Lagrangian and Hamiltonian structures in physics

 

Classical mechanics

Multiplicative form of the Lagrangian

(primary investigator: Mr Kittikun Surawuttinack)

 

We knew that the Lagrange function (Lagrangian) is not unique (normally called a nonuniqueness property). This means that we can multiply a constant or add the total derivative terms into the Lagrangian resulting the same equation(s) of motion. Usually, the Lagrangian takes the form

 

L=T(velocity) - V(position) ,...........................(a)

 

where T is the kinetic energy and V is the potential energy of the system. Here we call Eq (a) as the additive form (standard form).

 

We now may ask: does there exist an alternative form of the Lagrangian apart from the standard from?

 

To answer the question, we propose the multiplicative form of the Lagrangian in the following

 

L=F(velocity)G(position)................................(b)

 

We sucessfully solve the Lagrangian for both the nonrelativistic system and the relativistic system with one degree of freedom. An interesting point is that the muliplicative form of the Lagrangian can be considered as a generating function for an infinite heirarchy of Lagrangians. According to our knowladge, this would be the first time that this heirarchy is systematically produced.

 

 

The next step along this line of the investigation is the quantisation of the system through the Feynman path integrals with the multiplicative Lagrangian. Furthermore, the systems with higher degreees of freedom are also worth to search the alternative form of the Lagrangian.

 

November 2015

 

Integrable systems

The well known standard criterion of integrability for the Hamiltonian systems is the Liouville'sintegrability. We knew that in classical and quantum physics we may choose to work with Hamiltonian approach or Lagrangian approach in order to understand the nature of the system in question, leading to the same result for describing the behavoir of the system. Then we could ask what is the Lagrangian analogue of the Liouville's integrability? Based on the pioneer work for the systems with infinite degrees of freedom by Sarah Lobb and Frank Nijhoff, an important feature for the Lagrangian systems called the closure relation was first established on the discrete level for the systems in ABS list.

 

Soon later, the systems with finite degrees of freedom namely Calogero-Moser and Ruijsenaars-Schneider systems, possesing multi-time structure, were investigated. The closure relation can sucessfully established both discrete and continous time levels. Let us define what we mean by the closure relation. According to the variational principle, if we vary the curve with respect to the dependent variables (space) we would get the Euler-Lagrange equation requiring the extremum of the action functional. For the systems with multi-independent variables, we may perform the variation on the space of independent variables leading to the closure relation which is the result of invariance of the action functional under local deformations.

From these results, we may consider the closure relation as the important property for integrable systems from the Lagrangian point of view.

 

With the sucessful piaoneer investigations for works mentioned above, later a number of works have been published along this line of research.

 

Calogero-Moser type systems

(primary investigator: Mr Umpon Jairuk)

 

The discrete-time rational Calogero’s goldfish system is obtained from the Ansatz Lax pair. The discrete-time Lagrangians of the system possess the discrete-time 1-form structure as those in the discrete-time Calogero-Moser system and discrete- time Ruijsenaars-Schneider system. Performing two steps of continuum limits, Lagrangian hierarchy for the system is obtained. Expectedly, the continuous-time Lagrange 1-form structure of the system holds. Furthermore, the connection to the lattice KP systems is also established. 

November 2015