Superstring theoryをめぐって(放談室)

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The inverse problem of the calculus of variations

Using the theory of optimal rocket trajectories in general relativity, we present a candidate for the minimum total integrated acceleration closed timelike curve in the Godel universe, and give evidence for its minimality. The total integrated acceleration of this curve is lower than Malament’s conjectured value (Malament, 1984); however, Malament’s conjecture does seem to hold for periodic closed timelike curves.

/pdf/optimal-time-travel-in-the-godel-universe-1tsilbfh6f.pdf

Optimal time travel in the Gödel universe

The optimisation of low-acceleration interstellar relativistic rocket trajectories using genetic algorithms

The study of well-posed problem with mixed endpoint conditions is presented up to the second variation. Conditions are specifically formulated for the Lagrange with non-holonomic constraints to be a well-posed problem. Sufficient conditions for a critical curve f to be a local extremum are established for these problems. The construction of a field is analyzed and adjusted to these cases. Finally we apply the theory to a specific problem.

Well-posed variational problem with mixed endpoint conditions

We derive the covariant optimality conditions for rocket trajectories in general relativity, with and without a bound on the magnitude of the proper acceleration. The resulting theory is then applied to solve two specific problems: the minimum fuel consumption transfer between two galaxies in a FLRW model, and between two stable circular orbits in the Schwarzschild spacetime.

The Rocket Problem in General Relativity

A new approach to the study of variational problems defined through functionals given by multiple integrals is presented. This work is based on techniques from the theory of exterior differential systems. Different kinds of boundary conditions are formulate for well-posed variational problems. Sufficient conditions for an integral manifold to be a local extremum are established. Applications to the study of string theory as well as some other examples are discussed.

Calculus of variations in the context of exterior differential systems

P. A. Griffiths established the so-called mixed endpoint conditions for variational problems with non-holonomic constraints. We will present some results in this context and discuss the inverse problem of calculus of variations. Keywords:Inverse problem of calculus of variations.

https://www.ime.usp.br/~spjm/articlepdf/390.pdf

The inverse problem of variational calculus and the problem of mixed endpoint conditions

We will discuss the so-called mixed endpoint conditions for variational problems with non-holonomic constraints given by form actions of order greater than one. We will present some results and discuss the inverse problem of Calculus of Variations.

https://www.ime.usp.br/~spjm/articlepdf/450.pdf

Pedro Gonçalves Henriques

Papers

The Rocket Problem in General Relativity

Calculus of variations in the context of exterior differential systems

Well-posed variational problem with mixed endpoint conditions

The inverse problem of variational calculus and the problem of mixed endpoint conditions

Inverse Problem of Variational Calculus