Colloquim

Colloquim
Geometry, Mechanics, and Control of Many-body Systems revisited 岩井敏洋 2000年6月8日13:30 3F南演習室 Bundle picture or gauge theoretical treatment of many-body systems provides mathematical foundations of many-body mechanics. For a long period before the bundle picture was introduced, people had made a vain effort to separate rotational and vibrational motions. However, the separation was shown to be impossible by A. Guichardet by the use of the connection theory or gauge theory. Since then many-body mechanics has been described in the bundle picture. The center-of-mass system is viewed as a principal fiber bundle with structure group $SO(3)$, if collinear configurations of particles are gotten rid of from the center-of-mass system. With this constraint taken into account, we have formulated classical and quantum mechanics for many-body systems in the bundle picture. Quite recently, without the constraint, quantum mechanics has been reformulated. That is, the bundle picture of many-body systems is generalized so that it may deal with all configurations of particles without getting rid of collinear configurations. In the bundle picture, one can understand how vibrational motions can give rise to rotations. This talk is an invitation to this seemingly strange, but mathematically correct fact through some pictorial examples. First example is a planar many-body system. It starts a vibrational motion with an initial shape. After getting back to the original shape, it will have made a rotaion as a result. Second and third examples are concerned with vibrational motions of a system of two identical axially symmetric cylinders jointed together by a special type of joint. This system is a model of the falling cat. The cat can make a rotation to land on his legs when launched in the air, though he can make vibrational motions only. These examples shows vibrational motions accoding to laws of motion different to each other. In the last, it will be shown how the bundle picture is modified, if the many-body system consists of identical particles. 戻る

Geometry, Mechanics, and Control of Many-body Systems revisited

岩井敏洋

2000年6月8日13:30

3F南演習室

Bundle picture or gauge theoretical treatment of many-body systems provides mathematical foundations of many-body mechanics.
For a long period before the bundle picture was introduced, people had made a vain effort to separate rotational and vibrational motions.
However, the separation was shown to be impossible by A. Guichardet by the use of the connection theory or gauge theory.
Since then many-body mechanics has been described in the bundle picture.
The center-of-mass system is viewed as a principal fiber bundle with structure group $SO(3)$, if collinear configurations of particles are gotten rid of from the center-of-mass system. With this constraint taken into account, we have formulated classical and quantum mechanics for many-body systems in the bundle picture.
Quite recently, without the constraint, quantum mechanics has been reformulated.
That is, the bundle picture of many-body systems is generalized so that it may deal with all configurations of particles without getting rid of collinear configurations.
In the bundle picture, one can understand how vibrational motions can give rise to rotations.
This talk is an invitation to this seemingly strange, but mathematically correct fact through some pictorial examples. First example is a planar many-body system.
It starts a vibrational motion with an initial shape.
After getting back to the original shape, it will have made a rotaion as a result.
Second and third examples are concerned with vibrational motions of a system of two identical axially symmetric cylinders jointed together by a special type of joint. This system is a model of the falling cat. The cat can make a rotation to land on his legs when launched in the air, though he can make vibrational motions only. These examples shows vibrational motions accoding to laws of motion different to each other.
In the last, it will be shown how the bundle picture is modified, if the many-body system consists of identical particles.

戻る