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Difference between revisions of "Action"

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A functional expressed by the definite integral of a function, the stationary values of which determine a real motion of a mechanical system acted upon by given active forces, in the class of kinematically possible motions between some two final positions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105402.png" /> in space and satisfying certain conditions.
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A functional expressed by the definite integral of a function, the stationary values of which determine a real motion of a mechanical system acted upon by given active forces, in the class of kinematically possible motions between some two final positions $P_0$ and $P_1$ in space and satisfying certain conditions.
  
 
One distinguishes between Hamiltonian, Lagrangian and Jacobian actions, which appear in the corresponding principles of stationary action.
 
One distinguishes between Hamiltonian, Lagrangian and Jacobian actions, which appear in the corresponding principles of stationary action.
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The Hamiltonian action
 
The Hamiltonian action
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105403.png" /></td> </tr></table>
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$$\int\limits_{t_0}^{t_1}(T+U)dt$$
  
 
is defined in the class of kinematically possible motions of a holonomic system for which the initial and the final positions of the system, as well as the time of motion between them, are the same as the respective ones for real motion.
 
is defined in the class of kinematically possible motions of a holonomic system for which the initial and the final positions of the system, as well as the time of motion between them, are the same as the respective ones for real motion.
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The Lagrangian action
 
The Lagrangian action
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105404.png" /></td> </tr></table>
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$$\int\limits_{t_0}^t2Tdt$$
  
 
and the Jacobian action
 
and the Jacobian action
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105405.png" /></td> </tr></table>
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$$\int\limits_{P_0}^{P_1}\sqrt{2(U+h)}ds,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105406.png" /></td> </tr></table>
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$$ds^2=\sum_{i,j=1}^na_{ij}dq_idq_j,$$
  
are defined in the class of kinematically possible motions of a holonomic conservative system for which the initial and the final positions of the system, and the constant energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105407.png" />, are the same as the respective magnitudes for a real motion. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105408.png" /> is the kinetic energy of the system, and for a conservative system
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are defined in the class of kinematically possible motions of a holonomic conservative system for which the initial and the final positions of the system, and the constant energy $h$, are the same as the respective magnitudes for a real motion. Here $T$ is the kinetic energy of the system, and for a conservative system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a0105409.png" /></td> </tr></table>
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$$T=\frac12\sum_{i,j=1}^na_{ij}\dot q_i\dot q_j,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a01054010.png" /> are the generalized Lagrange coordinates, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010540/a01054011.png" /> is the force function of active forces.
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where $\dot q_i$ are the generalized Lagrange coordinates, and $U(q)$ is the force function of active forces.
  
 
For more details see [[Variational principles of classical mechanics|Variational principles of classical mechanics]]; [[Hamilton–Ostrogradski principle|Hamilton–Ostrogradski principle]]; [[Lagrange principle|Lagrange principle]]; [[Jacobi principle|Jacobi principle]].
 
For more details see [[Variational principles of classical mechanics|Variational principles of classical mechanics]]; [[Hamilton–Ostrogradski principle|Hamilton–Ostrogradski principle]]; [[Lagrange principle|Lagrange principle]]; [[Jacobi principle|Jacobi principle]].

Latest revision as of 15:03, 3 August 2014

A functional expressed by the definite integral of a function, the stationary values of which determine a real motion of a mechanical system acted upon by given active forces, in the class of kinematically possible motions between some two final positions $P_0$ and $P_1$ in space and satisfying certain conditions.

One distinguishes between Hamiltonian, Lagrangian and Jacobian actions, which appear in the corresponding principles of stationary action.

The Hamiltonian action

$$\int\limits_{t_0}^{t_1}(T+U)dt$$

is defined in the class of kinematically possible motions of a holonomic system for which the initial and the final positions of the system, as well as the time of motion between them, are the same as the respective ones for real motion.

The Lagrangian action

$$\int\limits_{t_0}^t2Tdt$$

and the Jacobian action

$$\int\limits_{P_0}^{P_1}\sqrt{2(U+h)}ds,$$

$$ds^2=\sum_{i,j=1}^na_{ij}dq_idq_j,$$

are defined in the class of kinematically possible motions of a holonomic conservative system for which the initial and the final positions of the system, and the constant energy $h$, are the same as the respective magnitudes for a real motion. Here $T$ is the kinetic energy of the system, and for a conservative system

$$T=\frac12\sum_{i,j=1}^na_{ij}\dot q_i\dot q_j,$$

where $\dot q_i$ are the generalized Lagrange coordinates, and $U(q)$ is the force function of active forces.

For more details see Variational principles of classical mechanics; Hamilton–Ostrogradski principle; Lagrange principle; Jacobi principle.

How to Cite This Entry:
Action. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Action&oldid=13315
This article was adapted from an original article by V.V. Rumyantsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article