Geometric Analysis of Hamiltonian Mechanics using Connections

Sprache des Vortragstitels:

Englisch

Original Tagungtitel:

GAMM Annual Meeting 2006

Sprache des Tagungstitel:

Englisch

Original Kurzfassung:

From a geometric point of view connections appear naturally in an intrinsic description of mechanics, such that the equations remain their structure, even when time dependent transformations are applied. This is obvious for time variant systems, but nevertheless also interesting in the time invariant case, from a control point of view, if stabilization of trajectories is the demand. The choice of connections is of course not unique and physical considerations are essential for the proper selection. When considering the equations of motion with respect to an observer, the correct interpretation of the velocity and the linear momentum requires a space-time connection, which takes into account the velocity of the observer. To derive the change of the momentum in an intrinsic way the introduction of a so called dynamic connection is important. The Hamiltonian approach also makes essential use of the space-time connection and furthermore there exists the Hamiltonian connection which gives us the possibility to split the Hamiltonian vector field into a vertical and a horizontal part, respectively. We will show how this splitting of the Hamiltonian vector field can be used to give a geometric interpretation of the power flows with respect to the Hamitlonian description.

Sprache der Kurzfassung:

Englisch

Englischer Vortragstitel:

Geometric Analysis of Hamiltonian Mechanics using Connections

Englischer Tagungstitel:

GAMM Annual Meeting 2006

Englische Kurzfassung:

From a geometric point of view connections appear naturally in an intrinsic description of mechanics, such that the equations remain their structure, even when time dependent transformations are applied. This is obvious for time variant systems, but nevertheless also interesting in the time invariant case, from a control point of view, if stabilization of trajectories is the demand. The choice of connections is of course not unique and physical considerations are essential for the proper selection. When considering the equations of motion with respect to an observer, the correct interpretation of the velocity and the linear momentum requires a space-time connection, which takes into account the velocity of the observer. To derive the change of the momentum in an intrinsic way the introduction of a so called dynamic connection is important. The Hamiltonian approach also makes essential use of the space-time connection and furthermore there exists the Hamiltonian connection which gives us the possibility to split the Hamiltonian vector field into a vertical and a horizontal part, respectively. We will show how this splitting of the Hamiltonian vector field can be used to give a geometric interpretation of the power flows with respect to the Hamitlonian description.