to stay perpendicular, circular motion occurs. An example provided is a spaceship using thrusters to maintain constant speed while navigating a wiggling path. The discussion emphasizes the feasibility of constant velocity with directional changes through specific force applications.
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Can there be any other case than rotating when the velocity of a body never becomes zero and still constantly changes in such a way that rate of change of magnitude of velocity is always zero(dv/dt = 0), i.e. the magnitude of velocity is constant?
 
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For velocity to change its direction without changing its magnitude it must experience a net force that is perpendicular to its direction of motion. If this force remains constant in direction, then parabolic motion (such as projectile motion) is the result.

If the force also changes its direction so as to constantly stay perpendicular to motion, then circular motion is the result. One could get clever and make a spaceship that has many thrusters mounted around the ships middle, all pointing perpendicular to the ships heading. If a computer managed to fire these such that only one was on at any moment, the ship could take a wiggling path, following many arcs through space, but always at a constant speed.
 
well it is easy, take a bike try to do all the maneuver u can while keeping the speed constant but changing direction.
 
I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...

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