Lagrangians (Lagrangians) are mathematical expressions that contain almost all of the information we care about in a physical system. Lardrate quantities are usually symmetrical, which means that they do not change when we turn or move them in a certain way. Symmetry and Lattice quantities are very important because we can use them to construct conservation quantities.
Conservation quantities are observable quantities that remain constant throughout the evolution of a physical system.
Physicists like to look for conservation quantities because they are not only profoundly philosophical, but also very useful in solving equations. When you know that some quantities remain the same, you can use them to simplify the solution of the equations.
Rotate so that the "smooth" symmetry is continuous symmetry. Nordisk theory states that for every continuous symmetry we can construct a conservation quantity. For example, if a system has rotational symmetry, we can get conservation of angular momentum.
Even more surprisingly, Nostra's theorem could show that conservation of energy is the result of the symmetry of time translation, and that the invariance of time translation means that the Lattice quantity itself does not contain time.
In other words, if the background in which the physical system is located does not change over time, then the total energy of the system will not change over time.
By Konrad Jacobs, Erlangen — CC BY-SA 2.0 de
The concept of symmetry is widespread in mechanics, classical and modern physics. For example, in quantum physics, the symmetry of a quantum mechanical system can correspond to conservation of quantum angular momentum. In electron theory, the charge and spin conservation of an electron stems from the symmetry that the electron follows.
How can symmetry be described in detail in mathematics? First, the principle of least action needs to be explained, and if we know the Radzi quantity, how we can use it to calculate the behavior of the field.
Amount of action and amount of Radze
Suppose there is a particle or field that evolves between two predetermined points in time, t1 and t2. If it is a particle, we can depict the evolution of the particle by drawing a path that extends through space, starting at time t1 and ending at time t2. If it is a field, we can imagine a heat map slowly evolving over time.
What can we know through the behavior of these particles and fields? How do we know what path the particle will take? In physics, we start with a model that can describe a physical system, one of which is typically a Ragenite quantity. A Radze is a mathematical quantity, which is usually written as the difference between kinetic energy and potential energy, and a Radze can give a specific number at any point in time. We like to use the Lattice quantity because it is independent of the observer and does not change with the reference frame.
It doesn't matter whether the observer is upright or handstanding, or moving at a speed close to the speed of light. Usually, the numerical value of a physical quantity varies depending on the choice of coordinates; however, the Rastens quantity does not change with the choice of coordinates, and its value is the same regardless of which observer. This property, which has nothing to do with the frame of reference, is very useful because it allows us to perform clear calculations.
To understand exactly what's going on, we need to construct a quantity called an action. For example, if a Ralph is known, we can calculate the integral of the Ralph between two points in time:
Integral means adding the value of the Ralphrate quantity at multiple time points. The total integral from t to t is called the action. It is usually represented by the uppercase letter S. The vertical curve ∫ before the Radzi quantity represents the integral.
The expression above is a mathematical definition of a activism. The Lardorean is usually a function of the first derivative of position and position. The Greek letter φ indicates the position of a particle in space; the second term φ is a first derivative of the particle's position, representing the rate of change in the particle's position over time.
What does the Rastensite quantity look like geometrically? We can illustrate it with some illustrations through which we can understand the general concepts about it. If the Lascelle quantity contains only kinetic energy in free space, a greater amount of action is often obtained for paths that differ from straight lines. The plot shows the magnitude of the particle's action corresponding to the different paths taken between time t and t. As you can see, the most complex paths have the most action. The path with the least amount of action is a straight path.
How do you derive the laws of physics?
In our eyes, Radzi quantities are mathematical objects, and we only regard the action quantities as physical. There is a philosophical reason for this. The results show that different amounts of Radze can produce the same amount of action. So, in some cases, there are two Radzels, but only one action. This means that we can derive the same laws of physics from two different Lattice quantities.
Why is that? The reason is that when we integrate some mathematical expression called a total derivative, the result is zero.
In the following formula, we have a acting amount, which is written as a specific Radzi amount and a total differential term. However, we can split the integral into two different parts. Once we separate it, we eliminate the full differential term because it becomes zero when we integrate.
It's an exciting thing! This means that there are two different Radzelides, under a less restrictive limit, that can be considered "equivalent". We don't need to make them exactly equivalent to arrive at the same physical phenomenon. If the Ralphian quantities differ only on the "total differential" term, they can be considered to be mutually equivalent. For example, in the following figure, the functions f , g , and h are all related to the total differential term , and all three of them produce the same amount of action. (I've written these three functions in different colors to express this point.) ）
Mathematically, we can use the following expression to express the "equivalence" between Lascelletian quantities, although the difference between them is a full differential term. In the following expression, the function f is a differentiable function.
If the concept of "rate of change" can be used for a function, then the function is differentiable. If the value of a function jumps in some place, sharp inflection points, or is not well defined, it is possible that the concept of "rate of change" cannot be used, in which case the concept of "rate of change" becomes acceptable only when many strict mathematical conditions are met. The set of all differentiable functions is C. Research on whether operations such as differentiation and integration have good definitions is called mathematical analysis and is a fascinating area of study.
Euler-Lagrange formula
The "principle of least action" tells us that the behavior of a field or particle is precisely the act of making the action take a minimum value. So if we know this action, we can use some mathematical operations to find the behavior of making this action take the minimum value of the time field. There is a branch of mathematics called variational methods that study the "rate of change of functions". (Translator's note: Variational methods tell us that the behavior of a field or particle can be derived from the Euler-Lagrange equation.) ）
The Particle Version of the Euler-Lagrange Equation is shown below. On the left side of the equation, we first take the partial derivative of the Lardorean quantity to the velocity, and then proceed to the derivative of time. On the right side of the equation, we derive the Lattice quantity in space. Then let the left side of the equation equal the right side, and you can get a path that minimizes the action.
The Field Theory version of the Euler-Lagrange equation is very similar to the Particle Version, and the equations are as follows:
It can give appearances a way of evolving in space-time.
The following is the translator's note:
Conserved quantity
Earlier we introduced that conservation quantities can be derived using symmetry, and then we will show how to do this. Nordic theory tells us that each symmetry corresponds to a conservation quantity.
If the physical system has time translation invariance, that is, the Radzi quantity does not contain time, then the expression can be obtained:
In parentheses on the left side of the equation is energy, and its derivative of zero over time indicates that it does not change over time.
If the physical system has spatial translation invariance, that is, the Lattice quantity does not contain spatial coordinates, then the expression can be obtained:
Inside the left parenthesis of the equation is the momentum, which does not change with time, which is the conservation of momentum.
By Afiq Hatta
Translation: Nothing
Reviewer: Zhenni
Original link:
https://www.cantorsparadise.com/noethers-theorem-and-the-principle-of-least-action-c84b789c51b6
The translated content represents the author's views only
Does not represent the position of the Institute of Physics, Chinese Academy of Sciences
Edit: zhenni