Lagrangian field theory is a formalism in classical field theory.It is the field-theoretic analogue of Lagrangian mechanics.Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom.Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.
The Lagrangian of a complex scalar field (x)⌘(φ1 +iφ2)/p 2 is L = ⌘μ⌫@μ @⌫⇤- m2 ⇤, (1.1.14) where ⇤is the complex conjugate of. Written in terms of the real and imaginary parts of the field,φ1 and φ2, this would become just two copies of the Lagrangian (1.1.7).
Writing coordinates (t, x) = (x 0, x 1, x 2, x 3) = x μ, this form of the Euler–Lagrange equation is [3] where a sum over μ is performed according to the rules of Einstein notation. By solving this equation, one arrives at the "equations of motion" of the field. [3]
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Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, ...
The time integral of the Lagrangian is called the action denoted by S. In field theory, a distinction is occasionally made between the Lagrangian L, of which the time integral is the action
Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom. One motivation for the development of the Lagrangian formalism on fields, and more generally, ...
The sigma model describes the motion of a scalar point particle constrained to move on a Riemannian manifold, such as a circle or a sphere. It generalizes the case of scalar and vector fields, that is, fields constrained to move on a flat manifold. The Lagrangian is commonly written in one of three equivalent forms:
The Einstein field equations are a set of 10 simultaneous, non-linear, differential equations. The solutions of the field equations are the components of the metric tensor of spacetime. A metric tensor describes a geometry of spacetime. The geodesic paths for a spacetime are calculated from the metric tensor.
Equivalence principle. The equivalence principle, explored by a succession of researchers including Galileo, Loránd Eötvös, and Einstein, expresses the idea that all objects fall in the same way, and that the effects of gravity are indistinguishable from certain aspects of acceleration and deceleration.
Gravity (from Latin gravitas 'weight' ), or gravitation, is a natural phenomenon by which all things with mass or energy —including planets, stars, galaxies, and even light —are attracted to (or gravitate toward) one another.
Gravity is most accurately described by the general theory of relativity (proposed by Albert Einstein in 1915), which describes gravity not as a force, but as a consequence of masses moving along geodesic lines in a curved spacetime caused by the uneven distribution of mass.
Gravity is the weakest of the four fundamental interactions of physics, approximately 10 38 times weaker than the strong interaction, 10 36 times weaker than the electromagnetic force and 10 29 times weaker than the weak interaction. As a consequence, it has no significant influence at the level of subatomic particles.
The ancient Greek philosopher Archimedes discovered the center of gravity of a triangle. He also postulated that if two equal weights did not have the same center of gravity, the center of gravity of the two weights together would be in the middle of the line that joins their centers of gravity.
The force of gravity is weakest at the equator because of the centrifugal force caused by the Earth's rotation and because points on the equator are furthest from the center of the Earth. The force of gravity varies with latitude and increases from about 9.780 m/s 2 at the Equator to about 9.832 m/s 2 at the poles.