Is there a modified Least Action Principle for nonholonomic systems?
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Hey mate ^_^
Your conjecture is reasonable but false. What you get under your suggested principle is known as vakonomic mechanics proposed by Kozlov in 1980s....
It has been argued, by Arnold among others, that vakonomics is a better approximation for certain constraints than the standard non-holonomic mechanics.....
#Be Brainly❤️
Your conjecture is reasonable but false. What you get under your suggested principle is known as vakonomic mechanics proposed by Kozlov in 1980s....
It has been argued, by Arnold among others, that vakonomics is a better approximation for certain constraints than the standard non-holonomic mechanics.....
#Be Brainly❤️
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Hello mate here is your answer.
We know that one can treat nonholonimic (but differential) constraints in the same manner as holonimic constraints. With a given Lagrange Function LL, the equations of motion for a holonomic constraint g(x⃗ )=0g(x→)=0follow as: ddt∂L∂x⃗ ˙−∂L∂x⃗ =−λ∂g∂x⃗ ddt∂L∂x→˙−∂L∂x→=−λ∂g∂x→
You can derive these either by looking at the constraint-forces, or by a somewhat enhanced least-action principle: The physical Path x⃗ (t)x→(t) is the one with stationary action, with respect to all variations δx⃗ g(t)δx→g(t)that satisfy the boundary condition gg.For non-holonomic constraints, given as a⃗ (x⃗ )x⃗ ˙=0a→(x→)x→˙=0, the equations of motion are given as: ddt∂L∂x⃗ ˙−∂L∂x⃗ =−λa⃗
It obviously helps our observation of exoplanets if they transit their star from our point-of-view. I would guess that the chances of this alignment are better than if their orbital plane was randomly oriented. Gravitational interaction between stars, planets, and the galaxy would increase the likelihood of inter-system alignment just as moons in our system orbit their planets in general alignment of the planets orbit around the Sun. Is this the case? On the other hand, moons in our system generally formed at the same time as the planets, which would not be the case for star systems.
Hope it helps you.
We know that one can treat nonholonimic (but differential) constraints in the same manner as holonimic constraints. With a given Lagrange Function LL, the equations of motion for a holonomic constraint g(x⃗ )=0g(x→)=0follow as: ddt∂L∂x⃗ ˙−∂L∂x⃗ =−λ∂g∂x⃗ ddt∂L∂x→˙−∂L∂x→=−λ∂g∂x→
You can derive these either by looking at the constraint-forces, or by a somewhat enhanced least-action principle: The physical Path x⃗ (t)x→(t) is the one with stationary action, with respect to all variations δx⃗ g(t)δx→g(t)that satisfy the boundary condition gg.For non-holonomic constraints, given as a⃗ (x⃗ )x⃗ ˙=0a→(x→)x→˙=0, the equations of motion are given as: ddt∂L∂x⃗ ˙−∂L∂x⃗ =−λa⃗
It obviously helps our observation of exoplanets if they transit their star from our point-of-view. I would guess that the chances of this alignment are better than if their orbital plane was randomly oriented. Gravitational interaction between stars, planets, and the galaxy would increase the likelihood of inter-system alignment just as moons in our system orbit their planets in general alignment of the planets orbit around the Sun. Is this the case? On the other hand, moons in our system generally formed at the same time as the planets, which would not be the case for star systems.
Hope it helps you.
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