# Lie Symmetry Methods for Solving Nonlinear ODEs

Around 1870, Marius Sophus Lie realized that many of the methods for solving differential equations could be unified using group theory. Lie symmetry methods are central to the modern approach for studying nonlinear ODEs. They use the notion of symmetry to generate solutions in a systematic manner. Here is a brief introduction to Lie's approach that provides some examples that are solved in this way by DSolve.

A key notion in Lie's method is that of an *infinitesimal generator* for a symmetry group. This concept is illustrated in the following example.

In[1]:= |

In[2]:= |

For a fixed value of , the point (in blue) can be obtained by rotating the line joining (in red) to the origin through an angle of in the counterclockwise direction.

In[3]:= |

Out[3]= |

A rotation through can be represented by the matrix

In[3]:= |

In[4]:= |

Out[4]= |

In[5]:= |

Out[5]= |

In[6]:= |

Out[6]= |

In[7]:= |

Out[7]= |

The Lie symmetry method requires calculating a first-order approximation to the expressions for the group. This approximation is called an *infinitesimal generator*.

In[8]:= |

Out[8]= |

In[9]:= |

Out[9]= |

In[10]:= |

*Lie equations.*For the group of rotations, the Lie equations are given by the first argument to DSolve shown here.

In[11]:= |

Out[11]= |

The rotation group arises in the study of symmetries of geometrical objects; it is an example of a *symmetry group*. The infinitesimal generator, a differential operator, is a convenient local representation for this symmetric group, which is a set of matrices.

An expression that reduces to 0 under the action of the infinitesimal generator is called an* invariant *of the group.

In[12]:= |

In[13]:= |

Out[13]= |

In the following examples, these ideas are applied to differential equations.

In[14]:= |

In[15]:= |

In[16]:= |

In[17]:= |

Out[17]= |

In[18]:= |

Out[18]= |

In[19]:= |

Now, the Riccati equation depends on three variables: , , and . Hence, the infinitesimal generator must be *prolonged* to act on all three variables in this first-order equation.

In[20]:= |

In[21]:= |

In[22]:= |

Out[22]= |

Depending on the order of the given equation, the knowledge of a symmetry (in the form of an infinitesimal generator) can be used in three ways.

- If the order of the equation is 1, it gives an integrating factor for the ODE that makes the equation
*exact*and hence solvable.

- It
*reduces*the problem of solving an ODE of order to that of solving an ODE of order , which is typically a simpler problem.

The DSolve function checks for certain standard types of symmetries in the given ODE and uses them to return a solution. Following are three examples of ODEs for which DSolve uses such a symmetry method.

In[23]:= |

In[24]:= |

In[25]:= |

Out[25]= |

In[26]:= |

Out[26]= |

In[27]:= |

In[28]:= |

In[29]:= |

*scaling symmetry*allows DSolve to find new coordinates in which the independent variable is not explicitly present. Hence the problem is solved easily.

In[30]:= |

Out[30]= |

In[31]:= |

Out[31]= |

In[32]:= |

In[33]:= |

In[34]:= |

Out[34]= |

In[35]:= |

Out[35]= |

In[36]:= |

This concludes the discussion of ordinary differential equations.