Variational Methods and Complementary Formulations in Dynamics

Tabarrok, B.; Rimrott, F. P. J.

doi:10.1007/978-94-015-8259-9

Cited by 37 publications

(20 citation statements)

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“…However, for mechanical systems, the co-Lagrangian and co-Hamiltonian equations are sometimes invoked to model special problems; see [15] for examples. Besides mechanical systems, the dual formulation can be insightful and sometimes even necessary for describing systems from other engineering domains.…”

Section: Limitations and Generalizationsmentioning

confidence: 99%

“…while the form of the co-Hamiltonian is given in (15). Hence, substituting (33) into (29) and (30) yields the equations of motion…”

Section: Examples 1 and 2 Revisited: The Brayton-moser Equationsmentioning

confidence: 99%

“…energy as defined and used in [3] and [9]- [15]. To elucidate the distinction between energy and co-energy, consider a point mass M > 0 moving in the x-direction.…”

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confidence: 99%

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Multidomain modeling of nonlinear networks and systems

Jeltsema

Scherpen²

2009

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Many physical systems, including mechanical, electrical, electromechanical, fluid, and thermal systems, can be modeled by the Lagrangian and Hamiltonian equations of motion [1]- [5]. A key aspect of the Lagrangian and Hamiltonian frameworks is the role of energy storage. Apart from the fact that energy is a fundamental concept in physics, there are several motivations for adopting an energybased perspective in modeling physical systems. First, since a physical system can be viewed as a set of simpler subsystems that exchange energy among themselves and the environment, it is common to view dynamical systems as energy-transformation devices. Second, energy is neither allied to a particular physical domain nor restricted to linear elements and systems. In fact, energies from different domains can be combined simply by adding up the individual energy contributions. Third, energy can serve as a lingua franca to facilitate communication among scientists and engineers from different fields. Lastly, the role of energy and the interconnections between subsystems provide the basis for various control strategies [4], [6]-[8].In multidomain Lagrangian and Hamiltonian modeling it is necessary to distinguish between two types of energies, energy and co-energy. Energy is the ability to do work, while co-energy is the complement of energy as defined and used in [3] and [9]-[15]. To elucidate the distinction between energy and co-energy, consider a point mass M > 0 moving in the x-direction. In the nonrelativistic case, the momentum p is related to the velocity v ¼ dx=dt by the linear constitutive relationship p ¼ Mv; and Newton's second law is given by F ¼ dp=dt, where F is the force acting on the mass. If the mass is moved by the force, then the increment of work done by the force is Fdx, which can be written as Fdx ¼ dp dt dx ¼ dp dx dt ¼ vdp ¼ p M dp:Since the kinetic energy of the mass is given by the integral of the work done by the force, integrating v ¼ p=M from zero to p results inWhen plotted in the v-versus-p plane as Figure 1(a), T (p) represents the area of the triangular region below the line p ¼ Mv and to the left of the dashed vertical line. On the other hand, the complementary kinetic energy, that is, the kinetic co-energy, T Ã is defined to be the area of the triangular region above the line p ¼ Mv and below the horizontal dashed line. The kinetic co-energy cannot be obtained directly from work but rather is defined in a complementary or dual fashion as the integral of the momentum with respect to the velocity, which, referring to Figure 1(a), is tantamount to extracting the triangular area defined by T (p) from the total square area defined by the product pv, that is,Note that the kinetic energy is quadratic in p, while the kinetic co-energy is quadratic in v, and, furthermore, T (p) ¼ T Ã (v). Because of this equality, it is traditional to not make a distinction between T (p) and T Ã (v), and as a result T Ã (v) is commonly called the kinetic energy rather than the kinetic co-energy. When the momentum p and ...

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Section: Limitations and Generalizationsmentioning

confidence: 99%

“…while the form of the co-Hamiltonian is given in (15). Hence, substituting (33) into (29) and (30) yields the equations of motion…”

Section: Examples 1 and 2 Revisited: The Brayton-moser Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Multidomain modeling of nonlinear networks and systems

Jeltsema

Scherpen²

2009

IEEE Control Syst.

114

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“…We refer simulation techniques that solve motion equations in vector form as vectorial methods [Tabarrok & Rimrott 1994]. Cloth is easy to fold but highly resistance to stretching.…”

Section: Why Energy Minimization?mentioning

confidence: 99%

“…In Physics nomenclature, energy minimization belongs to variational methods [Tabarrok & Rimrott 1994]. While vectorial methods deal with the acceleration and velocity of objects at specific time moment, variational methods talks about changes in properties of objects over an arbitrary amount of time.…”

Section: Why Energy Minimization?mentioning

confidence: 99%