- #1
Himanshu
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What does the term "Action" means in physics?
What does the term "Action" means in physics?
What does the term "Action" means in physics?
What does the term Action means in physics?
In summary, the term "action" in physics refers to a quantity that describes the behavior of a physical system. It can be defined in different ways depending on the field of physics, but it often involves integrating a Lagrangian along a path. Varying the action with respect to the system's trajectory can lead to the Euler-Lagrange equations, which describe the system's motion. The concept of action is often used to avoid explicitly dealing with forces in complex systems.
- #2
Hootenanny
Staff Emeritus
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I don't know, but someone usually says it after "lights, camera..."Himanshu said:
The specific meaning of an 'action' in physics depends on what field one is working in. However, roughly speaking an 'action' is some quantity in a particular [physical] system that can be used to describe how that system behaves.
- #3
f95toli
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In mechanics (classical and quantum) the usual meaning is the integral of the Lagrangian between two points. It is usually denoted S. There are a few other versions as well, but they all involve the integral of a "path" (in generalized coordinates) between two points.
- #4
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Some reading material:
http://www.eftaylor.com/leastaction.html
http://www.eftaylor.com/leastaction.html
- #5
Himanshu
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What is meant by varying an action ?
I've heard about somthing like " varying Einstien-Hilbert Action to get the field equations".
What does that mean?
I've heard about somthing like " varying Einstien-Hilbert Action to get the field equations".
What does that mean?
- #6
Mute
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It means you take a variational derivative of the action. (A variational derivative is essentially the derivative of a functional with respect to a function. A functional is like a function, except that the input is a function and the output is another function). So, if you have an action that is a function of particle trajectory q(t), you can vary the action with respect to q(t): [itex]\delta \mathcal{S}[q(t)]/\delta q[/itex] is the standard notation for a variational derivative.
The "Principle of Least Action" states that variations in the action should be zero (the name is really a misnomer, as it implies the action is a minimum when this condition is satisfied, but the action could be a maximum or a saddle).
In Lagrangian Mechanics, the action is defined
[tex]\mathcal{S} = \int_{t_1}^{t_2}dt L(q,\dot{q},t)[/tex]
where q(t) is the trajectory of the system. Varying the action with respect to the particle trajectory q(t), after some integration by parts you arrive at
[tex]\delta \mathcal{S} = \delta\left(\int_{t_1}^{t_2}dt L(q,\dot{q},t)\right) = \int_{t_1}^{t_2}dt\left(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} \right) \delta q[/tex]
where L is called the Lagrangian, and is assumed to be a function of t (either explicity or implicitly through the q(t)), q(t) and [itex]\dot{q} = dq/dt[/itex]. We want this to be zero, and since [itex]\delta q[/itex] is an arbitrary variation of q, it follows that what's in the brackets must be zero. This relation in the brackets is called the Euler-Lagrange Equations. (Plural because q(t) can really be a vector, and you have an Euler-Lagrange equation for each [itex]q_i[/itex]).
It turns out that the Euler-Lagrange (EL) equations reproduce Newton's Equations when [itex]L = T - V[/itex], the kinetic energy minus the potential energy (it gets somewhat more complicated for things like General Relativity, which is the example you gave). So, if you can write down the Lagrangian for your system, then given the action defined as above, by varying it you would arrive at the EL equations, and plugging your Lagrangian into that you would derive the equations of motion for your system.
(As a last note, the EL equations above assume the Lagrangian does not depend on any time derivatives of q higher than [itex]\dot{q}[/itex]. They become modified if it does).
The "Principle of Least Action" states that variations in the action should be zero (the name is really a misnomer, as it implies the action is a minimum when this condition is satisfied, but the action could be a maximum or a saddle).
In Lagrangian Mechanics, the action is defined
[tex]\mathcal{S} = \int_{t_1}^{t_2}dt L(q,\dot{q},t)[/tex]
where q(t) is the trajectory of the system. Varying the action with respect to the particle trajectory q(t), after some integration by parts you arrive at
[tex]\delta \mathcal{S} = \delta\left(\int_{t_1}^{t_2}dt L(q,\dot{q},t)\right) = \int_{t_1}^{t_2}dt\left(\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} \right) \delta q[/tex]
where L is called the Lagrangian, and is assumed to be a function of t (either explicity or implicitly through the q(t)), q(t) and [itex]\dot{q} = dq/dt[/itex]. We want this to be zero, and since [itex]\delta q[/itex] is an arbitrary variation of q, it follows that what's in the brackets must be zero. This relation in the brackets is called the Euler-Lagrange Equations. (Plural because q(t) can really be a vector, and you have an Euler-Lagrange equation for each [itex]q_i[/itex]).
It turns out that the Euler-Lagrange (EL) equations reproduce Newton's Equations when [itex]L = T - V[/itex], the kinetic energy minus the potential energy (it gets somewhat more complicated for things like General Relativity, which is the example you gave). So, if you can write down the Lagrangian for your system, then given the action defined as above, by varying it you would arrive at the EL equations, and plugging your Lagrangian into that you would derive the equations of motion for your system.
(As a last note, the EL equations above assume the Lagrangian does not depend on any time derivatives of q higher than [itex]\dot{q}[/itex]. They become modified if it does).
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- #7
rbj
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i was surprized to see the word "force" so carefully omitted in the explanations. not to equate the two, but they're related.
- #8
belliott4488
- 662
- 1
Actually, when I studied Hamilitonian Mechanics, it was presented as a way of avoiding explicit enumeration of forces, which gets pretty complicated for complex systems. The point is that by expressing the fundamental principles in terms of the action, you don't have to mentions forces at all if you don't want to.rbj said:
- #9
CPL.Luke
- 441
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actually that's not entirely true for systems with constraints, the lagrange multipliers actually become the forces of constraint, and partial L withrespect to q are the forces related to the potential energy.
- #10
belliott4488
- 662
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Yes, that's why I said "if you don't want to" - I was sweeping constraint forces under the rug, since I don't think mentioning them really clarifies the topic of this thread, i.e. the use of the action in Physics.CPL.Luke said:
Related to What does the term Action means in physics?
What does the term Action mean in physics?
In physics, the term Action refers to a fundamental concept that describes the behavior of a physical system over time. It is a numerical quantity that represents the total effect of all forces acting on a system.
How is Action related to other fundamental concepts in physics?
Action is closely related to other fundamental concepts in physics, such as energy, momentum, and force. In fact, the principle of least action, which states that a physical system will take the path of least action, is a fundamental principle in classical mechanics.
Can you provide an example of how the term Action is used in physics?
One example of the use of the term Action in physics is in the Lagrangian formulation of classical mechanics. In this formulation, the Action is defined as the difference between the kinetic and potential energies of a system, and is used to derive the equations of motion.
What is the SI unit for Action?
The SI unit for Action is joule-second (J·s), which is equivalent to the unit of energy multiplied by time. This unit is commonly used in quantum mechanics, where Action is often expressed in terms of the Planck constant (h).
How does the concept of Action apply to quantum mechanics?
In quantum mechanics, the concept of Action is used to describe the behavior of particles at the quantum level. The principle of least action is also applicable in quantum mechanics, and is used to derive the Schrödinger equation, which describes the dynamics of quantum systems.
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