scholarly journals Approximate symmetries and conservation laws for Itô and Stratonovich dynamical systems

2004 ◽  
Vol 297 (1) ◽  
pp. 152-168 ◽  
Author(s):  
Nail H Ibragimov ◽  
Gazanfer Ünal ◽  
Claes Jogréus
Keyword(s):  
Dynamical Systems ◽  
Conservation Laws ◽  
Approximate Symmetries

Lagrangian Mechanics

2019 ◽  
pp. 53-82
Author(s):  
David D. Nolte
Keyword(s):  
Dynamical Systems ◽  
Conservation Laws ◽  
Euler Equations ◽  
Virial Theorem ◽  
Variational Calculus ◽  
Mechanical Action ◽  
Lagrange Equations ◽  
Time Average ◽  
The Difference ◽  
Kinetic And Potential Energies

Dynamical systems follow trajectories for which the mechanical action integrated along the trajectory is an extremum. The action is defined as the time average of the difference between kinetic and potential energies, which is also the time average of the Lagrangian. Once a Lagrangian has been defined for a system, the Euler equations of variational calculus lead to the Euler–Lagrange equations of dynamics. This chapter explores applications of Lagrangians and the use of Lagrange’s undetermined multipliers. Conservation laws, central forces, and the virial theorem are developed and explained.

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